arXiv:1701.08350v1 [math.DS] 29 Jan 2017

FURSTENBERG ENTROPY OF INTERSECTIONAL
INVARIANT RANDOM SUBGROUPS
YAIR HARTMAN AND ARIEL YADIN
Abstract. We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free
groups, any a-priori possible entropy value can be realized as the entropy of an
ergodic stationary action. This generalizes results of Bowen.
The stationary actions we construct arise via invariant random subgroups
(IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under
some algebraic conditions, which gives a continuum of entropy values. Our tools
apply for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for
these groups.
For the free group, we construct the IRSs via a geometric construction of
subgroups, by describing their Schreier graphs. The analysis of the entropy
of these spaces is obtained by studying the random walk on the appropriate
Schreier graphs.

1. Introduction
In this paper we study the Furstenberg-entropy realization problem, which we
will describe below. We propose a new generic construction of invariant random
subgroups and we are able to analyse the Furstenberg-entropy via the random
walk and its properties. We apply this construction to free groups and lamplighter groups, generalizing results of Bowen [4] and of the first named author
with Tamuz [16]. In addition, we apply it to the group of finitely supported
infinite permutations and to certain extensions of this group, establishing a full
realization result for this class of groups. Let us introduce the notions and results
precisely.
Let G be a countable discrete group and let µ be a probability measure on
G. We will always assume that µ is generating and of finite entropy; that is, the
support of P
µ generates G as a semigroup and the Shannon entropy of µ is finite,
H(µ) := − g µ(g) log µ(g) < ∞.
2010 Mathematics Subject Classification. Primary .
Y. Hartman is supported by the Fulbright Post-Doctoral Scholar Program. A. Yadin is
supported by the Israel Science Foundation (grant no. 1346/15).
We would like to thank Lewis Bowen and Yair Glasner for helpful discussions.
1

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YAIR HARTMAN AND ARIEL YADIN

Suppose that G acts measurably on a standard probability space (X, ν). This
−1
induces an action on the probability measure ν given
P by gν(·) = ν(g ·). The
action G y (X, ν) is called (G, µ)-stationary if g∈G µ(g)gν = ν. Thus, the
measure ν is not necessarily invariant, but only “invariant on average”. An important quantity associated to a stationary space is the Furstenberg-entropy, given
by
Z
X
dg −1 ν
hµ (X, ν) := −
µ(g)
log
dν(x)
dν
X
g∈G
It is easy to see that hµ (X, ν) ≥ 0 and that equality holds if and only if the
measure ν is invariant (invariant means gν = ν for all g ∈ G). A classical result
of Kaimanovich-Vershik [20] asserts that the Furstenberg-entropy of any (G, µ)stationary space is bounded above by the random walk entropy
1
H(µn ),
n→∞ n
where µn denotes the n-th convolution power of µ. As a first step of classification of
the possible (G, µ)-stationary actions for a given µ, one may consider the following
definitions.
hRW (G, µ) := lim

Definition 1.1. We say that (G, µ) has an entropy gap if there exists some
c > 0 such that whenever hµ (X, ν) < c for an ergodic (G, µ)-stationary action
(X, ν) then hµ (X, ν) = 0. Otherwise we say that (G, µ) has no-gap.
We say that (G, µ) admits a full realization if any number in [0, hRW (G, µ)]
can be realized as the Furstenberg-entropy of some ergodic (G, µ)-stationary space.
Let us remark that choosing only ergodic actions G y (X, ν) is important.
Otherwise the definitions are non-interesting, since by taking convex combinations
one can always realize any value.
The motivation for the gap definition comes from Nevo’s result in [24] showing
that any discrete group with property (T) admits an entropy gap, for any µ. The
converse however is not true (see e.g. Section 7.2 in [7]) and it is interesting to
describe the class of groups (and measures) with no-gap.
Our main result is the following.
Theorem 1.2. Let µ be a finitely supported, generating probability measure on
the free group Fr on 2 ≤ r < ∞ generators. Then (Fr , µ) admits a full realization.
As will be explained below (see Remark 4.9), the restriction to finitely supported measures µ is technical and it seems that the result should hold in a wider
generality. The case where µ is the simple random walk (uniform measure on
the generators and their inverses) was done by Bowen [4]. We elaborate on the
similarity and differences with Bowen’s result below.
Our next result provides a solution for lamplighter groups.

