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the notion of co - hopficity plays an important role in group theory . recall that a group @xmath0 is said to be _ co - hopf _ if @xmath0 is not isomorphic to a proper subgroup of itself , that is , if every injective homomorphism @xmath2 is surjective . a group @xmath0 is _ almost co - hopf _ if for every injective homomorphism @xmath3 we have @xmath4<\infty$ ] . clearly , being co - hopf implies being almost co - hopf . the converse is not true : for example , for any @xmath5 the free abelian group @xmath6 is almost co - hopf but not co - hopf . it is easy to see that any freely decomposable group is not co - hopf . in particular , a free group of rank at least 2 is not co - hopf . it is also well - known that finitely generated nilpotent groups are always almost co - hopf and , under some additional restrictions , also co - hopf  @xcite . an important result of sela  @xcite states that a torsion - free non - elementary word - hyperbolic group @xmath0 is co - hopf if and only if @xmath0 is freely indecomposable . partial generalizations of this result are known for certain classes of relatively hyperbolic groups , by the work of belegradek and szczepaski  @xcite . co - hopficity has also been extensively studied for 3-manifold groups and for kleinian groups . delzant and potyagailo  @xcite gave a complete characterization of co - hopfian groups among non - elementary geometrically finite kleinian groups without 2-torsion . a counterpart agebraic notion is that of hopficity . a group @xmath0 is said to be _ hopfian _ if every surjective endomorphism @xmath2 is necessarily injective , and hence is an automorphism of @xmath0 . this notion is also extensively studied in geometric group theory . in particular , an important result of sela @xcite shows that every torsion - free word - hyperbolic group is hopfian . the notion of hopficity admits a number of interesting `` virtual '' variations . thus a group @xmath0 is called _ cofinitely hopfian _ if every endomorphism of @xmath0 whose image is of finite index in @xmath0 , is an automorphism of @xmath0 , see , for example  @xcite .    a key general theme in geometric group theory is the study of `` large - scale '' geometric properties of finitely generated groups . recall that if @xmath7 and @xmath8 are metric spaces , a map @xmath9 is called a _ coarse embedding _ if there exist monotone non - decreasing functions @xmath10 such that @xmath11 , that @xmath12 , and such that for all @xmath13 we have @xmath14 if @xmath15 is a path metric , then for any coarse embedding @xmath9 the function @xmath16 can be chosen to be affine , that is , of the form @xmath17 for some @xmath18 . a coarse map @xmath19 is called a _ coarse equivalence _ if @xmath19 is _ coarsely surjective _ , that is , if there is @xmath20 such that for every @xmath21 there exists @xmath22 with @xmath23 . a map @xmath9 is called a _ quasi - isometric embedding _ if @xmath19 is a coarse embedding and the functions @xmath24 in @xmath25 can be chosen to be affine , that is , of the form @xmath26 , @xmath27 where @xmath28 , @xmath29 . finally , a map @xmath9 is a _ quasi - isometry _ if @xmath19 is a quasi - isometric embedding and @xmath19 is coarsely surjective . the notion of co - hopficity has the following natural counterpart for metric spaces . we say that a metric space @xmath30 is _ quasi - isometrically co - hopf _ if every quasi - isometric embedding @xmath31 is coarsely surjective , that is , if every quasi - isometric embedding @xmath31 is a quasi - isometry . more generally , a metric space @xmath30 is called _ coarsely co - hopf _ if every coarse embedding @xmath31 is coarsely surjective . clearly , if @xmath30 is coarsely co - hopf then @xmath30 is quasi - isometrically co - hopf . if @xmath0 is a finitely generated group with a word metric @xmath32 corresponding to some finite generating set of @xmath0 , then every injective homomorphism @xmath2 is a coarse embedding . this easily implies that if @xmath33 is coarsely co - hopf then the group @xmath0 is almost co - hopf .    