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for any two non - zero integers @xmath0 and @xmath1 the baumslag solitar group @xmath2 is given by the presentation @xmath3 . these groups were introduced by baumslag and solitar in @xcite , who identified @xmath4 as the first example of a two - generator one - relator non - hopfian group and thus answered a question by b. h. neumann , see @xcite . later on , it was shown that @xmath2 is hopfian if and only if @xmath5 or @xmath6 or @xmath7 , where @xmath8 denotes the set of prime divisors of @xmath9 , see @xcite and @xcite . after reviewing some fundamental properties of baumslag solitar groups , we shall consider random walks on these groups . such a random walk is constructed as follows . first , we choose a probability measure @xmath10 on @xmath2 such that the support of @xmath10 generates @xmath2 as a semigroup . then , the random walk starts at the identity element and proceeds with independent @xmath10-distributed increments each of which is multiplied from the right to the current state . the poisson  furstenberg boundary was introduced by furstenberg in @xcite and @xcite . it is a probabilistic model for the long - time behaviour of the random walk , and simultaneously provides a way to represent all bounded harmonic functions on the state space . in the early 1990s , kaimanovich considered random walks on @xmath11 . under the assumption of finite first moment , he identified their poisson  furstenberg boundary geometrically , see ( * ? ? ? * theorem  5.1 ) . in particular , he showed that the latter is trivial if the random walk has no vertical drift .    for random walks on non - amenable groups the situation is different because their poisson  furstenberg boundary can never be trivial . this motivates the present paper , in which we study random walks on non - amenable baumslag solitar groups and their poisson  furstenberg boundary . the paper is organised as follows . in section [ sec : bsgroups ] , we discuss some algebraic and geometric properties of baumslag solitar groups @xmath2 with @xmath12 . we explain how these groups can be understood through their natural projections to the bass  serre tree @xmath13 and the hyperbolic plane @xmath14 . afterwards , we recall the construction of the space of ends @xmath15 and the boundary @xmath16 . they shall later be used to associate a geometric boundary to @xmath2 . in sections [ sec : randomwalks ] and [ sec : pfboundary ] , we discuss random walks on countable groups and define the notion of poisson  furstenberg boundary . we outline a few classical results and state kaimanovich s strip criterion , which is an important tool to identify the poisson  furstenberg boundary geometrically . in section [ sec : identification ] , we consider random walks on @xmath2 with @xmath12 . in order to ensure that the natural projections of the random walk to @xmath14 and @xmath13 converge almost surely to random elements in @xmath16 and @xmath15 respectively , we need to make suitable assumptions on the moments . if the random walk has vertical drift , we need to assume finite first moment . otherwise , the situation is much more subtle , and we need to assume finite ( @xmath17)-th moment . the fact that the projections converge almost surely allows us to endow the cartesian product @xmath18 , or occasionally just its second component @xmath15 , with the borel @xmath19-algebra and a hitting measure . finally , kaimanovich s strip criterion shows that the resulting probability space is isomorphic to the poisson  furstenberg boundary . for the part of the paper up to lemma [ lem : zerodrifttree ] , we will assume that the two non - zero integers @xmath0 and @xmath1 satisfy @xmath12 . in lemma [ lem : zerodrifttree ] and in the subsequent results , we have to restrict ourselves to the non - amenable subcase @xmath20 . the main result is theorem [ thm : identification ] on page . in the , we will explain how to obtain similar results for the remaining non - amenable cases . we would like to thank wolfgang woess . he suggested this problem and supported us with references and ideas while the research was carried out . moreover , we are grateful to vadim kaimanovich , who made a couple of comments that led to a substantial improvement of the results . the structure of baumslag  solitar groups can be studied by means of hnn extensions . indeed , @xmath2 is precisely the hnn extension @xmath21 with isomorphism @xmath22 given by @xmath23 . this fact allows us to use the respective machinery , such as britton s lemma , see @xcite , which implies that a freely reduced non - empty word @xmath24 over the letters @xmath25 and @xmath26 and their formal inverses can only represent the identity element @xmath27 if it contains @xmath28 with @xmath29 or @xmath30 with @xmath31 as a subword . now , one can easily conclude that , if neither @xmath5 nor @xmath6 , the elements @xmath32 and @xmath33 generate a non - abelian free subgroup . so , @xmath2 is non - amenable . on the other hand , if @xmath5 or @xmath6 , a simple calculation shows that the normal subgroup @xmath34 is abelian with quotient isomorphic to @xmath35 . in this case , @xmath2 is solvable and therefore amenable .    as we will discuss in section  [ sub : cratagi ] , the distinction between the two cases is of importance when working with random walks . assume first that @xmath12 . the cayley graph @xmath36 of the group @xmath37 with respect to the standard generators @xmath25 and @xmath26 is the directed multigraph with vertex set @xmath38 , edge set @xmath39 , source function @xmath40 given by @xmath41 , and target function @xmath42 given by @xmath43 . recall that a graph is just a pair consisting of a vertex set and an edge set with the property that every edge is a two - element subset of the vertex set . every directed multigraph can be converted into a graph by ignoring the direction and the multiplicity of the edges and deleting the loops . for the purpose of this paper it is sufficient to think of @xmath36 as a graph , and we shall tacitly do so . consider the illustration of @xmath36 in figure  [ fig : cayley ] . intuitively speaking , we may look at it from the side to see the associated bass serre tree . formally , let @xmath44 and let @xmath13 be the graph with vertex set @xmath45 and edge set @xmath46 . this graph is actually a tree ; it is obviously connected and , by britton s lemma , it does not contain any cycle . note that the canonical projection @xmath47 given by @xmath48 is a weak graph homomorphism from @xmath36 to @xmath13 , i.e.  whenever the vertices @xmath49 and @xmath50 are adjacent in @xmath36 , their images @xmath51 and @xmath52 either agree or they are adjacent in @xmath13 . [ rem : levelmaps ] consider the infinite cyclic group @xmath35 and the map @xmath53 given by @xmath54 and @xmath55 . the latter can be uniquely extended to a group homomorphism @xmath56 . indeed , the equation @xmath57 holds in @xmath35 so that we can apply von dyck s theorem to extend @xmath58 , see e.g.  ( * ? ? ? * , fn.2 ) . since @xmath59 , the group homomorphism @xmath56 is constant on the cosets from @xmath60 and therefore induces a well - defined map @xmath61 given by @xmath62 . we shall think of @xmath58 and @xmath63 as level functions , they assign a level to every vertex of @xmath36 and @xmath13 , respectively .    [ lem : neighbours ] every vertex @xmath51 of @xmath13 has exactly @xmath64 neighbours ; @xmath0 of them are one level below and @xmath1 of them are one level above the vertex .    by construction , the levels of two adjacent vertices always differ exactly by @xmath65 . the defining relation @xmath66 ( `` @xmath67 '' ) and britton s lemma ( `` @xmath68 '' ) imply that @xmath69 if and only if @xmath31 , whence the vertex @xmath51 has exactly @xmath1 neighbours above . similarly , it has exactly @xmath0 neighbours below because @xmath70 if and only if @xmath29 .     of @xmath11 with respect to the standard generators @xmath25 and @xmath26 . ] ( 0,0 ) ( -180.6,30)@xmath25 ( -149,36.5)@xmath25 ( -174,12)@xmath26 ( -158,15.5)@xmath26 ( -157,48.5)@xmath26      the second projection captures the information that is obtained by looking at @xmath36 from the front . in order to construct it , we introduce another group . let @xmath71 be the set of all affine transformations of the real line that preserve the orientation , i.e.  all maps @xmath72 of the form @xmath73 with @xmath74 and @xmath75 . this set endowed with the composition @xmath76 forms a group . as in the construction of the level function @xmath56 in remark [ rem : levelmaps ] , consider the map @xmath77 given by @xmath78 and @xmath79 . due to von dyck s theorem , it can be uniquely extended to a group homomorphism @xmath80 . the group @xmath71 has a geometric interpretation . in order to describe it , let @xmath14 be the hyperbolic plane as per the poincar half - plane model , i.e.  @xmath81 , endowed with the standard metric @xmath82 the elements of @xmath71 can be thought of as isometries of the hyperbolic plane @xmath14 , which are precisely the maps @xmath83 of the form @xmath84 see e.g.  ( * ? ? ? * theorem 7.4.1 ) . correspond to the orientation - preserving isometries of @xmath14 that fix @xmath85 , which is defined in section [ sub : compact ] . the orientation - reversing isometries of @xmath14 that fix @xmath85 will be crucial for the investigation of baumslag solitar groups @xmath2 with @xmath86 in the . ] now , we are ready to construct the second projection @xmath87 . pick an element @xmath88 , map it via @xmath89 to @xmath71 , think of the latter as an isometry of @xmath14 , and evaluate it at @xmath90 . the following lemma illustrates this construction . [ lem : lookingfromfront ] for every @xmath88 the point @xmath91 is above the point @xmath92 ; the two points have the same real part and their distance is @xmath93 . similarly , for every @xmath88 the point @xmath94 is right from the point @xmath92 ; the two points have the same imaginary part and their distance is @xmath95 . so , in some way , we are actually looking at @xmath36 from the front . this is clear for @xmath96 . now , pick an arbitrary element @xmath88 . the points @xmath91 and @xmath92 are obtained by applying @xmath97 to the points @xmath98 and @xmath99 . remains true when replacing @xmath100 by @xmath101 . therefore , we may actually conclude that @xmath102 . ] but since @xmath97 is the composition of a dilation @xmath103 and a translation @xmath104 , the relative position of the two points is preserved . the same argument works for the second assertion , which completes the proof .      here and throughout the paper , we use the symbol @xmath105 to denote the non - negative integers and the symbol @xmath106 to denote the strictly positive integers .    given a graph with vertex set @xmath107 , we consider finite paths @xmath108 , infinite paths @xmath109 , and doubly infinite paths @xmath110 . in any case , being a path means that for every possible choice of @xmath111 the vertices @xmath112 and @xmath113 are adjacent in the graph . moreover , we say that a path is reduced if for every possible choice of @xmath111 the vertices @xmath112 and @xmath114 are distinct . fix an ascending doubly infinite path @xmath115 in the tree @xmath13 . ascending refers to the level function defined in remark [ rem : levelmaps ] , and it means that for every @xmath116 the vertex @xmath112 is located above the preceding vertex @xmath117 . now , let @xmath118 be the full @xmath119-preimage of @xmath120 , i.e.  the set consisting of all @xmath88 such that the image @xmath121 is traversed by @xmath122 . the subgraph @xmath123 spanned by @xmath118 , see in figure  [ fig : hyperbolic ] , is obviously connected so that the graph distance @xmath124 becomes a metric . this subgraph is sometimes referred to as discrete hyperbolic plane or plane of bricks , which makes particular sense in light of proposition [ prop : discretehyperbolicplane ] . variations of the latter have already been used in the literature , e.g.  in @xcite , @xcite , and @xcite . concerning @xcite , see also the remark in @xcite . ( right ) and its projection to @xmath14 ( left ) . ]    ( 0,0 ) ( -255,32)@xmath125 ( -366,104)@xmath126 ( -189,27 ) ( -321,80 ) ( -243,92.5 )    [ prop : discretehyperbolicplane ] the restriction @xmath127 is a quasi - isometry , even a bi - lipschitz map , between the graph @xmath128 endowed with the graph distance @xmath124 and the hyperbolic plane @xmath14 endowed with the standard metric @xmath129 .    