ENTROPY OF INTERSECTIONAL IRSS

3

Theorem 1.3. Let G = ⊕B L o B be a lamplighter group where L and B are some
non trivial countable discrete groups. Let µ be a generating probability measure
with finite entropy on G and denote by µ̄ the projection of µ onto the quotient
B∼
= G/(⊕B L o {e}).
Then, whenever (B, µ̄) is Liouville, (G, µ) admits a full realization.
Here, Liouville is just the property that the random walk entropy is zero. For
example, if the base group B is virtually nilpotent (equivalently, of polynomial
growth in the finitely generated context), then B is Liouville for any measure µ
(see Theorem 2.4 below).
In [16], the first named author and Tamuz prove that a dense set of values can
be realized, under an additional technical condition on the random walk.
In addition, our tools apply also to nilpotent extensions of Sym∗ (X), the group
of finitely supported permutations of an infinite countable set X. For simplicity,
we state here the result for Sym∗ (X) itself, and refer the reader to Section 3.4 for
the precise statement and proof of the general result.
Theorem 1.4. Let µ be a generating probability measure with finite entropy on
Sym∗ (X). Then (Sym∗ (X), µ) admits a full realization.
Following the basic approach of Bowen in [4], our realization results use only a
specific type of stationary actions known as Poisson bundles that are constructed
out of invariant random subgroups. An invariant random subgroup, or IRS,
of G, is random subgroup whose law is invariant to the natural G-action by conjugation. IRSs serve as a stochastic generalization of a normal subgroup and arise
naturally as the stabilizers of probability measure preserving actions. In fact, any
IRS is obtained in that way (see Proposition 14 in [3] for the discrete case, and [1]
for the general case). Since the term IRS was coined in [2], they have featured in
many works and inspired much research.
In this paper we construct explicit families of IRSs for the free group, the lamplighter group and Sym∗ (X). For further discussion about the structure of IRSs on
these groups the reader is referred to the works of Bowen [5] for the free group,
Bowen, Grigorchuk and Kravchenko [6] for lamplighter groups and Vershik [28]
for the Sym∗ (X) case.
Given an ergodic IRS of a group G, one can construct an ergodic stationary
space which is called a Poisson bundle. These were introduced by Kaimanovich
in [19] and were further studied by Bowen in [4]. In particular, it was shown in
[4] that the Furstenberg-entropy of a Poisson bundle constructed using an IRS,
equals the random walk entropy of the random walk on the associated coset space
(this will be made precise below). Hence our main results can be interpreted as
realization results for the random walk entropies of coset spaces associated with
ergodic IRSs.
Given a subgroup with an infinite conjugacy class, we develop a general tool
to construct a family of ergodic IRSs, that we call intersectional IRSs. This

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YAIR HARTMAN AND ARIEL YADIN

family of IRSs is obtained by randomly intersecting subgroups from the conjugacy
class (see Section 3). Furthermore, we prove that under a certain condition related to the measure µ and the conjugacy class, the random walk entropy varies
continuously along this family of IRSs (see Section 3.2). In fact, in some cases we
may find a specific conjugacy class that satisfies the above mentioned condition
for any measure µ. An example of such, is a conjugacy class that we call “locally
co-nilpotent”. These are conjugacy classes such that that their associated Schreier
graphs satisfy some geometric condition which we describe in Section 4.5.
Using such families of IRSs we get our main tool for constructing a continuum
of entropy values.
Proposition 1.5. Let G be a countable discrete group. Assume that there exists a
conjugacy class of subgroups which is locally co-nilpotent. (See just before Corollary
3.6 and Section 4.5 for the precise definition of locally co-nilpotent.) Denote its
normal core by N C G.
Then, for any finite entropy, generating, probability measure µ on G, if G/N
has positive µ-random walk entropy, then (G, µ) has no-gap.
Furthermore, if the normal core N is trivial, then (G, µ) admits a full realization.
Note that this proposition reveals a structural condition on G and its subgroups
that allows to conclude realization results for many different measures µ at the
same time. Furthermore, these realization results do not depend on the description
of the Furstenberg-Poisson boundary.
By constructing conjugacy classes in lamplighter groups and in Sym∗ (X) with
the relevant properties, we obtain Theorems 1.3 and 1.4. However, the case of the
free group is more complicated. In Section 4.3 we show how to construct many
subgroups of the free group with a conjugacy class which is locally co-nilpotent,
and such that G/N has positive random walk entropy where N is the normal core
of the conjugacy class. This is enough to realize an interval of entropy values
around 0, showing in particular, that the free group admits no-gap for any µ with
finite entropy. The fact that the free group admits no-gap for any finite first
moment µ was proved in [16].
Theorem 1.6. Let µ be a finite entropy generating probability measure on the
free group Fr on 2 ≤ r < ∞ generators. Then (Fr , µ) has an interval of the form
[0, c] of realizable values.
However, there is no single (self-normalizing) subgroup of the free group which
has a locally co-nilpotent conjugacy class and a trivial normal core (see Lemma 3.7).
The importance of the requirement that the normal core N is trivial, is to ensure
that the random walk entropy on G/N is equal to the full random walk entropy.
Hence, to prove a full realization for the free group, we approximate this property.
That is, we find a sequence of conjugacy classes which are all locally co-nilpotent,