the real line @xmath34 is coarsely co - hopf ( and hence quasi - isometrically co - hopf ) . this follows from the fact that any coarse embedding must send the ends of @xmath34 to distinct ends . since @xmath34 has two ends , a coarse embedding induces a bijection on the set of ends of @xmath34 . it is then not hard to see that a coarse embedding from @xmath34 to @xmath34 must be coarsely surjective . see @xcite for the formal definition of ends of a metric space . [ ex : rooted ] the rooted regular binary tree @xmath35 is not quasi - isometrically co - hopf . we can identify the set of vertices of @xmath35 with the set of all finite binary sequences . the root of @xmath35 is the empty binary sequence @xmath36 and for a finite binary sequence @xmath37 its left child is the sequence @xmath38 and the right child is the sequence @xmath39 . consider the map @xmath40 which maps @xmath35 isometrically to a copy of itself that `` hangs below '' the vertex @xmath41 . thus @xmath42 for every finite binary sequence @xmath37 . then @xmath19 is an isometric embedding but the image @xmath43 is not co - bounded in @xmath35 since it misses the entire infinite branch located below the vertex @xmath44 . [ ex : free ]    consider the free group @xmath45 on two generators . then @xmath46 is not quasi - isometrically co - hopf . the cayley graph @xmath30 of @xmath46 is a regular @xmath47-valent tree with every edge of length @xmath44 . we may view @xmath30 in the plane so that every vertex has one edge directed upward , and three downward . picking a vertex @xmath48 of @xmath30 , denote its left branch by @xmath49 and the remainder of the tree by @xmath50 . we have @xmath51 , and @xmath49 is a rooted ternary tree . define a quasi - isometric embedding @xmath52 by taking @xmath19 to be a shift on @xmath49 ( defined similarly to example [ ex : rooted ] ) and the identity on @xmath50 . the map @xmath19 is not coarsely surjective , but it is a quasi - isometric embedding . moreover , for any vertices @xmath53 of @xmath30 we have @xmath54 . one can also see that @xmath45 is not quasi - isometrically co - hopf for algebraic reasons . let @xmath55 with @xmath56\ne 1 $ ] . then there is an injective homomorphism @xmath57 such that @xmath58 and @xmath59 . this homomorphism @xmath19 is always a quasi - isometric embedding of @xmath60 into itself .    if , in addition , @xmath61 and @xmath62 are chosen so that @xmath63 then @xmath64=\infty$ ] and the image @xmath65 is not co - bounded in @xmath60 . thus , the group @xmath46 is not almost co - hopf and not quasi - isometrically co - hopf . there do exist finitely generated groups that are algebraically co - hopf but not quasi - isometrically co - hopf . the simplest example of this kind is the solvable baumslag - solitar group @xmath66 . it is well - known that @xmath67 is co - hopf .    to see that @xmath67 is not quasi - isometrically co - hopf we use the fact that @xmath67 admits an isometric properly discontinuous co - compact action on a proper geodesic metric space @xmath30 that is `` foliated '' by copies of the hyperbolic plane @xmath68 . we refer the reader to the paper of farb and mosher  @xcite for a detailed description of the space @xmath30 , and will only briefly recall the properties of @xmath30 here . topologically , @xmath30 is homeomorphic to the product @xmath69 where @xmath70 is an infinite 3-regular tree ( drawn upwards ) : there is a natural projection @xmath71 whose fibers are homeomorphic to @xmath72 . the boundary of @xmath70 is decomposed into two sets : the `` lower boundary '' consisting of a single point @xmath61 and the `` upper boundary '' @xmath73 which is homeomorphic to the cantor set ( and can be identified with the set of dyadic rationals ) . for any bi - infinite geodesic @xmath74 in @xmath70 from @xmath61 to a point of @xmath73 the full-@xmath75-preimage of @xmath74 in @xmath30 is a copy of the hyperbolic plane @xmath68 ( in the upper - half plane model ) . the @xmath75-preimage of any vertex of @xmath70 is a horizontal horocycle in the @xmath68-``fibers '' . any two @xmath76-fibers intersect along a complement of a horoball in @xmath68 . similar to the above example for @xmath60 , we can take a quasi - isometric embedding @xmath77 whose image misses an infinite subtree in @xmath70 and such that @xmath78 for any vertices @xmath53 of @xmath70 . it is not hard to see that this map @xmath19 can be extended along the @xmath75-fibers to a map @xmath79 such that @xmath80 is a quasi - isometric embedding but not coarsely surjective . since @xmath30 is quasi - isometric to @xmath67 , it follows that @xmath67 is not quasi - isometrically co - hopf . grigorchuk s group @xmath0 of intermediate growth provides another intersting example of a group that is not quasi - isometrically co - hopf . this group @xmath0 is finitely generated and can be realized as a group of automorphisms of the regular binary rooted tree @xmath35 . the group @xmath0 has a number of unusual algebraic properties : it is an infinite 2-torsion group , it has intermediate growth , it is amenable but not elementary amenable and so on . viii in @xcite for detailed background on the grigorchuk group . it is known that there exists a subgroup @xmath81 of index 16 in @xmath0 such that @xmath82 is isomorphic to a subgroup of index 64 in @xmath0 . the map @xmath83 , @xmath84 is clearly a quasi - isometric embedding which is not coarsely surjective . since both @xmath81 and @xmath82 are quasi - isometric to @xmath0 , it follows that @xmath0 is not quasi - isometrically co - hopf .    