we realise the edges of the graph @xmath128 geometrically . whenever two vertices @xmath130 are adjacent , we connect their images @xmath92 and @xmath131 by a geodesic in @xmath14 . in order to avoid confusion , we refer to these images as @xmath14-vertices and to the geodesics between them as @xmath14-edges . by the proof of lemma [ lem : neighbours ] and by lemma [ lem : lookingfromfront ] , the @xmath14-vertices and @xmath14-edges yield a tessellation of the hyperbolic plane with isometric bricks of the following shape . the @xmath14-vertices of each brick are located on two distinct horizontal lines ; on the upper one there are @xmath132 and on the lower one there are @xmath133 . in either case , the @xmath14-vertices are connected by @xmath14-edges of length @xmath134 to form a chain ( = piecewise geodesic curve ) . due to the curvature , both the two leftmost and the two rightmost @xmath14-vertices are located precisely above each other and connected by vertical @xmath14-edges of length @xmath135 , see in figure [ fig : hyperbolic ] and figure [ fig : approximate ] . since the bricks are uniformly bounded and cover the hyperbolic plane @xmath14 , the restriction @xmath127 is certainly quasi - surjective . pick any two vertices @xmath130 . we aim to estimate the distances @xmath136 and @xmath137 by multiples of each other . first , choose a path of minimal length from @xmath49 to @xmath50 in @xmath128 . it corresponds to a chain of @xmath14-edges from @xmath138 to @xmath139 , see in figure [ fig : hyperbolic ] . this chain consists of @xmath136 many @xmath14-edges , each of which has length at most @xmath140 . hence , @xmath141 for the converse estimate , let us make the following auxiliary definition . every point @xmath142 that is not in the interior of a brick is either an @xmath14-vertex , in which case we define @xmath143 to be @xmath9 , or it is in the interior of an @xmath14-edge , in which case we define @xmath143 to be one of the endpoints of the @xmath14-edge , whichever is closer . in the case that @xmath9 is precisely in the middle of the @xmath14-edge , we choose the left endpoint rather than the right one and the lower endpoint rather than the upper one . with this notion in mind , consider the geodesic  @xmath144 from @xmath138 to @xmath139 , see figure  [ fig : approximate ] . whenever @xmath144 traverses the interior of a brick @xmath145 , it enters the interior at some point @xmath146 and leaves it at some other point @xmath147 . in this situation , approximate the part of @xmath144 from @xmath9 to @xmath148 by a chain of @xmath14-edges from @xmath143 to @xmath149 . we may choose this chain such that , whenever @xmath150 , the chain has no @xmath14-edge at all and , otherwise , the number of @xmath14-edges in the chain is at most @xmath151 . but , by a compactness argument , there is an @xmath152 such that if the part of @xmath144 has length smaller than @xmath153 , then @xmath150 and the chain has no @xmath14-edge at all . therefore , we may conclude that @xmath154 it is not hard to see that if we do this for every brick @xmath145 whose interior is traversed by @xmath144 , we finally obtain a chain of @xmath14-edges from @xmath138 to @xmath139 . depending on whether a part of @xmath144 originally traversed the interior of a brick or ran along an @xmath14-edge , we may estimate the number of @xmath14-edges approximating it by @xmath155 , by @xmath156 , or by @xmath157 times its length . hence , @xmath158     in the case @xmath159 and @xmath160 . ] ( 0,0 ) ( -251,109)upper ( -251,16)lower ( -9,69)@xmath135 ( -49,19)@xmath134 ( -162,5)@xmath144 ( -162,35)@xmath9 ( -141,34)@xmath143 ( -63,130)@xmath148 ( -111,126)@xmath149    [ rem : exponentialline ] note that the horizontal lines mentioned in the proof of proposition [ prop : discretehyperbolicplane ] are horospheres , and by no means geodesics . for example , one may pick such a horizontal line and observe that the part of the line contained in the closed disc @xmath161 with centre @xmath90 and radius @xmath162 has a length growing exponentially in @xmath163 , see also figure [ fig : shortline ] on page . [ rem : treebolic ] the level of a vertex @xmath88 can be recovered both from @xmath164 and from @xmath92 . in fact , the image of @xmath165 is contained in the horocyclic product of the tree @xmath13 and the hyperbolic plane @xmath14 , which is sometimes referred to as treebolic space , see @xcite for details . both the tree @xmath13 and the hyperbolic plane @xmath14 have a natural compactification . in case of @xmath13 , it is the end compactification , which can be constructed as follows . fix a base point , say @xmath166 , and consider the set @xmath167 of all reduced paths that start in @xmath145 , be they finite or infinite . the endpoint map yields a one - to - one correspondence between the finite paths and the vertices @xmath60 . we may therefore think of @xmath60 as a subset of @xmath167 . the set @xmath167 can be endowed with the metric @xmath168 here , the symbol @xmath169 denotes the number of edges the two paths run together until they separate , i.e.  @xmath170 , see in figure  [ fig : compactifications ] . hence , the later the paths separate the closer they are . the set @xmath167 endowed with the metric @xmath171 is a compact metric space that contains @xmath60 as a discrete and dense subset . the complement of @xmath60 is the set of infinite paths , it is usually denoted by @xmath15 and called the space of ends .    in case of @xmath14 , we temporarily switch to the poincar disc model . more precisely , instead of working in the half - plane @xmath81 , we consider the open unit disc @xmath172 . the cayley transform @xmath173 given by @xmath174 is one possibility to convert between the two models . since we are currently interested in the topological structure , let us highlight that the hyperbolic topology on @xmath175 is the one induced by the cayley transform , i.e.  the one that turns the cayley transform into a homeomorphism . it happens to agree with the standard topology on @xmath175 . so , topologically speaking , the hyperbolic plane in the poincar disc model is just a subspace of the complex plane @xmath176 . we may therefore compactify it by taking the closed unit disc @xmath177 , see figure  [ fig : compactifications ] . in order to translate this compactification back to the poincar half - plane model , we first extend both the domain and the codomain of the cayley transform so that we obtain a bijection @xmath178 , and then apply its inverse . the resulting space @xmath179 is our compactification . it is , once again , endowed with the induced topology , and thus a compact space that contains @xmath14 as a dense subset . the complement of @xmath14 is the union @xmath180 , it is usually denoted by @xmath16 and called the hyperbolic boundary . having introduced the hyperbolic boundary this way , the following lemma gives us a helpful criterion for convergence . its proof is elementary and we leave it to the reader . [ lem : convergence ] a sequence @xmath181 in @xmath14 converges to @xmath85 if and only if the absolute values @xmath182 tend to infinity . moreover , it converges to a point @xmath183 if and only if it does with respect to the standard topology on the complex plane @xmath176 . ( 0,0 ) ( -258,97)@xmath9 ( -250,82 ) ( -196,72)@xmath148 ( -240,45)@xmath145 the aim of the current work is to study random walks on baumslag  solitar groups . before doing so , we fix the notation . given a countable state space @xmath184 , an initial probability measure @xmath185 $ ] , and transition probabilities @xmath186 $ ] , we are interested in the markov chain @xmath187 that starts according to @xmath188 and proceeds according to @xmath0 . formally , we construct the probability space @xmath189 , where @xmath190 is the set of trajectories , @xmath191 is the product @xmath19-algebra , and @xmath192 is the probability measure induced by @xmath188 and @xmath0 . the projections @xmath193 given by @xmath194 become random variables that constitute the markov chain . for details on the terminology used above , see e.g.  @xcite , and for a gentle introduction to discrete markov chains , see e.g.  @xcite . we will use the term random walk instead of markov chain . next , let us assume that @xmath184 is a countable group @xmath38 , in which case we may study random walks whose transition probabilities are adapted to the group structure . in order to do so , we first pick a probability measure @xmath195 $ ] whose support @xmath196 generates @xmath38 as a semigroup , see also remark  [ rem : recurrenceandtransience ] below . then , we consider the random walk given by the following data . the initial probability measure @xmath197 $ ] puts all mass on the identity element @xmath198 and the transition probabilities @xmath199 $ ] are given by @xmath200 . we could also have said @xmath201 , which leads to a handy interpretation . the random walk starts at the identity element and has independent @xmath10-distributed increments each of which is multiplied from the right to the current state . therefore , @xmath202 a.s . ( = almost surely ) and for every @xmath162 we may decompose @xmath203 , where @xmath204 is a sequence of independent @xmath10-distributed random variables , see the right - hand side of figure  [ fig : randomwalk ] .    . ]    ( 0,0 ) ( -287,60)@xmath205 ( -406,94 ) ( -380.5,108.5 ) ( -347,7)@xmath189 ( -176.5,1)@xmath206 ( -163,148)@xmath207 ( -28,132)@xmath208    [ rem : recurrenceandtransience ] since we assume that @xmath209 generates @xmath38 as a semigroup , the random walk is irreducible , i.e.  any two states can be reached from each other with positive probability . in particular , the following dichotomy holds . either every state is recurrent , i.e.  the return probability is equal to 1 , or every state is transient , i.e.  the return probability is smaller than 1 . in the latter case , the probability that every finite set of states will eventually be left and the random walk escapes to infinity is equal to 1 . we need to assume that the probability of huge jumps is sufficiently small . the notion of moments helps us to make this assumption rigorous . given a probability space , e.g.  @xmath189 introduced in section [ sub : bn ] , and a real valued random variable @xmath210 , the latter has finite first moment if @xmath211 . in this case , both @xmath212 and @xmath213 , and we can define the expectation @xmath214 . of course , the difference would still make sense if only one of the two integrals was finite . but this is not of relevance for us and when writing @xmath215 we implicitly mean that @xmath216 . more generally , given any non - negative @xmath217 , a real valued random variable @xmath210 has finite @xmath111-th moment if @xmath218 . in our setting , the increments @xmath204 are not real valued random variables but take values in @xmath38 , whence we need to specify the hugeness of a jump before talking about finite moments . if @xmath38 is a finitely generated group and @xmath219 is a finite generating set , then the word metric @xmath220 on @xmath38 is given by @xmath221 note that the word metric coincides with the distance in the respective cayley graph . a random variable @xmath222 has finite @xmath111-th moment if the image @xmath223 has finite @xmath111-th moment in the classical sense , i.e.  if @xmath224 . we leave it to the reader to verify that this property does not depend on the choice of the finite generating set @xmath219 , see also ( * ? ? ? * lemma 11.37 ) . let us now return to the situation we are interested in , namely that @xmath225 with @xmath12 . when working with the projection @xmath226 , we often consider the imaginary parts @xmath227 and the real parts @xmath228 separately , and it is convenient to abbreviate the former by @xmath229 and the latter by @xmath230 . occasionally , we do not need to assume that @xmath231 has some finite moment but impose this assumption on the images @xmath232 and @xmath233 . the following lemma relates the two situations . [ lem : assumptions ] if @xmath231 has finite @xmath111-th moment , then @xmath232 and @xmath233 have finite @xmath111-th moment , too . [ rem : multiple ] before we prove lemma [ lem : assumptions ] , let us note that for every @xmath88 the imaginary part @xmath229 can be expressed in terms of the level @xmath234 , namely as @xmath235 . this formula can be shown using either the multiplication in @xmath71 or lemma [ lem : lookingfromfront ] . taking the logarithm on both sides yields @xmath236 . so , instead of thinking of@xmath237 we may think of a multiple of @xmath234 . let @xmath238 be the standard generating set . then @xmath239 concerning the second assertion , observe that @xmath240 , which can be shown by the same argument as in the proof of proposition [ prop : discretehyperbolicplane ] . this observation allows us to estimate @xmath241 by a multiple of @xmath242 . indeed , @xmath243 & \leq\max{\ensuremath{\{\ , \ell_{a},\ell_{b } \,\}}}\cdot\operatorname{\textit{d}_{s}}(1,g)+\ln\left(\frac{q}{p}\right)\cdot|\lambda(g)|\leq\max{\ensuremath{\{\ , \ell_{a},\ell_{b } \,\}}}\cdot\operatorname{\textit{d}_{s}}(1,g)+\ln\left(\frac{q}{p}\right)\cdot\operatorname{\textit{d}_{s}}(1,g)\,.\end{aligned}\ ] ] therefore , @xmath244 which proves the claim . it is easy to construct examples showing that the converse of lemma  [ lem : assumptions ] does not hold . in addition to the moments of @xmath232 and @xmath233 , we will use the notion of vertical drift . consider a random walk @xmath245 on @xmath38 and its pointwise projection @xmath246 to the levels . since @xmath247 , these projections constitute a random walk on the integers with i.i.d . ( = independent and identically distributed ) increments . [ def : verticaldrift ] if@xmath232 has finite first moment , then @xmath248 has finite first moment and we can define the expectation @xmath249 . the latter is called the vertical drift and denoted by  @xmath250 . we will distinguish between positive vertical drift , i.e.  @xmath251 , negative vertical drift , i.e.  @xmath252 , and no vertical drift , i.e.  @xmath253 , which is the most subtle of the three cases . the poisson  furstenberg boundary is a probabilistic model for the long - time behaviour of a random walk . in order to define it , we need to ensure that we are working with lebesgue  rohlin spaces , which are also known as standard probability spaces . for definitions and basic examples we refer to @xcite , @xcite , and @xcite . moreover , let us mention the collection of facts in ( * ? ? ? * appendix ) and the more informal introduction in @xcite . the most prominent examples of lebesgue rohlin spaces are discrete probability spaces and the unit interval @xmath254 $ ] endowed with the lebesgue @xmath19-algebra @xmath255 and the lebesgue measure @xmath58 . in fact , every lebesgue rohlin space is isomorphic . so , we actually mean isomorphic @xmath256 . recall that two probability spaces @xmath257 and @xmath258 are isomorphic @xmath256 if there are null sets @xmath259 with @xmath260 and a bijection @xmath261 which is measurable and measure preserving in both directions . ] either to one of these examples or to the disjoint union of an interval @xmath262 $ ] with @xmath263 and countably many atoms with total mass @xmath264 , see @xcite and ( * ? ? ? * proposition 6 ) . [ rem : polish ] a polish space is a topological space that is separable , i.e.  contains a countable and dense subset , and completely metrisable , i.e.  there is a metric that induces the topology and turns the space into a complete metric space . all polish spaces endowed with the borel @xmath19-algebra  @xmath265 and a borel measure @xmath10 become , after completion , examples of lebesgue  rohlin spaces , see @xcite and ( * ? ? ? * , example 1 ) .      in light of remark [ rem : polish ] , we may observe that the space of trajectories @xmath266 introduced in section [ sub : bn ] is the product @xmath267 and can therefore be endowed with the product topology . one can show that the latter is actually a polish space , see e.g.  ( * ? ? ? * theorem 24.11 ) . since its borel @xmath19-algebra agrees with the product @xmath19-algebra @xmath191 , the completion of @xmath189 is a lebesgue  rohlin space . from now on , let us assume that , as soon as a measurable space is endowed with a probability measure , we are working with its completion . we may therefore say that @xmath189 is a lebesgue  rohlin space .    since we are interested in the long - time behaviour of the trajectories @xmath268 , we identify those pairs of trajectories whose tails sooner or later behave identically . more precisely , we define an equivalence relation @xmath269 on @xmath266 by @xmath270 note that we allow the times @xmath271 and @xmath272 to be different . if we did not , we would end up with the tail boundary instead of the poisson  furstenberg boundary . consider the partition @xmath273 of @xmath266 into equivalence classes mod @xmath269 , see in figure [ fig : randomwalk ] . this partition induces a sub-@xmath19-algebra @xmath274 of @xmath191 , consisting of all @xmath275 which are compatible with the partition @xmath273 , i.e.  which are unions of equivalence classes mod  @xmath269 , see in figure [ fig : randomwalk ] . the poisson  furstenberg boundary @xmath276 is the quotient of @xmath189 with respect to the induced sub-@xmath19-algebra @xmath274 . more precisely , it is the lebesgue  rohlin space @xmath277 that consists of the measurable hull @xmath278 of @xmath273 , the induced sub-@xmath19-algebra @xmath279 , and the restriction @xmath280 of the probability measure @xmath192 to @xmath279 . concerning the measurable hull , see @xcite and @xcite . moreover , compare ( * ? ? ? * proposition 11 ) .    the map from the trajectory space @xmath266 to the poisson  furstenberg boundary @xmath145 that assigns to every trajectory @xmath281 the respective element of the partition @xmath278 is called the boundary map @xmath282 . note that the above is not the only possible definition of the poisson  furstenberg boundary , further equivalent ones are given in @xcite . one important feature of the poisson  furstenberg boundary is that it can be used to describe all bounded harmonic functions on the state space @xmath184 . assume we are given a countable state space  @xmath184 and transition probabilities @xmath283 $ ] as introduced in section [ sub : bn ] . a function @xmath284 is called harmonic if for every element @xmath285 the equation @xmath286 holds . in other words , being at @xmath285 , the value of @xmath287 today is exactly as large as the expected value of @xmath287 tomorrow . the initial probability measure of a random walk is denoted by @xmath185 $ ] . first , we pick some reference measure @xmath188 with @xmath288 . then , we consider the random walk @xmath245 that starts according to @xmath188 , has probability measure @xmath289 and poisson  furstenberg boundary @xmath290 . all other initial probability measures , in particular the dirac measures @xmath291 at points @xmath285 , are absolutely continuous with respect to @xmath188 . therefore , the measures @xmath292 are absolutely continuous with respect to @xmath289 , which implies that we may endow @xmath293 with measures @xmath294 in order to obtain the respective poisson  furstenberg boundaries .    from this point of view , it would have made sense to define the poisson  furstenberg boundary as a measurable space @xmath293 endowed with a family of measures . a first step decomposition shows that for every two points @xmath295 the equation @xmath296 holds . hence , given an essentially bounded function @xmath297 mapping from the poisson  furstenberg boundary @xmath290 to the real numbers @xmath298 , we can construct a bounded harmonic function @xmath299 given by the poisson integral representation formula @xmath300 . there is also a way back from @xmath287 to @xmath297 using martingale convergence so that , in the end , one obtains a one - to - one correspondence , even an isometry of banach spaces , between the space @xmath301 of equivalence classes of essentially bounded functions and the space @xmath302 of bounded harmonic functions , see e.g.  ( * ? ? ? * section 2.1 ) .      given a random walk , be it on a generic state space or on a group , a challenging problem is to decide whether the poisson  furstenberg boundary is trivial or not . in the latter case , one may wonder how to identify it geometrically . we shall only outline a few results about the poisson  furstenberg boundary of random walks on countable groups . a recent survey has been given by erschler in @xcite . as before , let @xmath245 be a random walk on a countable group @xmath38 driven by the probability measure  @xmath10 . we assume that the support @xmath209 generates @xmath38 as a semigroup , see section [ sub : bn ] . if @xmath38 is abelian , then the poisson  furstenberg boundary is always trivial , see @xcite and @xcite . the same holds true for all groups of polynomial growth , and for groups of subexponential growth endowed with a probability measure @xmath10 with finite first moment . for the special case of probability measures with finite support , see @xcite , and for the general case , see e.g.  ( * ? ? ? * theorem 5.3 ) and @xcite . moreover , it was shown in @xcite , that the assumption of finite first moment can not be dropped . if @xmath38 is amenable , then one can show that there is at least one symmetric probability measure @xmath10 such that the poisson  furstenberg boundary is trivial , see the conjecture in @xcite . the proof of the conjecture has been announced in ( * ? ? ? * theorem 4 ) and given in @xcite and @xcite . in case of the baumslag solitar group @xmath303 , the poisson  furstenberg boundary may or may not be trivial depending on the vertical drift , see definition  [ def : verticaldrift ] . more precisely , for random walks on @xmath303 with finite first moment the poisson furstenberg boundary is isomorphic to @xmath298 for @xmath252 and trivial for @xmath253 and isomorphic to @xmath304 for @xmath251 , see ( * ? ? ? * theorem  5.1 ) . we may think of @xmath304 as the space of upper ends of the corresponding bass  serre tree @xmath13 . further results about random walks on rational affinities are given in @xcite . for the poisson  furstenberg boundary of lamplighter random walks , see @xcite , @xcite , @xcite , and also @xcite . if @xmath38 is non - amenable , then the poisson  furstenberg boundary is always non - trivial , see @xcite . in particular , this implies that the poisson  furstenberg boundary of random walks on non - amenable baumslag solitar groups can never be trivial , also when @xmath253 . kaimanovich s strip criterion is a tool for identifying the poisson  furstenberg boundary geometrically . the strategy is to choose a suitable @xmath10-boundary as a candidate . our one will be given in terms of the boundaries @xmath16 and @xmath15 . the strip criterion then enables us to prove that our candidate is indeed isomorphic to the poisson  furstenberg boundary . let us first recall the strip criterion . for a proof we refer to @xcite . [ thm : strip ] let @xmath245 be a random walk on a countable group @xmath38 driven by a probability measure @xmath10 with finite entropy @xmath305 . moreover , let @xmath306 and @xmath307 be @xmath308- and @xmath10-boundaries , respectively . if there exist a gauge @xmath309 on @xmath38 with associated gauge function @xmath310 and a measurable @xmath38-equivariant map @xmath311 assigning to pairs of points @xmath312 non - empty strips @xmath313 such that for every @xmath88 and @xmath314-almost every @xmath312 @xmath315 then the @xmath10-boundary @xmath307 is maximal . [ rem : removeg ] the proof shows that it is not even necessary to verify the convergence for every @xmath88 . it suffices to consider the special case @xmath96 as long as we can ensure that a random strip contains the identity element @xmath198 with positive probability , i.e.  that @xmath316 . the following four notions have not yet been introduced .    * ( a ) `` entropy '' *    the entropy of the probability measure @xmath10 is the expected amount of information contained in the outcome of a random variable that is distributed according to @xmath10 . more precisely , it is the real number given by @xmath317 . here , as usual , one defines @xmath318 . for us , the assumption of finite entropy will be no issue because baumslag solitar groups are finitely generated and the increments under investigation have finite first moment . this implies that their probability measures  @xmath10 have finite entropy , as shown by the following well - known lemma . [ lem : entropy ] let @xmath38 be a finitely generated group and a probability measure @xmath10 with infinite entropy supported on the generating set @xmath319 . then , @xmath184 has finite first moment with respect to the word metric @xmath320 , but @xmath10 has infinite entropy . ] and let @xmath321 $ ] be a probability measure . if a random variable @xmath222 distributed according to @xmath10 has finite first moment , then @xmath10 has finite entropy . let @xmath219 be a non - empty finite generating set . moreover , let @xmath322 , whence @xmath323 . in this proof , we shall use the shorthand notation @xmath324 instead of @xmath220 to denote the word metric on @xmath38 . we have to show that the entropy @xmath325 is finite . first , we change the base of the logarithm @xmath326 and split the summands appropriately @xmath327 then , we recall that the function @xmath328 is increasing on the interval @xmath329 $ ] , and conclude that @xmath330     & \leq\sum_{g\in g}\operatorname{\textit{d}}(1,g)\cdot b^{-\operatorname{\textit{d}}(1,g)}\leq\sum_{n=0}^{\infty}(2\cdot|s|)^{n}\cdot n\cdot b^{-n}<\infty\,.\end{aligned}\ ] ] on the other hand , since @xmath184 has finite first moment , @xmath331 so , both sums are finite , whence @xmath305 must be finite , too .    * ( b ) `` @xmath10-boundary '' *    two equivalent definitions of a @xmath10-boundary can be found in @xcite . for us , it suffices to record that every lebesgue rohlin space @xmath307 endowed with a left @xmath38-action and a boundary map @xmath332 that is measurable , @xmath269-invariant , and @xmath38-equivariant is a @xmath10-boundary .    * ( c ) `` @xmath308-boundary '' *    while @xmath10 is the probability measure driving the random walk , the symbol @xmath308 denotes the reflected probability measure given by @xmath333 . accordingly , a @xmath308-boundary is a space that satisfies the requirements of a @xmath10-boundary when replacing @xmath10 by @xmath308 .    * ( d ) `` gauge '' *    a gauge @xmath334 is an exhaustion @xmath309 of the group @xmath38 , i.e.  a sequence of subsets @xmath335 which is increasing @xmath336 and whose union @xmath337 is the whole group @xmath38 . given a gauge @xmath334 and an element @xmath88 , we may ask for the minimal index @xmath338 with the property that @xmath339 . this index is the value of the associated gauge function @xmath310 at @xmath49 . kaimanovich distinguishes between various kinds of gauges , see @xcite . for example , a gauge @xmath334 is subadditive if any two group elements @xmath340 satisfy @xmath341 and it is temperate if all gauge sets @xmath342 are finite and grow at most exponentially . even though these two properties do play a crucial role in the corollaries to the strip criterion given in @xcite , they are not required in the strip criterion itself . and , in fact , not all of our gauges will have these two properties . * ( e ) `` measurable strips '' *    the power set @xmath343 is naturally endowed with the product @xmath19-algebra , which enables us to talk about measurability of the map @xmath344 .    * ( f ) `` maximal '' *    the poisson  furstenberg boundary @xmath276 inherits a left @xmath38-action from the space of trajectories , and every @xmath10-boundary @xmath307 is a @xmath38-invariant measurable quotient of @xmath276 . @xmath345_{\operatorname{bnd } } \ar[dr]^{\operatorname{bnd}_{+ } } \ar@{}[d]|/.28868\einheit/{\circlearrowleft } & \\ \makebox[30pt][r]{$(b,\mathcal{b},\nu)$ } \ar[rr]_{\vphantom{b}\pi } & & \makebox[30pt][l]{$(b_{+},\mathcal{b}_{+},\nu_{+})$ } } \end{xy}\ ] ] a @xmath10-boundary @xmath307 is called maximal if the projection @xmath346 is a measurable isomorphism between lebesgue  rohlin spaces . we still assume that @xmath12 and consider a random walk @xmath245 on @xmath225 . moreover , recall the abbreviations @xmath347 and @xmath348 introduced in section  [ sub : rpipavd ] .      the following lemmas concern the behaviour of the projections @xmath349 . they seem to be well - known and we do not claim originality . but , for the sake of completeness , we give rigorous proofs . [ lem : positivedrifthyp ] assume that @xmath232 has finite first moment . if the vertical drift is positive , i.e.  @xmath251 , then the projections @xmath349 converge a.s .  to @xmath85 . we can use the strong law of large numbers to obtain @xmath350{~n\to\infty~}\,\mathbb{e}(\lambda(x_{1}))=\delta>0\,.\ ] ] hence , the projections @xmath351 tend a.s .  to infinity and , by remark [ rem : multiple ] , the imaginary parts @xmath352 do . this , of course , implies that the absolute values @xmath353 tend a.s .  to infinity . now , we can use lemma  [ lem : convergence ] to complete the proof . [ lem : negativedrifthyp ] assume that both @xmath232 and @xmath233 have finite first moment . if the vertical drift is negative , i.e.  @xmath252 , then the projections @xmath349 converge a.s .  to a random element @xmath183 . note that the argument given in the proof of lemma [ lem : positivedrifthyp ] can be adapted to show that the imaginary parts @xmath352 converge a.s .  to @xmath354 , whence we only need to understand the behaviour of the real parts @xmath355 . the equation @xmath356 yields @xmath357 , and in light of the multiplication in @xmath71 we obtain @xmath358 & = \big(\pi_{\operatorname{aff\,^{+}}(\mathbb{r})}(x_{1})\circ\dotsc\circ\pi_{\operatorname{aff\,^{+}}(\mathbb{r})}(x_{n})\big)(i ) \\ & = a_{x_{1}}\cdot\dotsc\cdot a_{x_{n}}\cdot i+\sum_{k=1}^{n}a_{x_{1}}\cdot\dotsc\cdot a_{x_{k-1}}\cdot b_{x_{k}}\,.\end{aligned}\ ] ] hence , the real parts @xmath355 are partial sums of the infinite series @xmath359 with @xmath360 . in order to verify a.s .  convergence of the latter , we apply cauchy s root test , @xmath361{~k\to\infty~}\,\left(\frac{q}{p}\right)^{\delta}<1\,.\ ] ] for the convergence claimed in the first factor we can use the strong law of large numbers , for the one claimed in the second factor the borel  cantelli lemma . indeed , let us write @xmath362 for the quotient @xmath363 . in order to show that @xmath364 a.s . , recall that @xmath233 has finite first moment . so , for every @xmath152 we may estimate @xmath365 therefore , the borel cantelli lemma yields @xmath366 . replacing @xmath153 by @xmath367 , we obtain a countable family of null sets whose union is , of course , again a null set that consists of all @xmath368 with @xmath369 . hence , @xmath364 a.s . , see also ( * ? ? ? * exercise 5.1.3 ) . so , we have finally convinced ourselves that @xmath370 a.s . , whence @xmath359 converges a.s .  to a random element @xmath371 . the natural question that remains is the one asking for the driftless case . [ quote : brofferio ] an answer has been given by brofferio in ( * ? ? ? * theorem 1 ) . it says that under the same mild assumptions , namely that @xmath232 and @xmath233 have finite first moment , the projections @xmath349 converge a.s .  to @xmath85 . but , for us , a result of slightly different flavour will be of relevance . [ lem : zerodrifthyp ] assume that @xmath232 has finite second moment and there is an @xmath152 such that @xmath233 has finite @xmath372-th moment . if there is no vertical drift , i.e.  @xmath253 , then the projections @xmath349 have sublinear speed , i.e. @xmath373{~n\to\infty~}\,0\,.\ ] ]    the proof is based on ideas that go back to lie in ( * ? ? ? * lemme 5.49 ) and have also been used by cartwright , kaimanovich , and woess in ( * ? ? ? * proposition 4b ) . we first adapt these ideas to our situation by stating and proving lemma [ lem : elie ] , and then deduce lemma  [ lem : zerodrifthyp ] .    by assumption , there is no vertical drift so that the pointwise projection @xmath246 is recurrent , see plya s theorem for the simple random walk and the chung fuchs theorem in @xcite for the general case . in particular , we know that there exists a.s .  a strictly increasing sequence @xmath374 given by @xmath375 and by @xmath376 for all @xmath162 . we call @xmath377 the @xmath163-th ladder time , see figure [ fig : upwardstopping ] for an illustration of the first ladder times @xmath378 and @xmath379 . the following lemma concerns the random variable @xmath380 with @xmath381 .     and @xmath379 . ]    ( 0,0 ) ( -116,26)@xmath382 ( -184,86)@xmath383    [ lem : elie ] under the same assumptions as in lemma [ lem : zerodrifthyp ] , namely that @xmath232 has finite second moment , there is an @xmath152 such that @xmath233 has finite @xmath372-th moment , and there is no vertical drift , i.e.  @xmath253 , the random variable @xmath380 has finite first moment .    adapting the proof of ( * lemme 5.49 ) , we begin with some preliminaries . pick an @xmath152 that satisfies the requirements of lemma [ lem : elie ] and let @xmath384 . since @xmath232 has finite second moment , we know that also @xmath248 has finite second moment and @xmath385 , see @xcite referring to @xcite . using this asymptotics , we obtain @xmath386 in particular , there is a @xmath387 such that for all @xmath388 the summands @xmath389 are strictly smaller than @xmath390 . since @xmath391 , we know that @xmath392 . moreover , note that , by construction of the ladder times @xmath374 , the differences @xmath393 are i.i.d . , whence the fact that @xmath394 , which implies that @xmath395 , and the strong law of large numbers yield @xmath396{~n\to\infty~}\,\mathbb{e}\left(\tau^{\beta}\right)\ , , \\ \longrightarrow~\limsup_{n\to\infty}\frac{{\tau(n)}^{\beta}}{n}<\infty~~\text{a.\,s . } \label{eqn : preliminary } \tag{$\ast$}\end{gathered}\ ] ] now , we are prepared for the main argument . recall that we aim to show that @xmath380 has finite first moment . the sums @xmath397 are i.i.d .  