ENTROPY OF INTERSECTIONAL IRSS

5

and such that their normal cores satisfy that the random walk entropy on the
quotients tends to hRW (G, µ).
Finding such a sequence is quite delicate, and proving it is a bit technical.
The difficulty comes from the fact that entropy is semi-continuous, but in the
“wrong direction” (see Theorem 2.2 and the discussion after). It is for this analysis
that we need to assume that µ is finitely supported, although it is plausible that
this condition can be relaxed. By that we mean that the very same conjugacy
classes we construct in this paper might satisfy that random walk entropy on the
quotients by their normal cores tends to hRW (G, µ), even for infinitely supported
µ as well (perhaps under some further regularity conditions, such as some moment
conditions).
1.1. Intersectional IRSs. We now explain how to construct intersectional IRSs.
Let K ≤ G be a subgroup with an infinite conjugacy class |K G | = ∞ (we
use the notation K g = g −1 Kg for conjugation). We may index this conjugacy
class by the integers: K G = (Kz )z∈Z . It is readily checked that the G-action (by
conjugation) on the conjugacy class K G is translated to an action by permutations
on Z. Namely, for g ∈ G, z ∈ Z, we define g.z to be the unique integer such that
(Kz )g = Kg.z . Thus, G also acts on subsets S ⊂ Z element-wise.
T Given a non-empty subset S ⊂ Z, we may associate to S a subgroup CoreS (K) =
s∈S Ks . If S is chosen to be random, with a law invariant to the G-action, then
the random subgroup CoreS (K) is an IRS. Moreover, it is quite simple to get such
a law: for a parameter p ∈ (0, 1] let S be p-percolation on Z; that is, every element
z ∈ Z belongs to S with probability p independently.
There are few issues with this construction.
• It is not immediately clear what should be associated to the “empty intersection” when S = ∅. In Section 3.1 we give an apropriate definition,
which follows from a construction of an associated “semi-norm” on the
group G. This will turn out to be a normal subgroup of G that we denote
by Core∅ (K).
• Note that p = 1 corresponds to the non-random subgroup CoreG (K), which
is just the normal core of the conjugacy class K G . This is also a normal
subgroup, and CoreG (K) C Core∅ (K).
• It seems very reasonable that as p varies, certain quantities related to the
IRS should vary continuously. In Section 3.2 we provide a condition on
the conjugacy class K G that guarantees that the Furstenberg-entropy is
indeed continuous in p.
After establishing continuity of the Furstenberg-entropy in p, one wishes to
show that the full interval [0, hRW (G, µ)] of possible entropy values is realized.
As mentioned above, in the free group we must use a sequence of subgroups
Kn (or conjugacy classes), such that the random walk entropy of the quotient

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YAIR HARTMAN AND ARIEL YADIN

G/CoreG (Kn ) is large, and push this quotient entropy all the way up to the full
random walk entropy hRW (G, µ).
This leads to a study of CoreG (Kn ), the normal cores of a sequence of subgroups
(Kn )n , with the objective of showing that CoreG (Kn ) are becoming “smaller”
in some sense. As will be discussed below, a naive condition such as having
CoreG (Kn ) converge to the trivial group does not suffice. What is required, is to
approximate the Furstenberg-Poisson boundary of the random walk on the full
group by the boundaries of the quotients. By that we mean that we need to find a
sequence such that one can recover the point in the boundary of the random walk
on the free group, by observing the projections in the boundaries of the quotients
G/CoreG (Kn ).
However, we construct the subgroups Kn geometrically, by specifying their associated Schreier graphs (and not by specifying the algebraic properties of the
subgroup). Hence the structure of CoreG (Kn ) and in particular the FurstenbergPoisson boundary of G/CoreG (Kn ) is somewhat mysterious. A further geometric
argument using the random walk on the aforementioned Schreier graph (specifically transience of this random walk) allows us to “sandwich” the entropy of the
G/CoreG (Kn ) and thus show that these entropies tend to the correct limiting
value. This is done in Section 4.
It may be interesting for some readers to note that in the simple random walk
case considered by Bowen [4], he also shows a full realization by growing intervals
of possible entropy values. However, Bowen obtains the intervals of realization differently than us. In fact, he proves that the entropy is continuous when restricting
to a specific family of subgroups which are “tree-like”. This is a global property
of the Schreier graph that enables him to analyze the random walk entropy of
the simple random walk and to prove a continuity result. Next, within this class
of tree-like subgroups he finds paths of IRSs obtaining realization of intervals of
the form [εn , hRW (G, µ)] for εn → 0 (“top down”). While it is enough for a full
realization for the simple random walk, the condition of being tree-like is quite
restrictive, and difficult to apply to other measures µ on the free group.
In contrast, based on Proposition 1.5, our results provide realization on intervals
of the form [0, hRW (G, µ) − εn ] for εn → 0 (“bottom up”). In order to “push” the
entropy to the top we also use a geometric condition, related to trees. However,
this condition is more “local” in nature, and considers only finite balls in the
Scherier graph. In particular, we show in Section 4.6 that it is easy to construct
many subgroups that satisfy our conditions. Also, it enables us to work with any
finitely supported measure.
1.2. Related results. The first realization result is due to Nevo-Zimmer [25],
where they prove that connected semi-simple Lie groups with finite center and
R-rank at least 2 admit only finitely many entropy values under some mixing
condition on the action. They also show that this fails for lower rank groups,