for gromov - hyperbolic groups and spaces quasi - isometric co - hopficity is closely related to the properties of their hyperbolic boundaries . we say that a compact metric space @xmath81 is _ topologically co - hopf _ if @xmath81 is not homeomorphic to a proper subset of itself . we say that @xmath81 is _ quasi - symmetrically co - hopf _ if every quasi - symmetric map @xmath85 is surjective . note that for a compact metric space @xmath81 being topologically co - hopf obviously implies being quasi - symmetrically co - hopf . [ ex : merenkov ] a recent important result of merenkov  @xcite shows that the converse implication does not hold . he constructed a round sierpinski carpet @xmath86 such that @xmath86 is quasi - symmetrically co - hopf . since @xmath86 is homeomorphic to the standard `` square '' serpinski carpet , clearly @xmath86 is not topologically co - hopf . it is well - known ( see , for example , @xcite ) that if @xmath87 are proper gromov - hyperbolic geodesic metric spaces , then any quasi - isometric embedding @xmath88 induces a quasi - symmetric topological embedding @xmath89 between their hyperbolic boundaries . it is then not hard to see that if @xmath0 is a word - hyperbolic group whose hyperbolic boundary @xmath90 is quasi - symmetrically co - hopf ( e.g. if it is topologically co - hopf ) , then @xmath0 is quasi - isometrically co - hopf . this applies , for example , to any word - hyperbolic groups whose boundary @xmath90 is homeomorphic to an @xmath91-sphere ( with @xmath5 ) , such as fundamental groups of closed riemannian manifolds with all sectional curvatures @xmath92 . the main result of this paper is the following :    [ thm : a ] let @xmath0 be a non - uniform lattice in a rank - one semi - simple real lie group other than @xmath93 . then @xmath0 is quasi - isometrically co - hopf . thus , for example , if @xmath94 is a complete finite volume non - compact hyperbolic manifold of dimension @xmath95 then @xmath96 is quasi - isometrically co - hopf . note that if @xmath0 is a non - uniform lattice in @xmath93 then the conclusion of theorem  [ thm : a ] does not hold since @xmath0 is a virtually free group . if @xmath0 is a uniform lattice in a rank - one semi - simple real lie group ( including possibly a lattice in @xmath93 ) then @xmath0 is gromov - hyperbolic with the boundary @xmath90 being homeomorphic to @xmath97 ( for some @xmath5 ) . in this case it is easy to see that @xmath0 is also quasi - isometrically co - hopf since every topological embedding from @xmath97 to itself is necessarily surjective . [ conv : x ] from now on and for the remainder of this paper let @xmath98 be a rank - one negatively curved symmetric space with metric @xmath15 ( or just @xmath99 in most cases ) . namely , @xmath30 is isometric to a hyperbolic space @xmath100 ( with @xmath95 ) , @xmath101 ( with @xmath102 ) , @xmath103 over the reals , complexes , or quaternions , or to the octonionic plane @xmath104 . if @xmath0 is as in theorem  [ thm : a ] , then @xmath0 acts properly discontinuously ( but with a non - compact quotient ) by isometries on such a space @xmath30 and there exists a @xmath0-invariant collection @xmath105 of disjoint horoballs in @xmath30 such that @xmath106 is compact . the `` truncated '' space @xmath107 , endowed with the induced path - metric @xmath108 is quasi - isometric to the group @xmath0 by the milnor - schwartz lemma . thus it suffices to prove that @xmath109 is quasi - isometrically co - hopf . richard schwartz  @xcite established quasi - isometric rigidity for non - uniform lattices in rank - one semi - simple lie groups and we use his proof as a starting point . first , using coarse cohomological methods ( particularly techniques of kapovich - kleiner  @xcite ) , we prove that spaces homeomorphic to @xmath110 with `` reasonably nice '' metrics are coarsely co - hopf . this result applies to the euclidean space @xmath110 itself , to simply connected nilpotent lie groups , to the rank - one symmetric spaces @xmath30 mentioned above , as well as to the horospheres in @xmath30 . let @xmath111 be a quasi - isometric embedding . schwartz work implies that for every peripheral horosphere @xmath112 in @xmath113 there exists a unique peripheral horosphere @xmath114 of @xmath30 such that @xmath115 is contained in a bounded neighborhood of @xmath114 . using coarse co - hopficity of horospheres , mentioned above , we conclude that @xmath19 gives a quasi - isometry ( with controlled constants ) between @xmath112 and @xmath114 . then , following schwartz , we extend the map @xmath19 through each peripheral horosphere to the corresponding peripheral horoball @xmath116 in @xmath30 . we then argue that the extended map @xmath117 is a coarse embedding . using coarse co - hopficity of @xmath30 , it follows that @xmath118 is coarsely surjective , which implies that the original map @xmath111 is coarsely surjective as well . it seems likely that the proof of theorem  [ thm : a ] generalizes to an appropriate subclass of relatively hyperbolic groups . however , a more intriguing question is to understand what happens for higher - rank lattices :    let @xmath0 be a non - uniform lattice in a semi - simple real lie group of rank @xmath119 . is @xmath0 quasi - isometrically co - hopf ?    