and non - negative with the additional property that they are not a.s . equal to @xmath354 . hence , by ( * ? ? ? * lemme 5.23 ) , @xmath398 it thus suffices to verify the right - hand side . in order to do so , we would like to estimate @xmath399 a priori , it might be the case that @xmath400 and the second factor in the rightmost term is @xmath354 , in which case the product would not make sense . we claim that @xmath401 is a.s . finite , which does not only legitimate the above estimate but also completes the proof . indeed , observe that @xmath402 & \leq\underbrace{\limsup_{n\to\infty}\frac{\ln\left(\tau(n)\right)}{{\tau(n)}^{\beta}}}_{=\;0}\,+\limsup_{n\to\infty}\frac{\ln\left(1+\max_{1\leq k\leq\tau(n)}{\ensuremath{\{\ , |b_{x_{k}}| \,\}}}\right)}{{\tau(n)}^{\beta } } \\ & = \limsup_{n\to\infty}\left(\frac{\max_{1\leq k\leq\tau(n)}\left\{\,\ln\left(1+|b_{x_{k}}|\right)^{\frac{1}{\beta}}\,\right\}}{\tau(n)}\right)^{\beta}\leq\limsup_{n\to\infty}\left(\underbrace{\frac{\sum_{k=1}^{\tau(n)}\ln\left(1+|b_{x_{k}}|\right)^{\frac{1}{\beta}}}{\tau(n)}}_{=:\;m_{n}}\right)^{\beta}\,.\end{aligned}\ ] ] now , recall that @xmath403 . so , by the strong law of large numbers , @xmath404{~n\to\infty~}\,\mathbb{e}\left(\ln(1+|b_{x_{1}}|)^{\frac{1}{\beta}}\right)\,.\ ] ] this implies that @xmath405 a.s . , and completes the proof . recall from @xcite and @xcite that @xmath385 with a strictly positive constant . in particular , there is a @xmath387 such that for all @xmath388 the summands @xmath406 are strictly larger than @xmath407 and we obtain @xmath408 as we have mentioned in the proof of lemma [ lem : elie ] , the differences @xmath393 are i.i.d . , and they are non - negative . so , we may deduce from the strong law of large numbers that @xmath409{~n\to\infty~}\,\infty\quad\text{and}\quad\frac{n}{\tau(n)}\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,0\,.\ ] ] this can be used to estimate the distance between @xmath410 and @xmath349 from above . first , for every @xmath411 let @xmath412 be the unique element with @xmath413 . this element exists a.s .  because the ladder times @xmath414 do . now , we may estimate @xmath415{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}}+\underbrace{\frac{\operatorname{\textit{d}_{\mathbb{h}}}(a_{z_{\tau(m)}}\cdot i , a_{z_{\tau(m)}}\cdot i+b_{z_{n}})}{n}}_{{\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 2};}}}+\underbrace{\frac{\operatorname{\textit{d}_{\mathbb{h}}}(a_{z_{\tau(m)}}\cdot i+b_{z_{n}},a_{z_{n}}\cdot i+b_{z_{n}})}{n}}_{{\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 3};}}}\,.\ ] ] the numbers refer to figure [ fig : distancedecomposition ] . we will consider the three summands separately and show that each of them converges a.s .  to @xmath354 . for and this is straightforward . indeed , @xmath416{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}=\frac{|\ln(a_{z_{\tau(m)}})|}{n}\leq\frac{|\ln(a_{z_{\tau(m)}})|}{\tau(m)}=\ln\left(\frac{q}{p}\right)\cdot\left|\frac{\lambda(x_{1})+\dotsc+\lambda(x_{\tau(m)})}{\tau(m)}\right|\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,\ln\left(\frac{q}{p}\right)\cdot|\mathbb{e}(\lambda(x_{1}))|=\ln\left(\frac{q}{p}\right)\cdot|\delta|=0\ ] ] and similarly @xmath416{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 3};}}\leq\frac{\operatorname{\textit{d}_{\mathbb{h}}}(a_{z_{\tau(m)}}\cdot i , i)}{n}+\frac{\operatorname{\textit{d}_{\mathbb{h}}}(i , a_{z_{n}}\cdot i)}{n}={\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}+\frac{|\ln(a_{z_{n}})|}{n}={\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}+\ln\left(\frac{q}{p}\right)\cdot\left|\frac{\lambda(x_{1})+\dotsc+\lambda(x_{n})}{n}\right|\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,0\,.\ ] ] for recall from the proof of lemma [ lem : negativedrifthyp ] that @xmath417 and observe that for every @xmath418 with @xmath419 the following estimate holds @xmath420 & = \sum_{k=\tau(m)+1}^{\ell}\underbrace{a_{x_{\tau(m)+1}}\cdot\dotsc\cdot a_{x_{k-1}}}_{\leq\,1}\cdot\,|b_{x_{k}}|\leq\sum_{k=\tau(m)+1}^{\ell}|b_{x_{k}}|~~\text{a.\,s . } \tag{$\ast$}\end{aligned}\ ] ] hence , using that @xmath421 and @xmath422 , we obtain @xmath423{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 2 } ; } } & = \frac{1}{n}\cdot\operatorname{arcosh}\left(1+\frac{1}{2}\cdot\left(\frac{|b_{z_{n}}|}{a_{z_{\tau(m)}}}\right)^{2}\right)=\frac{1}{n}\cdot\ln\left(1+\frac{1}{2}\cdot\left(\frac{|b_{z_{n}}|}{a_{z_{\tau(m)}}}\right)^{2}+\sqrt{\left(1+\frac{1}{2}\cdot\left(\frac{|b_{z_{n}}|}{a_{z_{\tau(m)}}}\right)^{2}\right)^{2}-1}\;\right ) \\[3pt ] & \leq\frac{1}{n}\cdot\left(\ln(2)+\ln\left(1+\left(\frac{|b_{z_{n}}|}{a_{z_{\tau(m)}}}\right)^{2}\right)\right)\leq\frac{1}{n}\cdot\left(\ln(2)+2\cdot\ln\left(1+\frac{|b_{z_{n}}|}{a_{z_{\tau(m)}}}\right)\right ) \\[10pt ] & \leq\frac{1}{n}\cdot\left(\ln(2)+2\cdot\ln\left(1+\frac{|b_{z_{\tau(1)}}-b_{z_{\tau(0)}}|}{a_{z_{\tau(0)}}}+\frac{|b_{z_{\tau(2)}}-b_{z_{\tau(1)}}|}{a_{z_{\tau(1)}}}+\dotsc+\frac{|b_{z_{\tau(m)}}-b_{z_{\tau(m-1)}}|}{a_{z_{\tau(m-1)}}}+\frac{|b_{z_{n}}-b_{z_{\tau(m)}}|}{a_{z_{\tau(m)}}}\right)\right)\,,\end{aligned}\ ] ] which allows us to apply ( [ eqn : observation ] ) and finally conclude @xmath424 & \leq\underbrace{\frac{\ln(2)}{n\vphantom{()}}}_{\to\;0}+\,2\cdot\underbrace{\frac{\ln\left(1+\sum_{k=\tau(0)+1}^{\tau(1)}|b_{x_{k}}|\right)+\dotsc+\ln\left(1+\sum_{k=\tau(m)+1}^{\tau(m+1)}|b_{x_{k}}|\right)}{m+1\vphantom{()}}}_{\to\;\mathbb{e}(\ln(1+\sum_{k=1}^{\tau}|b_{x_{k}}|))~\text{a.\,s.~by lemma \ref{lem : elie}}}\cdot\underbrace{\frac{m+1}{\tau(m)}}_{\substack{\to\;0 \\ \text{a.\,s.}}}\cdot\underbrace{\frac{\tau(m)}{n\vphantom{()}}}_{\substack{\leq\,1 \\ \text{a.\,s.}}}\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,0\,.\hspace*{38pt}\end{aligned}\ ] ]     and @xmath349 . ]    ( 0,0 ) ( -132,49 ) ( -74,80.5 ) ( -45,41 ) ( -109,94)@xmath425 ( -43,15)@xmath349 ( -114,31)@xmath410      unlike the ones considered in section [ sub : hyperbolicconvergence ] , the projections @xmath426 do not need to satisfy the markov property . consider , for example , the random walk @xmath245 driven by the uniform measure on the standard generators and their formal inverses . then , given @xmath427 and @xmath428 , the projection @xmath429 comes back to @xmath145 with probability @xmath430 . on the other hand , coming back to @xmath145 in a single step would not be possible if the history was @xmath431 and @xmath432 . despite of this subtlety , the following lemmas yield almost sure convergence of the projections @xmath426 to a random end . [ lem : nonzerodrifttree ] assume that @xmath231 has finite first moment . if the vertical drift is different from @xmath354 , i.e.  @xmath433 , then the projections @xmath426 converge a.s .  to a random end @xmath434 .    by lemma [ lem : assumptions ] , the assumption that @xmath231 has finite first moment implies that @xmath232 has also finite first moment . for the proof of the lemma  [ lem : nonzerodrifttree ] , we give an argument using the notion of regular sequences , see @xcite . one difference to @xcite is that we do not fix any particular end @xmath435 . therefore , we replace the busemann function  @xmath50 , which depends on the choice of @xmath435 , by the graph distance to the basepoint  @xmath145 . the other difference is that we work with the limit inferior instead of the limit in order to be prepared to deal with the driftless case , too . let @xmath436 be the graph distance in the tree @xmath13 . accordingly , the symbol @xmath437 denotes the graph distance @xmath438 from the basepoint @xmath145 to the vertex @xmath9 . we call a sequence @xmath439 of vertices regular if @xmath416{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}\quad\liminf_{n\to\infty}\;\frac{|x_{n}|_{\mathbb{t}}}{n}>0\quad\text{and}\quad{\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 2};}}\quad\frac{\operatorname{\textit{d}_{\mathbb{t}}}(x_{n},x_{n+1})}{n}\,\xrightarrow{~n\to\infty~}\,0\,.\ ] ] in order to prove lemma  [ lem : nonzerodrifttree ] , we pursue a two - step strategy . first , we show that every regular sequence converges to an end and , second , that the projections @xmath426 constitute a.s .  a regular sequence . concerning the first part , we pick an arbitrary regular sequence @xmath439 and claim that there is an end @xmath434 such that for every @xmath152 the open ball @xmath440 contains infinitely many  @xmath441 . assume there was no such end . then , we know that for every @xmath434 there is an @xmath442 such that @xmath443 contains only finitely many @xmath441 , whence there is also an @xmath444 such that @xmath445 does not contain any @xmath441 at all . the open balls @xmath445 with @xmath434 and the singletons @xmath446 of vertices @xmath9 form an open covering of @xmath167 . by compactness , it contains a finite subcovering . but since the constants @xmath444 have been chosen in such a way that the sequence @xmath439 does not enter any of the open balls @xmath445 , it must remain in a finite subset of the tree , which contradicts . next , we pick such an end @xmath434 and claim that the sequence @xmath439 converges to @xmath447 . let @xmath152 . we define @xmath448 . by , the constant @xmath449 is strictly positive . moreover , all but finitely many @xmath450 are strictly greater than @xmath451 . the elements in @xmath452 are characterised by the property that the paths starting in @xmath145 and representing them must have a certain finite initial piece . let @xmath453 be the length of this piece , i.e.  @xmath454 , see figure [ fig : jumpaway ] . by the above and by , there is an @xmath455 such that for all @xmath456 the inequalities @xmath457 and @xmath458 hold . since @xmath452 contains infinitely many @xmath441 , we can even find an @xmath459 such that @xmath460 . it turns out that not just for @xmath461 but for all @xmath462 we have @xmath463 . indeed , if there was an @xmath462 such that @xmath463 and @xmath464 , we would know that @xmath465 the latter , of course , contradicts @xmath458 , see figure [ fig : jumpaway ] . so , the two claims show that every regular sequence converges to an end . concerning the second part , we aim to prove that @xmath416{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}\quad\liminf_{n\to\infty}\;\frac{|\pi_{\mathbb{t}}(z_{n})|_{\mathbb{t}}}{n}>0~~\text{a.\,s.}\quad\text{and}\quad{\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 2};}}\quad\frac{\operatorname{\textit{d}_{\mathbb{t}}}(\pi_{\mathbb{t}}(z_{n}),\pi_{\mathbb{t}}(z_{n+1}))}{n}\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,0\,.\ ] ] recall from remark [ rem : multiple ] that not just @xmath231 and @xmath232 have finite first moment but also @xmath248 has . so , the strong law of large numbers yields @xmath466{~n\to\infty~}\,|\mathbb{e}(\lambda(x_{1}))|=|\delta|>0\ ] ] which implies that @xmath467 next , let @xmath238 be the standard generating set . the numerators @xmath468 of the fraction considered in are i.i.d . , and the first one satisfies @xmath469 so , again , by the strong law of large numbers @xmath470{~n\to\infty~}\,\mathbb{e}(\operatorname{\textit{d}_{\mathbb{t}}}(\pi_{\mathbb{t}}(z_{0}),\pi_{\mathbb{t}}(z_{1})))\,,\ ] ] from where a simple calculation yields .    . ]    ( 0,0 ) ( -2,92)@xmath447 ( -71,107)@xmath452 ( -26,88.5)@xmath471 ( -37,35)@xmath472 ( -143,38)@xmath473 ( -198,52)@xmath145 ( -127,71)@xmath453 ( -221.5,110)eventually left    for the driftless case , the situation is not as easy and in order to show almost sure convergence of the projections @xmath426 to a random end , we restrict ourselves to the non - amenable subcase @xmath20 . [ lem : zerodrifttree ] let @xmath20 . assume that @xmath231 has finite first moment , @xmath232 has finite second moment , and there is an @xmath152 such that @xmath233 has finite @xmath372-th moment . if there is no vertical drift , i.e.  @xmath253 , then the projections @xmath426 converge a.s .  to a random end @xmath434 . again , we claim that the projections @xmath426 constitute a.s .  a regular sequence . but since @xmath253 , we need to modify the argument from the proof of lemma [ lem : nonzerodrifttree ] that showed . by assumption , @xmath38 is non - amenable and , in particular , the spectral radius @xmath474 of the random walk @xmath187 is strictly smaller than 1 , see e.g.  ( * ? ? ? * corollary 12.5 ) . this , together with the fact that the random walk is uniformly irreducible yields that @xmath475 for a proof of this statement , see e.g.  ( * ? ? ? * proposition 8.2 ) . in order to estimate the numerators @xmath476 from above , we apply an auxiliary result : _ there are @xmath477 such that for every element @xmath88 the inequality @xmath478 holds . _ let us postpone the proof and record that , using this auxiliary result and lemma [ lem : zerodrifthyp ] , we obtain @xmath479 this is . concerning , note that the respective argument from the proof of lemma [ lem : nonzerodrifttree ] did not use the assumption that @xmath480 , and therefore does also works for @xmath253 . so , we know that the projections @xmath426 constitute a.s .  a regular sequence , which converges to a random end by the proof of lemma [ lem : nonzerodrifttree ] .    it remains to show the auxiliary result . in order to do so , we construct a path from @xmath65 to @xmath49 in the cayley graph @xmath36 with at most @xmath481 many edges , where the values of @xmath449 and @xmath482 are to be determined uniformly , i.e.  not depending on @xmath49 . first , we aim to adjust the tree component . either combinatorially using the defining relation @xmath66 or geometrically using the properties of the cayley graph @xmath36 , we can find a path from @xmath65 to the coset @xmath51 with at most @xmath483 many edges . let @xmath484 be the endpoint of such a path . next , recall the notion of discrete hyperbolic plane from section [ sub : dhp ] . we pick an arbitrary ascending doubly infinite path @xmath485 in the tree @xmath13 that traverses the vertex @xmath51 , consider the discrete hyperbolic plane @xmath128 , and take a shortest path from @xmath50 to @xmath49 in @xmath128 . by the proof of proposition [ prop : discretehyperbolicplane ] , its length @xmath486 can be estimated from above by @xmath487 with @xmath488 . we may continue this estimate and finally obtain @xmath489 & \leq\kappa\cdot\operatorname{\textit{d}_{\mathbb{h}}}(\pi_{\mathbb{h}}(h),\pi_{\mathbb{h}}(1))+\kappa\cdot\operatorname{\textit{d}_{\mathbb{h}}}(\pi_{\mathbb{h}}(1),\pi_{\mathbb{h}}(g ) ) \\ & \leq\kappa\cdot\max{\ensuremath{\{\ , \ell_{a},\ell_{b } \,\}}}\cdot\left(\left\lfloor\frac{q}{2}\right\rfloor+1\right)\cdot|\pi_{\mathbb{t}}(g)|_{\mathbb{t}}+\kappa\cdot\operatorname{\textit{d}_{\mathbb{h}}}(\pi_{\mathbb{h}}(1),\pi_{\mathbb{h}}(g))\,.\end{aligned}\ ] ] so , the concatenation of the two paths considered above has at most @xmath490 many edges with @xmath491 and @xmath492 .      resuming sections [ sub : hyperbolicconvergence ] and [ sub : treeconvergence ] , we may formulate the following theorem . [ thm : convergence ] let @xmath187 be a random walk on a non - amenable baumslag solitar group @xmath225 with @xmath20 . suppose that the increment @xmath231 has finite first moment .    1 . if the vertical drift is * positive * , i.e.  @xmath251 , then the projections @xmath349 converge a.s .  to @xmath85 and the projections @xmath426 converge a.s . to a random element @xmath434 . 2 .   if the vertical drift is * negative * , i.e.  @xmath252 , then the projections @xmath349 converge a.s .  to a random element @xmath183 and the projections @xmath426 converge a.s .  to a random element @xmath434 . 3 .   if there is * no vertical drift * , i.e.  @xmath253 , and , in addition , @xmath232 has finite second moment and there is an @xmath152 such that @xmath233 has finite @xmath372-th moment , which is certainly the case if the increment @xmath231 has finite @xmath372-th moment , then the projections @xmath349 have sublinear speed and the projections @xmath426 converge a.s .  to a random element @xmath434 . the boundaries @xmath16 and @xmath15 are endowed with their borel @xmath19-algebras @xmath493 and @xmath494 . under suitable assumptions on the moments , the projections @xmath349 and @xmath426 converge a.s .  to a random element in the respective boundary and we may consider the boundary maps @xmath495 and @xmath496 , defined almost everywhere , assigning to a trajectory @xmath497 the limits @xmath498 even though the boundary maps are only defined almost everywhere , they are measurable in the sense that the preimages of measurable sets are measurable . given @xmath499 and @xmath500 , we may construct their product map @xmath501 . it is measurable with respect to the product @xmath19-algebra @xmath502 . since both @xmath16 and @xmath15 are metrisable and separable topological spaces , it is not hard to see that the product @xmath19-algebra @xmath502 agrees with the borel @xmath19-algebra @xmath503 , see e.g.  ( * ? ? ? * appendix m.10 ) .    in the following , one should keep in mind that @xmath499 , @xmath500 , and @xmath504 are only defined if the respective projections @xmath349 and @xmath426 converge a.s .  to a random element in the boundary . the three pushforward probability measures @xmath505 , @xmath506 , @xmath507 on the measurable spaces @xmath508 , @xmath509 , @xmath510 are called the hitting measures . note that we may again , tacitly , complete the probability spaces with respect to @xmath511 , @xmath512 , @xmath513 . each of the boundaries @xmath16 and @xmath15 is endowed with a left @xmath38-action . the one on @xmath16 is induced by the action @xmath514 on @xmath14 and the one on @xmath15 is induced by the action @xmath515 on  @xmath13 . let us describe them in more detail . the former is an action by isometries on @xmath14 , and in light of their classification mentioned in section [ sub : ptthp ] , we can also evaluate them at @xmath16 .    for the latter , recall that ends are infinite reduced paths that start in @xmath145 . the coordinatewise action on the ends maps every such path @xmath434 to some other path that need not start in @xmath145 any more . the end @xmath516 is obtained by connecting @xmath145 with the initial vertex of this path and reduce the concatenation . this way , it is not hard to see that we can map every @xmath434 to an end with an arbitrarily chosen finite initial piece.consider such a finite initial piece , i.e.  a finite reduced path from @xmath145 to some vertex @xmath51 . given the end @xmath434 , we construct its image @xmath516 . it will have the correct finite initial piece unless cancellation takes place . but then , consider the image @xmath517 instead . since @xmath518 and @xmath519 , cancellation will take place in at most one of the two cases . ] in particular , every orbit @xmath520 is infinite and dense in @xmath15 .    by changing the initial probability measure of the random walk @xmath245 as in section [ sub : pfbasoip ] , we obtain stationarity of the measure @xmath512 . more precisely , for every measurable set @xmath521 @xmath522 the same result holds true for @xmath16 and the product @xmath18 , which is endowed with the componentwise left @xmath38-action . these observations will be helpful in a moment , when we show that the hitting measures are either dirac measures or non - atomic . our proof is based on ( * ? ? ? * lemma 3.4 ) , which is much more general . the original idea for our special case might be older . [ lem : nonatomic ] the hitting measure @xmath512 is non - atomic . moreover , if @xmath251 , then the hitting measure @xmath511 is the dirac measure at @xmath85 and , if @xmath252 , then @xmath511 is again non - atomic . let us first consider the hitting measure @xmath512 . suppose , there were elements of positive measure . then , we may choose such an element @xmath434 with maximal measure @xmath25 . in particular , for every element @xmath523 we know that @xmath524 . we claim that for every @xmath523 the equality @xmath525 holds . indeed , let us first suppose that there was an element @xmath526 with @xmath527 . then , @xmath528 this is a contradiction . due to the irreducibility of the random walk , @xmath529 does not only hold for all @xmath526 but inductively for all @xmath530 , which proves our claim . but the orbit @xmath520 is infinite , so @xmath531 . and this is , again , a contradiction .    if @xmath251 , the result that the hitting measure @xmath511 is the dirac measure at @xmath85 is an immediate consequence of lemma [ lem : positivedrifthyp ] . on the other hand , if @xmath252 , then @xmath532 by lemma [ lem : negativedrifthyp ] . now , we can repeat the above argument . suppose , there was an element of positive measure . then , we may choose such an element @xmath183 of maximal measure . again , all elements in its orbit @xmath533 must have the same measure and , since the orbit is infinite , this yields a contradiction . [ lem : support ] the hitting measure @xmath512 has full support , i.e.  every non - empty open subset @xmath521 has positive measure . moreover , if @xmath252 , the hitting measure @xmath513 on the cartesian product @xmath18 has full support . we classify the ends of the tree @xmath13 according to which neighbour of the vertex @xmath145 they first traverse . this yields a partition of the space of ends into @xmath64 open subsets , i.e.  the open balls of radius 1 . at least one of them must have positive measure , call it @xmath534 . now , let @xmath521 be an arbitrary non - empty open subset . in particular , there is a vertex @xmath51 , such that all ends traversing @xmath51 belong to @xmath535 . similarly to the argument given in footnote [ fn : endcorrection ] , either @xmath536 or @xmath537 is contained in @xmath535 , see figure [ fig : hittingtree ] . we may assume w.l.o.g . that @xmath538 . moreover , due to the irreducibility of the random walk , there is an @xmath162 such that @xmath539 , i.e.  the probability to reach @xmath49 in exactly @xmath163 steps is positive . hence , @xmath540 the proof of the second assertion is similar . since we have to keep track of two components , it is slightly more technical so that we give only a proof sketch . recall from above the set @xmath534 . given the random variable @xmath500 takes a value in @xmath541 , at least one open interval @xmath542 with @xmath116 will be hit by the random variable @xmath499 with positive probability , call it @xmath543 . so , we know that @xmath544 .    now , let @xmath545 be an arbitrary non - empty open subset . by definition of the product topology , the open set @xmath535 contains a rectangle of open sets @xmath546 and @xmath547 . we seek to construct an element @xmath530 such that both @xmath548 and @xmath549 , from where we may finally conclude as above that @xmath550 . again , there is a vertex @xmath51 of the tree , such that all ends traversing @xmath51 belong to @xmath551 . moreover , there are an @xmath371 and an @xmath152 such that the open interval @xmath552 is contained in @xmath553 . based on this data , we will find an element @xmath530 of the form @xmath554 with the desired properties . let us first look at the tree component . the exponent @xmath555 is either @xmath354 or @xmath65 , whichever ensures that the reduced path from @xmath145 to @xmath556 traverses @xmath51 . now , let us turn to the hyperbolic component . the image @xmath557 is a bounded open interval . the exponent @xmath558 is chosen in such a way that the length of the image @xmath559 is at most @xmath560 . finally , there is an integer @xmath116 such that both images @xmath561 and @xmath562 are contained in the interval @xmath552 and therefore both belong to @xmath553 . the exponent @xmath563 will be either @xmath111 or  @xmath564 . let us return to the tree and choose it in such a way that all ends in the image @xmath565 traverse @xmath566 . then , by construction , they also traverse the vertex @xmath51 and belong to @xmath551 . has full support . ] ( 0,0 ) ( -82,60)@xmath145 ( -56,35)@xmath51 ( 1,65)@xmath535 ( -161,120)@xmath541 ( -154,41)positive ( -150,29)measure    [ thm : identification ] let @xmath187 be a random walk on a non - amenable baumslag  solitar group @xmath225 with @xmath20 and suppose that the increment @xmath231 has finite first moment .    