unlike the groups considered in the present paper , higher - rank lattices are not relatively hyperbolic . quasi - isometric rigidity for higher - rank lattices is known to hold , by the result of eskin  @xcite , but the proofs there are quite different from the proof of schwartz in the rank - one case . another natural question is :    let @xmath0 be as in theorem  [ thm : a ] . is @xmath0 coarsely co - hopf ? our proof only yields quasi - isometric co - hopficity , and it is possible that coarse co - hopficity actually fails in this context . the result of merenkov ( example [ ex : merenkov ] ) produces the first example of a compact metric space @xmath81 which is quasi - symmetrically co - hopf but not topologically co - hopf . topologically , @xmath81 is homeomorphic to the standard sierpinkski carpet and there exists a word - hyperbolic group ( in fact a kleinian group ) with boundary homeomorphic to @xmath81 . however , the metric structure on the sierpinski carpet in merenkov s example is not `` group - like '' and is not quasi - symmetric to the visual metric on the boundary of a word - hyperbolic group . does there exist a word - hyperbolic group @xmath0 such that @xmath90 ( with the visual metric ) is quasi - symmetrically co - hopf ( and hence @xmath0 is quasi - isometrically co - hopf ) , but such that @xmath90 is not topologically co - hopf ? in particuar , do there exist examples of this kind where @xmath90 is homeomorphic to the sierpinski carpet or the menger curve ? the above question is particularly interesting for the family of hyperbolic buildings @xmath120 constructed by bourdon and pajot  @xcite . in their examples @xmath121 is homeomorphic to the menger curve , and it turns out to be possible to precisely compute the conformal dimension of @xmath121 . note that , similar to the sierpinski carpet , the menger curve is not topologically co - hopf . are the burdon - pajot buildings @xmath120 quasi - isometrically co - hopf ? equivalently , are their boundaries @xmath121 quasi - symmetrically co - hopf ? it is also interesting to investigate quasi - isometric and coarse co - hopficity for other natural classes of groups and metric spaces . in an ongoing work ( in preparation ) , jason behrstock , alessandro sisto , and harold sultan study quasi - isometric co - hopficity for mapping class groups and also characterize exactly when this property holds for fundamental groups of 3-manifolds . _ acknowledgement : _ the authors would like to thank misha kapovich for useful conversations . recall that , by convention  [ conv : x ] , @xmath30 is a rank one symmetric space different from @xmath68 . namely , @xmath30 is isometric to a hyperbolic space @xmath100 ( with @xmath95 ) , @xmath101 ( with @xmath122 ) , @xmath103 over the reals , complexes , or quaternions , or to the octonionic plane @xmath104 . we recall some properties of @xmath30 . see @xcite , chapter ii.10 , for details . let @xmath123 be a basepoint and @xmath124 a geodesic ray starting at @xmath41 . the associated function @xmath125 given by @xmath126 is known as a busemann function on @xmath30 . a _ horosphere _ is a level set of a busemann function . the set @xmath127 is a _ horoball_. up to the action of the isometry group on @xmath30 , there is a unique busemann function , horosphere , and horoball . a busemann function @xmath128 provides a decomposition of @xmath30 into _ horospherical coordinates _ , a generalization of the upper - halfspace model . namely , let @xmath129 and decompose @xmath130 as follows : given @xmath22 , flow along the gradient of @xmath131 for time @xmath128 to reach a point @xmath132 , and write @xmath133 . in horospherical coordinates , the @xmath112-fibers @xmath134 are geodesics , the @xmath135-fibers @xmath136 are horospheres , and the sets @xmath137 are horoballs . other horoballs appear as closed balls tangent to the boundary @xmath138 .    if @xmath139 is a metric space and @xmath20 , a path @xmath140\to m$ ] , parameterized by arc - length , is called a _ @xmath141-rough geodesic _ in @xmath94 , if for any @xmath142 $ ] we have @xmath143    if @xmath144 are metric spaces , a map @xmath145 is _ coarsely lipschitz _ if there exists @xmath146 such that for any @xmath147 we have @xmath148 . if @xmath149 is a path metric space then it is easy to see that @xmath145 is coarsely lipschitz if and only if there exist constants @xmath150 such that for any @xmath147 with @xmath151 we have @xmath152 . the following two lemmas appear to be well known folklore facts :    [ lemma : horoball1 ] there exists @xmath146 with the following property : let @xmath105 be a horoball in @xmath30 , @xmath153 and @xmath154 . let @xmath131 be the point in @xmath105 closest to @xmath155 . then the piecewise geodesic @xmath156 \cup [ b , x_2]$ ] is a @xmath141-rough geodesic . acting by isometries of @xmath30 , we may assume that @xmath105 is a fixed horoball that is tangent to the boundary of @xmath30 in the horospherical model . we may also assume that @xmath131 is the top - most point of @xmath105 , so that @xmath155 lies in the vertical geodesic passing through @xmath131 . see figure [ fig : horoball1 ] . consider the `` top '' of @xmath105 , i.e.  the maximal subset of @xmath157 that is a graph in horospherical coordinates . considering the riemannian metric on @xmath30 in horospherical coordinates , one sees that the geodesic @xmath158 $ ] must pass through the top of @xmath105 . setting @xmath141 to be the radius of the top of @xmath105 , centered at @xmath131 , completes the proof . [ lemma : horoball2 ] let @xmath159 be disjoint horoballs , and @xmath160 . let @xmath161 $ ] be the minimal geodesic between @xmath162 and @xmath163 . then @xmath164 \cup [ b_1 , b_2 ] \cup [ b_2 , x_2]$ ] is a @xmath141-rough geodesic , for the value of @xmath141 in lemma [ lemma : horoball1 ] .    the proof is analagous to that of lemma [ lemma : horoball1 ] . we may normalize the horoballs @xmath159 as in figure [ fig : horoball2 ] . the normalization depends only on the distance @xmath165 . any geodesic @xmath158 $ ] must then pass through compact regions near @xmath166 and @xmath167 . let @xmath168 be the radius of this region in @xmath162 . fixing @xmath162 and varying @xmath163 , set @xmath169 . the value @xmath168 remains bounded if the distance between the horoballs goes to infinity ( converging to the constant @xmath141 in lemma [ lemma : horoball1 ] ) . thus , the infimum is attained and @xmath170 . this completes the proof . [ lemma : horoballs3 ] let @xmath159 be disjoint horoballs , @xmath171 , @xmath172 . denote the minimal geodesic between @xmath162 and @xmath163 by @xmath161 $ ] . then @xmath173 . fix @xmath174 and allow @xmath175 and @xmath172 to vary with the restriction @xmath176 . define a function @xmath19 on the interval @xmath177 $ ] by @xmath178 where the supremum is over all combinations of the variables with the restriction stated above , and @xmath166 denotes the closest point of @xmath162 to @xmath163 . then @xmath19 is a decreasing function , since increasing @xmath179 pushes the horoballs farther apart and forces @xmath155 closer to @xmath180 . in particular , @xmath181 since necessarily @xmath182 . conversely , @xmath183 , taking @xmath184 . we then have for any choice of disjoint @xmath159 and @xmath185 in the corresponding horoballs , that @xmath186 as desired . let @xmath187 be a negatively curved rank one symmetric space . a _ truncated space _ @xmath113 is the complement in @xmath30 of a set of disjoint open horoballs . a truncated space is _ equivariant _ if there is a ( non - uniform ) lattice @xmath188 that leaves @xmath113 invariant , with @xmath189 compact . we will consider @xmath113 with the induced path metric @xmath108 from @xmath30 . under this metric , curvature remains negative in the interior of @xmath113 . the curvature on the boundary need not be negative . for an extensive treatment of truncated spaces , see @xcite . note that truncated spaces are , in general , not uniquely geodesic . specifically , if @xmath30 is not a real hyperbolic space , then components of @xmath190 ( which come from horospheres in @xmath30 ) are isometrically embedded in @xmath109 copies of non - uniquely - geodesic riemannian metrics on certain nilpotent groups . in particular , @xmath109 is not necessarily a @xmath191-space . let @xmath30 be a negatively curved rank one symmetric space and @xmath188 a non - uniform lattice . then @xmath192 is a finite - volume manifold with cusps . in @xmath30 , each cusp corresponds to a @xmath193-invariant family of horoballs . removing the horoballs produces an equivariant truncated space @xmath113 whose quotient @xmath189 is the compact core of @xmath192 . [ prop : uniform ] let @xmath30 be a negatively curved rank one symmetric space and @xmath194 an equivariant truncated space . then the inclusion @xmath195 is a coarse embedding . since @xmath108 and @xmath15 are path metrics with the same line element , we have @xmath196 to get the lower bound , define an auxilliary function @xmath197 let @xmath81 be a compact fundamental region for the action of @xmath193 on @xmath113 . because @xmath193 acts on @xmath113 