1 .   if the vertical drift is * positive * , i.e.  @xmath251 , then the poisson  furstenberg boundary is isomorphic to @xmath567 endowed with the boundary map @xmath496 . 2 .   if the vertical drift is * negative * , i.e.  @xmath252 , then the poisson  furstenberg boundary is isomorphic to @xmath568 endowed with the boundary map @xmath501 . 3 .   if there is * no vertical drift * , i.e.  @xmath253 , and , in addition , @xmath232 has finite second moment and there is an @xmath152 such that @xmath233 has finite @xmath372-th moment , which is certainly the case if the increment @xmath231 has finite @xmath372-th moment , then the poisson  furstenberg boundary is again isomorphic to @xmath567 endowed with the boundary map @xmath496 . [ rem : realnumbers ] in the case of negative vertical drift , we could also have used @xmath569 instead of @xmath18 . as already mentioned , we seek to apply the strip criterion , see theorem [ thm : strip ] . by lemma [ lem : entropy ] , the probability measure @xmath10 driving the random walk has finite entropy . moreover , it is not hard to see that @xmath16 and @xmath15 are polish spaces , and so is their product @xmath18 . therefore , by remark [ rem : polish ] , the probability spaces @xmath567 and @xmath568 are lebesgue  rohlin spaces . they are endowed with a left @xmath38-action and boundary maps @xmath496 and @xmath501 , defined almost everywhere . in order to show that they are @xmath10-boundaries , we have to ensure that the boundary maps are measurable , @xmath269-invariant , and @xmath38-equivariant . but all three properties are immediate by construction , compare also ( * ? ? ? * end of  1.5 ) . if the vertical drift is negative , i.e.  @xmath252 , let us take the @xmath10-boundary @xmath568 and the @xmath308-boundary @xmath570 . here , @xmath571 denotes the hitting measure of the pointwise projection of the random walk @xmath572 driven by the reflected probability measure @xmath308 to the bass  serre tree @xmath13 . next , we need to define gauges and strips . let @xmath238 be the standard generating set and define gauges @xmath573 . in other words , the gauges exhaust the group @xmath38 with balls centred at the identity @xmath198 , and the gauge function @xmath310 is nothing but the distance to @xmath65 with respect to the word metric @xmath220 .    by lemma [ lem : nonatomic ] , we know that @xmath574-almost every pair of points @xmath575 has distinct ends @xmath576 and a boundary value @xmath577 . in this situation , we may connect @xmath578 and  @xmath579 by a unique doubly infinite reduced path @xmath580 and define the strip @xmath581 as follows . it consists of all @xmath88 that are contained in the full @xmath119-preimage of @xmath120 , i.e.  their image @xmath121 is traversed by  @xmath122 , and have the property that the real part @xmath228 has minimal distance to @xmath577 among all real parts @xmath582 with @xmath484 , see the left - hand side of figure [ fig : strips ] . to all remaining pairs we assign the whole of @xmath38 as a strip . this way , the map @xmath311 becomes measurable and @xmath38-equivariant . by lemma [ lem : support ] , a random strip contains the identity element @xmath198 with positive probability , i.e.  the map @xmath311 satisfies the inequality of remark [ rem : removeg ] . so , it suffices to verify the following convergence for an arbitrary pair @xmath575 with distinct ends @xmath576 and @xmath577 ,     ( left ) and @xmath253 ( right ) . ]    ( 0,0 ) ( -286,-7)@xmath577    @xmath583{~n\to\infty~}\,0\,.\ ] ]    but the strip @xmath581 intersects the gauge @xmath584 in at most @xmath585 many cosets of the form @xmath60 , and each of them contains at most two elements of the strip . therefore , @xmath586{~n\to\infty~}\,0\,.\ ] ] in the final step of the above calculation , we use again that the increments @xmath231 have finite first moment . indeed , @xmath587{~n\to\infty~}\,\mathbb{e}(\operatorname{\textit{d}_{s}}(1,x_{1}))\,,\ ] ] from where we may first conclude that the sequence @xmath588 is a.s .  bounded and second that the sequence @xmath589 converges a.s . so , we can finally apply the strip criterion and obtain that @xmath568 is isomorphic to the poisson  furstenberg boundary . vice versa , if the vertical drift is positive , i.e.  @xmath251 , the same argument yields that @xmath567 is isomorphic to the poisson  furstenberg boundary . it remains to consider the driftless case , i.e.  @xmath253 . then , both @xmath10 and @xmath308 are driftless and there is no natural candidate for a real number that determines the horizontal position of the strip . but the fact that the projections @xmath349 have sublinear speed allows us to solve this issue . more precisely , take the @xmath10-boundary @xmath567 and the @xmath308-boundary @xmath570 . now , define gauges @xmath590 again , we know that @xmath591-almost every pair of points @xmath592 has distinct ends @xmath576 , which we may connect by a unique doubly infinite reduced path @xmath580 . let @xmath593 be the full @xmath119-preimage of @xmath120 , i.e.  the set of all group elements @xmath88 such that the image @xmath121 is traversed by @xmath122 , see the right - hand side of figure [ fig : strips ] . again , to all remaining pairs we assign the whole of @xmath38 as a strip . this way , the map @xmath311 becomes measurable , @xmath38-equivariant , and satisfies the inequality of remark [ rem : removeg ] . now , pick an arbitrary pair @xmath592 with distinct ends @xmath576 . we claim that @xmath594{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}}+\underbrace{\frac{|z_{n}|+2}{n}}_{{\tikz[baseline=-3.8pt]{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 2};}}}\,.\ ] ] indeed , the inequality holds for a similar reason as above ; the strip @xmath593 intersects the gauge @xmath584 in at most @xmath595 many cosets of the form @xmath60 . slightly more involved is the observation that each of them contains at most @xmath596 many elements of the gauge . fix a coset @xmath51 . the projections @xmath139 of the elements @xmath484 are located on the horizontal line @xmath597 with imaginary part @xmath598 . one necessary condition for such an element @xmath484 to be contained in the gauge @xmath584 is that the projection @xmath139 is contained in the closed disc @xmath599 . if @xmath600 is empty , then the coset  @xmath51 does not contain any element of the gauge and we are done otherwise , there is a unique @xmath100 with @xmath601 such that @xmath600 is the horizontal line between @xmath602 and @xmath603 , see figure [ fig : shortline ] . the projections @xmath139 with @xmath484 have the property that the real parts @xmath582 and @xmath604 differ exactly by @xmath148 . so , the horizontal line @xmath600 contains at most @xmath605 many of them . let us now estimate @xmath605 in terms of  @xmath606 . since @xmath607 and @xmath608 are both contained in @xmath609 , their distance is at most @xmath610 . therefore , @xmath611 and , in particular , @xmath612 so , the coset @xmath51 contains strictly fewer than @xmath596 elements of the gauge . we will now show that both summands and converge a.s .  to @xmath354 , which will complete the proof . let us first observe that @xmath613 concerning , we deduce from ( [ eqn : gaugefunctionestimate ] ) and the proof of proposition [ prop : discretehyperbolicplane ] that @xmath614 , and finally obtain by the same argument as above @xmath416{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 1};}}=\frac{\ln\left(2\cdot|z_{n}|^{2}+1\right)}{n}\leq\frac{\ln\left(2\cdot(\max{\ensuremath{\{\ , \ell_{a},\ell_{b},1 \,\}}}\cdot\operatorname{\textit{d}_{s}}(1,z_{n})+1)^{2}+1\right)}{n}\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,0\,.\ ] ] on the other hand , concerning , we apply ( [ eqn : gaugefunctionestimate ] ) and lemma [ lem : zerodrifthyp ] to obtain @xmath416{\node[shape = circle , draw , inner sep=0.5pt ] ( char ) { \scriptsize 2};}}=\frac{|z_{n}|+2}{n}\leq\frac{\max\left\{\,\operatorname{\textit{d}_{\mathbb{h}}}(\pi_{\mathbb{h}}(1),\pi_{\mathbb{h}}(z_{n})),\sqrt{\operatorname{\textit{d}_{s}}(1,z_{n})}\,\right\}+3}{n}\,\xrightarrow[\text{\raisebox{.5ex}{a.\,s.}}]{~n\to\infty~}\,0\,.\ ] ]    , the closed disc @xmath609 , and their intersection @xmath600 . ] ( 0,0 ) ( -114,33)@xmath615 ( -77,58)@xmath148 ( -2,65)@xmath401 ( -89,25)@xmath609 ( -195,86)@xmath616 ( -97,86)@xmath617 recall from section [ sub : aobsg ] that a baumslag solitar group @xmath2 is non - amenable if and only if neither @xmath5 nor @xmath6 . until now , we have only identified the poisson  furstenberg boundary for random walks on non - amenable baumslag solitar groups @xmath2 with @xmath20 . replacing one of the two generators by its inverse , it is easy to see that @xmath618 so , in order to investigate the remaining non - amenable baumslag  solitar groups , the only cases that we have to consider are @xmath86 and @xmath619 . in this appendix , we shall review our methods from the previous sections and explain how to adjust the arguments in order to obtain similar results for these cases . let us first assume that @xmath225 with @xmath86 . in order to define the projection @xmath87 back in section [ sub : ptthp ] , we considered the map @xmath77 given by @xmath78 and @xmath79 , and extended it to a homomorphism . now , we are assuming that @xmath86 , in which case the transformation @xmath620 is not orientation preserving any more . if we replaced @xmath1 by @xmath621 in the definition , then @xmath622 would be orientation preserving but @xmath623 , whence we could not apply von dyck s theorem any more . so , we have to change the approach . let @xmath624 be the set of all maps @xmath625 either of the form @xmath626 or of the form @xmath627 with @xmath74 and @xmath75 . this set endowed with the composition forms again a group . consider the map @xmath628 given by @xmath629 and @xmath630 . with this map , it is possible to apply von dyck s theorem and to extend it uniquely to a group homomorphism @xmath631 . finally , as in the case of @xmath71 , every @xmath632 can be thought of as an isometry of @xmath14 . so , we may again consider the projection @xmath87 given by @xmath633 . the following lemma illustrates this definition . [ lem : lookingfromfrontnew ] for every @xmath88 the point @xmath91 is above the point @xmath92 ; the two points have the same real part and their distance is @xmath634 . but for every @xmath88 the point @xmath94 is either right or left from the point @xmath92 depending on whether the level @xmath234 is even or odd ; in any case , the two points have the same imaginary part and their distance is @xmath635 . the proof is similar to the one of lemma [ lem : lookingfromfront ] . so , we only discuss the differences . let us consider the two points @xmath99 and @xmath636 . if @xmath637 is of the form @xmath638 , then it is again the composition of a dilation @xmath639 and a translation @xmath104 , whence the relative position of the two points is preserved . on the other hand , if @xmath637 is of the form @xmath640 , then it is the composition of a reflection at the imaginary axis @xmath641 , a dilation @xmath639 , and a translation @xmath104 , in which case the relative position of the two points is still preserved with the exception that right and left are switched .    in order to decide whether @xmath637 is of the first or the second form , we can write the element @xmath88 as a product over @xmath642 and @xmath643 . since @xmath631 is a homomorphism , the image @xmath637 can be written as the respective product over @xmath644 and @xmath645 . but each occurrence of @xmath644 yields one reflection . so , the image @xmath637 is of the first form if and only if the number of occurrences of @xmath644 is even , which is the case if and only if @xmath234 is even .    