by isometries with respect to both metrics @xmath15 and @xmath108 , we may equivalently define @xmath198 by @xmath199 because @xmath81 is compact and the metrics @xmath15 , @xmath108 are complete , @xmath200 for @xmath201 . furthermore , @xmath202 \rightarrow [ 0 , \infty]$ ] is continuous and increasing , with @xmath203 . because horospheres have infinite diameter for both @xmath15 and @xmath108 ( they are isometric to appropriate nilpotent lie groups with left - invariant riemannian metrics , see @xcite ) , we also have @xmath204 . let @xmath205 be an increasing homeomorphism of @xmath206 $ ] with @xmath207 for all @xmath208 and consider its inverse @xmath209 . for @xmath210 we then have @xmath211 this concludes the proof .    a more precise quantitative version of proposition [ prop : uniform ] can be obtained by studying geodesics in @xmath113 , see @xcite . for this section , let @xmath212 be a truncated space , with @xmath213 , and @xmath214 a @xmath108-quasi - isometric embedding . to ease the exposition , we refer to the target truncated space as @xmath215 . [ lemma : schwartz1 ] there exists @xmath146 so that for every boundary horosphere @xmath112 of @xmath113 , there exists a boundary horosphere @xmath114 of @xmath216 such that @xmath115 is contained in a @xmath141-neighborhood of @xmath114 . using nearest - point projection , we may assume @xmath217 . [ defi : extension ] let @xmath218 be horoballs with boundaries @xmath219 . a point in @xmath112 corresponds , in horospherical coordinates , to a geodesic ray in @xmath116 . a map @xmath220 then extends to a map @xmath221 in the obvious fashion .    in view of lemma [ lemma : schwartz1 ] , a @xmath108-quasi - isometric embedding @xmath222 likewise extends to a map @xmath223 by filling the map on each boundary horoball . [ lemma : schwartz2 ] a quasi - isometry @xmath224 induces a quasi - isometry @xmath221 , with uniform control on constants .    one considers the metric on the horospheres of @xmath105 parallel to @xmath112 , or alternately fixes a model horosphere and varies the metric . one then shows that if @xmath19 is a quasi - isometry with respect to one of the horospheres , it is also a quasi - isometry with respect to the horospheres at other horo - heights . one then decomposes the metric on @xmath105 into a sum of the horosphere metric and the standard metric on @xmath34 , in horospherical coordinates . this replacement is coarsely lipschitz , so the extended map is also coarsely lipschitz . taking the inverse of @xmath19 completes the proof . let @xmath30 be a simplicial complex and @xmath225 nested compacts with @xmath226 . compactly supported cohomology @xmath227 is defined by @xmath228    for a compact space @xmath30 , @xmath229 but the two do not generally agree for unbounded spaces . we have @xmath230 and @xmath231 for a non - trivial truncated space @xmath113 . in fact , one has the following lemma . [ lemma : everything ] let @xmath232 be a closed subset . then @xmath233 if and only if @xmath234 . it is well - known that the choice of nested compact sets does not affect @xmath235 . choose the sequence @xmath236 , the intersection of a closed ball and @xmath237 . with respect to the subset topology of @xmath237 , the boundary of @xmath238 is given by @xmath239 . we have by excision @xmath240 note that @xmath241 and @xmath242 , so @xmath243 . thus , if @xmath244 , then @xmath245 . that is , @xmath246 is a compact set in @xmath247 , and @xmath248 . thus , if @xmath234 , we have @xmath249 . otherwise , @xmath250 . compactly supported cohomology is not invariant under quasi - isometries or uniform embeddings . the remainder of this section is distilled from @xcite , where compactly supported cohomology is generalized to a theory invariant under uniform embeddings . for our purposes , the basic ideas of this theory , made explicit below , are sufficient . let @xmath30 be a simplicial complex with the standard metric assigning each edge length @xmath44 . recall that a _ chain _ in @xmath30 is a formal linear combination of simplices . the _ support _ of a chain is the union of the simplices that have non - zero coefficients in the chain . the _ diameter _ of a chain is the diameter of its support . an acyclic metric simplicial complex @xmath30 is @xmath251__-uniformly acyclic _ _ if there exists a function @xmath252 such that any closed chain with diameter @xmath99 is the boundary of a @xmath253-chain of diameter at most @xmath254 . if @xmath30 is @xmath251-uniformly acyclic for all @xmath251 , we say that it is _ uniformly acyclic_.    