using this projection @xmath87 and , of course , replacing @xmath1 by @xmath621 wherever it is necessary , we can repeat the arguments from the previous sections . the definition of the tree @xmath13 and the level functions @xmath58 and @xmath63 , including lemma [ lem : neighbours ] , as well as the definition of the discrete hyperbolic plane @xmath128 , including proposition [ prop : discretehyperbolicplane ] , can be adapted . recall that , in section [ sub : rpipavd ] , we considered the imaginary and real parts of @xmath92 separately , and introduced the shorthand notation @xmath347 and @xmath348 . let us highlight that we have @xmath646 , which allows us to adapt the proof of lemma [ lem : assumptions ] .    in order to identify the poisson  furstenberg boundary geometrically , we have to ensure convergence to the boundaries @xmath16 and @xmath15 . let us first consider the boundary @xmath16 . the proof of lemma [ lem : positivedrifthyp ] for @xmath251 can be adapted . the proof of lemma [ lem : negativedrifthyp ] for @xmath252 , in turn , deserves a bit of work . we have to show that the real parts @xmath355 converge a.s .  to a random element @xmath183 . in the original proof , we observed that @xmath647 and @xmath648 with @xmath360 . while the first formula remains true , the second one does not . we are now in a situation where not only the scaling but also the direction of the next horizontal increment depends on the current level . however , instead of the above , we obtain that @xmath649 with @xmath650 if @xmath234 is even and @xmath651 if @xmath234 is odd . this observation allows us to apply cauchy s root test precisely as in the proof of lemma [ lem : negativedrifthyp ] . for the same reason , namely because all the estimates are not in terms of the actual horizontal increments but of their absolute values , the proofs of lemmas [ lem : zerodrifthyp ] and [ lem : elie ] for @xmath253 can be adapted . the same holds , concerning the boundary @xmath15 , for the proofs of lemma [ lem : nonzerodrifttree ] for @xmath480 and lemma [ lem : zerodrifttree ] for @xmath253 . from these observations , we may deduce the following results .      1 . if the vertical drift is * positive * , i.e.  @xmath251 , then the projections @xmath349 converge a.s .  to @xmath85 and the projections @xmath426 converge a.s .  to a random element @xmath434 . 2 .   if the vertical drift is * negative * , i.e.  @xmath252 , then the projections @xmath349 converge a.s .  to a random element @xmath183 and the projections @xmath426 converge a.s .  to a random element @xmath434 . 3 .   if there is * no vertical drift * , i.e.  @xmath253 , and , in addition , @xmath232 has finite second moment and there is an @xmath152 such that @xmath233 has finite @xmath372-th moment , which is certainly the case if the increment @xmath231 has finite @xmath372-th moment , then the projections @xmath349 have sublinear speed and the projections @xmath426 converge a.s .  to a random element @xmath434 .      1 . if the vertical drift is * positive * , i.e.  @xmath251 , then the poisson  furstenberg boundary is isomorphic to @xmath567 endowed with the boundary map @xmath496 . if the vertical drift is * negative * , i.e.  @xmath252 , then the poisson  furstenberg boundary is isomorphic to @xmath568 endowed with the boundary map @xmath501 . 3 .   if there is * no vertical drift * , i.e.  @xmath253 , and , in addition , @xmath232 has finite second moment and there is an @xmath152 such that @xmath233 has finite @xmath372-th moment , which is certainly the case if the increment @xmath231 has finite @xmath372-th moment , then the poisson  furstenberg boundary is again isomorphic to @xmath567 endowed with the boundary map @xmath496 . let us now assume that @xmath225 with @xmath619 . this situation differs fundamentally from the ones discussed so far because the bricks introduced in the proof of proposition  [ prop : discretehyperbolicplane ] would now have equally many @xmath14-vertices on their upper and lower level . therefore , we use the euclidean plane @xmath652 instead of the hyperbolic plane @xmath14 . in order to construct a projection @xmath653 , let @xmath654 and consider the map @xmath628 given by @xmath655 ( ( x , y)\mapsto(-x , y+1))&\text{if}~q<0 \end{array}\right.\quad\text{and}\quad\pi_{{\ensuremath{m}}}(b):=((x , y)\mapsto(x+1,y))\,.\ ] ] in both cases , i.e.  @xmath656 and @xmath657 , it is possible to apply von dyck s theorem and to extend the map uniquely to a group homomorphism @xmath658 . now , we may consider the projection @xmath653 given by @xmath659 . the definition of the tree @xmath13 and the level functions @xmath58 and @xmath63 , including lemma [ lem : neighbours ] , remain the same . but instead of the discrete hyperbolic plane , we now obtain a discrete euclidean plane @xmath128 . the proof of proposition [ prop : discretehyperbolicplane ] can be adapted to the new situation and shows that the graph @xmath128 endowed with the graph distance @xmath124 is quasi - isometric , and even bi - lipschitz , to the euclidean plane @xmath652 endowed with the standard metric @xmath660 . we aim to show that , as soon as the projections converge to a random element in @xmath15 , independently of the vertical drift , the poisson  furstenberg boundary is isomorphic to @xmath567 . in particular , we do not need to introduce any boundary to capture the behaviour of the projections @xmath661 . concerning the projections @xmath426 , we distinguish between two cases . if the vertical drift is different from @xmath354 , i.e.  @xmath480 , then the proof of lemma [ lem : nonzerodrifttree ] can be adapted and we obtain that the projections @xmath426 converge a.s .  to a random end @xmath434 . but if there is no vertical drift , i.e.  @xmath253 , then the proof of lemma [ lem : zerodrifttree ] can not be adapted because it was based on the fact that the projections @xmath349 had sublinear speed ; and the projections @xmath661 do not need to have sublinear speed any more . in this situation , the following lemma may be used instead of lemma [ lem : zerodrifttree ] . let @xmath187 be a random walk on an arbitrary non - amenable baumslag solitar group @xmath225 . if the increment @xmath231 has finite support , then the projections @xmath426 converge a.s .  to a random end @xmath434 . recall from the beginning of section [ sub : treeconvergence ] that the projections @xmath426 do not need to satisfy the markov property . despite of this , we first show that they leave a.s . every finite ball with centre @xmath145 , i.e.  every set of vertices of the form @xmath662 . suppose they did not . then , there is a ball such that the probability to visit this ball infinitely often is strictly positive . now , it is not hard to see that the probability to visit the centre of this ball infinitely often is also strictly positive . in other words , @xmath663 but for every @xmath368 we obtain @xmath664 therefore , @xmath665 . we may read the random sequence @xmath666 with @xmath667 as the left random walk on the left homogeneous space @xmath60 driven by the reflected probability measure @xmath308 . by the above , @xmath401 is recurrent . but since the group @xmath38 is non - amenable and the subgroup @xmath668 is amenable , one can show that the homogeneous space @xmath60 is non - amenable , see @xcite referring to ( * ? ? ? * , exemple 2b ) . therefore , by ( * ? ? ? * satz ii.9 ) , @xmath401 must be transient , which is a contradiction . now , we can repeat the argument given in the proof of lemma  [ lem : nonzerodrifttree ] to show that for almost every trajectory @xmath368 there is an end @xmath669 such that for every @xmath152 the projections @xmath670 visit the open ball @xmath671 infinitely often . we know that for almost every trajectory @xmath368 both the projections @xmath670 leave every finite ball with centre @xmath145 and the graph distance of any two subsequent projections @xmath670 and @xmath672 is bounded by a constant . hence , for every @xmath152 the projections @xmath670 must eventually remain in the open ball @xmath673 . in other words , they converge to @xmath669 . now , we can show as in lemmas [ lem : nonatomic ] and [ lem : support ] that the hitting measure @xmath674 is again non - atomic and has full support . moreover , we obtain the following version of the identification theorem .      1 . if the * vertical drift * is different from @xmath354 , i.e.  @xmath480 , then the poisson  furstenberg boundary is isomorphic to @xmath567 endowed with the boundary map @xmath496 . if there is * no vertical drift * , i.e.  @xmath253 , and the increment @xmath231 has not just finite first moment but also finite support , then the poisson  furstenberg boundary is again isomorphic to @xmath567 endowed with the boundary map @xmath496 . again , we apply the strip criterion . as in the original proof of the identification theorem , we take the @xmath10-boundary @xmath567 and the @xmath308-boundary @xmath570 . next , we define gauges @xmath675 we know that @xmath591-almost every pair of points @xmath592 has distinct ends @xmath576 , which we may connect by a unique doubly infinite reduced path @xmath580 . let @xmath593 be the full @xmath119-preimage of @xmath120 . to all remaining pairs we assign the whole of @xmath38 as a strip . this way , the map @xmath311 becomes measurable , @xmath38-equivariant , and satisfies the inequality of remark [ rem : removeg ] . now , pick an arbitrary pair @xmath592 with distinct ends @xmath576 . we claim that @xmath676 indeed , the inequality is easy to see . the strip @xmath593 intersects the gauge @xmath584 in at most @xmath585 many cosets of the form @xmath60 , and each of them contains at most @xmath585 many elements of the gauge . now , it suffices to consider the standard generating set @xmath238 and to observe that @xmath677 . then , using the fact that @xmath588 is a.s . bounded , we may conclude that @xmath678{~n\to\infty~}\,0\,,\ ] ] which allows us to apply the strip criterion . to4em , _ poisson  furstenberg boundaries , large - scale geometry and growth of groups _ , proceedings of the international congress of mathematicians , vol . ii , hindustan book agency , new delhi , 2010 , pp .  681704 . to4em , _ boundary theory and stochastic processes on homogeneous spaces _ , harmonic analysis on homogeneous spaces ( proc . pure math . xxvi , williams coll . , williamstown , mass . , 1972 ) , soc . , providence , r.i . , 1973 , pp . 193229 . to4em , _ boundaries of invariant markov operators : the identification problem _ , ergodic theory of @xmath680 actions ( warwick , 19931994 ) , london math . soc . lecture note ser . 228 , cambridge univ . press , cambridge , 1996 , pp .
the aim of this paper is to give a description of the poisson  furstenberg boundary of random walks on non - amenable baumslag  solitar groups . after a short introduction to baumslag 
 solitar groups and their geometry , we change our focus to random walks on these groups . 
 the poisson  furstenberg boundary is a probabilistic model for the long - time behaviour of random walks . for random walks on non - amenable baumslag 
 solitar groups we identify the poisson  furstenberg boundary in terms of the boundary of the hyperbolic plane and the space of ends of the associated bass  serre tree using kaimanovich s strip criterion . 
 the precise statement can be found in theorem [ thm : identification ] on page . 
 * keywords : * random walk , poisson  furstenberg boundary , strip criterion , baumslag  solitar group , compactification . 
 * msc 2010 classes : * 60j50 ( primary ) , 60g50 , 20f65 , 50c63 .