likewise , we say that a metric simplicial complex @xmath30 is @xmath251__-uniformly contractible _ _ if there exists a function @xmath252 such that every continuous map @xmath255 with image having diameter @xmath99 extends to a map @xmath256 with diameter at most @xmath254 . if @xmath30 is @xmath251-uniformly contractible for all @xmath251 , we say it is _ uniformly contractible_. rank one symmetric spaces and nilpotent lie groups ( with left - invariant riemannian metrics ) are uniformly contractible and uniformly acyclic . [ lemma : approx ] let @xmath30 , @xmath149 be uniformly contractible and geometrically finite metric simplicial complexes and @xmath257 a uniform embedding . then there exists an iterated barycentric subdivision of @xmath30 and @xmath258 depending only on the uniformity constants of @xmath259 and @xmath149 such that @xmath19 is approximated by a continuous simplicial map with additive error of at most @xmath260 . we first approximate @xmath19 by a continuous ( but not simplicial ) map by working on the skeleta of @xmath30 . starting with the @xmath41-skeleton , adjust the image of each vertex by distance at most @xmath44 so that the image of each vertex of @xmath30 is a vertex of @xmath149 . next , assuming inductively that @xmath19 is continuous on each @xmath251-simplex of @xmath30 , we now extend to the @xmath253 skeleton using the uniform contractibility of @xmath149 . since error was bounded on the @xmath251-simplices , it remains bounded on the @xmath253-skeleton . now that @xmath19 has been approximated by a continous map , a standard simplicial approximation theorem replaces @xmath19 by a continuous simplicial map , with bounded error depending only on the geometry of @xmath30 and @xmath149 ( see for example the proof of theorem 2c.1 of @xcite ) . [ lemma : coho ] let @xmath30 and @xmath149 be uniformly acyclic simplicial complexes and @xmath257 a uniform embedding . suppose furthermore that @xmath19 is a continuous simplicial map . then if @xmath261 . we first consturct a left inverse @xmath262 of the map @xmath263 induced by @xmath19 on the chain complex of @xmath30 , up to a chain homotopy @xmath264 . that is , @xmath264 will be a map @xmath265 satisfying , for each @xmath266 , the homotopy condition @xmath267 and furthermore with diameter of @xmath268 controlled uniformly by the diameter of @xmath269 . we start with the @xmath41-skeleton . each vertex @xmath270 is the image of some vertex @xmath271 ( not necessarily unique ) . set @xmath272 , and extend by linearity to @xmath273 . to define @xmath264 , let @xmath62 be an arbitrary vertex in @xmath30 and note that @xmath274 . we have to satisfy @xmath275 . since @xmath30 is acyclic , there exists a 1-chain @xmath276 satisfying this condition . furthermore , note that @xmath277 is , by construction , a vertex such that @xmath278 . since @xmath19 is a uniform embedding , @xmath279 is uniformly bounded above . thus , @xmath276 may be chosen using _ uniform _ acyclicity so that its diameter is also uniformly bounded above . assume next that @xmath262 and @xmath264 are defined for all @xmath280 with uniform control on diameters . let @xmath112 be a @xmath251-simplex in @xmath30 . then @xmath281 is a chain in @xmath30 whose diameter is bounded independently of @xmath112 . then , by uniform acyclicity there is a chain @xmath114 with @xmath282 . we define @xmath283 . as before , we need to link @xmath114 back to @xmath112 . we have @xmath284 by the homotopy condition [ fla : homotopy ] , we further have @xmath285 thus , by bounded acyclicity there is a @xmath253 chain @xmath286 such that @xmath287 as desired . we extend both @xmath262 and @xmath264 by linearity to all of @xmath288 and @xmath289 , respectively .    to conclude the argument , let @xmath81 be a compact subcomplex of @xmath30 and consider the complex @xmath290 . the maps @xmath264 and @xmath291 on @xmath292 induce maps on @xmath293 , and the condition @xmath294 remains true for the induced maps and chains .    because chain - homotopic maps on @xmath295 induce the same maps on homology , we have , for @xmath296 , @xmath297 . conversely , @xmath298 is the identity on cell complexes , so still the identity on homology . thus , @xmath299 . by duality , @xmath300 . taking @xmath238 to be an exhaustion of @xmath30 by compact subcomplexes and taking a direct limit , we conclude that @xmath301 . [ cor : isomorphism ] let @xmath30 and @xmath149 be uniformly acyclic simplicial complexes and @xmath257 a uniform embedding . there exists an @xmath258 depending only on the uniformity constants of @xmath302 , and @xmath149 so that @xmath303 . lemma [ lemma : approx ] approximates @xmath19 by a continuous simplicial map , within uniform additive error . lemma [ lemma : coho ] shows that the resulting approximation induces an isomorphism on compactly supported cohomology . [ thm : coarse ] let @xmath304 be a manifold homeomorphic to @xmath247 , with @xmath15 a path metric that is uniformly acyclic and uniformly contractible . for each pair of non - decreasing functions @xmath305 with @xmath306 and @xmath307 , there exists a @xmath308 such that any @xmath309-coarse embedding @xmath52 is @xmath308-coarsely surjective .    by corollary [ cor : isomorphism ] , there is a uniform @xmath258 such that @xmath310 . by lemma [ lemma : everything ] , @xmath311 . taking @xmath312 completes the proof . let @xmath212 and @xmath215 be equivariant truncated spaces and @xmath313 a quasi - isometric embedding . then @xmath19 is coarsely surjective with respect to the truncated metric @xmath108 .    by lemma [ lemma : schwartz1 ] , we may assume that @xmath19 maps boundary horospheres of @xmath113 to boundary horospheres of @xmath216 . by theorem [ thm : coarse ] , @xmath19 is a surjection up to a constant independent of the boundary horosphere in question . we then have an extension @xmath314 , as in definition [ defi : extension ] .    by lemma [ lemma : schwartz2 ] , for each boundary horoball @xmath105 , the restriction @xmath315 is a quasi - isometry . by assumption , @xmath316 is a @xmath108-quasi - isometry , so @xmath316 is a @xmath99-uniform embedding by proposition [ prop : uniform ] . since @xmath30 is a path metric space , @xmath317 is then coarsely lipschitz on all of @xmath30 .    we now show that @xmath317 is a uniform embedding by establishing a lower bound for distances between image points . recall that all distances are measured with respect to @xmath318 unless another metric is explicitly mentioned . let @xmath319 so that @xmath317 is coarsely @xmath320-lipschitz and @xmath321 is coarsely @xmath320-co - lipschitz for every boundary horoball @xmath105 . let @xmath322 be increasing proper functions so that @xmath19 is an @xmath309-uniform embedding . let @xmath323 with @xmath324 . we need to provide a lower bound for @xmath325 in terms of @xmath326 . clearly , the lower bound will go to @xmath327 since @xmath317 is an isometry along vertical geodesics in horoballs . there are four cases to consider ; in all cases we can ignore additive noise by working with sufficiently large @xmath326 and slightly increasing @xmath320 .    1 . let @xmath328 for the same horoball @xmath105 . then @xmath329 . 2 .   let @xmath330 . this case is controlled by the uniform embeddings @xmath331 and @xmath332 ( proposition [ prop : uniform ] ) and the @xmath108-quasiisometry constants of @xmath19 . 3 .   let @xmath333 for a horoball @xmath105 . let @xmath334 be the closest point to @xmath155 . then by lemma [ lemma : horoball1 ] , @xmath156 \cup [ b , x_2]$ ] is a @xmath141-quasi - geodesic for a universal @xmath141 depending only on @xmath30 and @xmath335 ( see also figure [ fig : horoball1 ] ) . we consider two sub - cases : + suppose that @xmath336 . let @xmath337 be the closest point to @xmath338 . then by definition of @xmath131 , we have @xmath339 using lemma [ lemma : horoballs3 ] , we conclude @xmath340 + suppose , instead , that @xmath341 . then we have the estimate @xmath342 . we also have @xmath343 , so @xmath344 . consider now @xmath345 , the closest point to @xmath346 . by lemma [ lemma : horoball1 ] , thus , @xmath348 4 . let @xmath160 be in disjoint horoballs . this case is identical to the previous one , except one uses lemma [ lemma : horoball2 ] rather than [ lemma : horoball1 ] . we have then provided a lower bound for @xmath349 for any pair of points @xmath323 . thus , the extended map @xmath317 is a coarse embedding . by theorem [ thm : coarse ] , @xmath317 is then coarsely surjective . namely , there exists @xmath258 so that @xmath350 ( the neighborhood is taken with respect to @xmath99 ) . let @xmath351 be an arbitrary point . since @xmath317 is coarsely surjective , there exists @xmath352 so that @xmath353 . if @xmath354 , then we have shown that @xmath355 . otherwise , @xmath37 is contained in a horoball associated with @xmath113 . in appropriate horospherical coordinates , the horoball is given by @xmath356 and @xmath37 can be written as @xmath357 , with @xmath358 . likewise , @xmath359 has coordinates @xmath360 , with @xmath361 . furthermore , we have @xmath362 . now , @xmath351 , so it has horospherical coordinates @xmath363 with @xmath364 . it is easy to see that @xmath365 thus , for an arbitrary @xmath351 we have @xmath366 . because @xmath332 is a uniform embedding , this implies that @xmath222 is coarsely surjective . m.  bridson and a.  haefliger , _ metric spaces of non - positive curvature _ , grundlehren der mathematischen wissenschaften [ fundamental principles of mathematical sciences ] , vol . 319 , springer - verlag , berlin , 1999 .
we prove that if @xmath0 is a non - uniform lattice in a rank - one semi - simple lie group @xmath1 then @xmath0 is quasi - isometrically co - hopf . 
 this means that every quasi - isometric embedding @xmath2 is coarsely surjective and thus is a quasi - isometry . 
 [ multiblock footnote omitted ]