ENTROPY OF INTERSECTIONAL IRSS

7

such as P SL2 (R), by finding infinitely many entropy values. However, no other
guarantees are given regarding these values.
The breakthrough of the realization results is the mentioned result of Bowen [4],
where he makes a use of IRSs for realization purposes. He proves a full realization
for simple random walks on free groups Fr where 2 ≤ r < ∞.
In [16], the first named author and Tamuz prove a dense realization for lamplighter groups with Liouville base. Furthermore, it is shown that virtually free
groups, such as SL2 (Z), admit no-gap.
In a recent paper [8], Burton, Lupini and Tamuz prove that the Furstenbergentropy is an invariant of weak equivalence of stationary actions.
2. Preliminaries
2.1. Stationary spaces. Throughout the paper we assume that G is a discrete
countable group and that µ is a generating probability measure on G.
A measurable action G y (X,Pν) where (X, ν) is a standard probability spaces
is called (G, µ)-stationary if
g∈G µ(g)gν = ν. The basic example of noninvariant (G, µ)-stationary action is the action on the Furstenberg-Poisson boundary, Π(G, µ) (see e.g. [12–15] and references therein for more on this topic, which
has an extreme body of literature associated to it).
The Kakutani fixed point theorem implies that whenever G acts continuously on
a compact space X, there is always some µ-stationary measure on X. However,
there are not many explicit descriptions of stationary measures. One way to
construct stationary actions is via the Poisson bundle constructed from an IRS:
Denote by SubG the space of all subgroups of G (recall that G is discrete). This
space admits a topology, induced by the product topology on subsets of G, known
as the Chabauty topology. Under this topology, SubG is a compact space, with
a natural right action by conjugation (K g = g −1 Kg where K ∈ SubG ), which
is continuous. An Invariant Random Subgroup (IRS) is a Borel probability
measure λ on SubG which is invariant under this conjugation action. It is useful
to think of an IRS also as a random subgroup with a conjugation invariant law λ.
For more background regarding IRSs we refer the interested reader to [2, 3].
Fix a probability measure µ on G. Given an IRS λ of G, one can consider the
Poisson bundle over λ, denoted by Bµ (λ). This is a probability space with a (G, µ)
stationary action, and when λ is an ergodic IRS, the action on Bµ (λ) is ergodic.
Poisson bundles were introduced in a greater generality by Kaimanovich in
[19]. In [4] Bowen relates Poisson bundles to IRSs. Although the stationary
spaces that we construct in this paper are all Poisson bundles, we do not elaborate
regarding this correspondence, as this will not be important for understanding our
results. Rather, we refer the interested reader to [4, 16, 17, 19], and mention here
the important features of Poisson bundles that are relevant to our context. To
describe these features, we now turn to describe Schreier graphs.

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2.2. Schreier graphs. Since we will use Schreier graphs only in the context of
the free group, we assume in this section that G is a free group. To keep the presentation simple, we consider G = F2 the free group on two generators, generated
by a and b. The generalization to higher rank is straightforward.
Let K ∈ SubG . The Schreier graph associated with K is a rooted oriented
graph, with edges labeled by a and b, which is defined as follows. The set of
vertices is the coset space K\G and the set of edges is {(Kg, Kga), (Kg, Kgb) :
Kg ∈ K\G}. Edges of the type (Kg, Kga) are labeled by a and of the type
(Kg, Kgb) are labeled by b. It may be the case that there are multiple edges
between vertices, as well as self loops. The root of the graph is defined to be the
trivial coset K.
From this point on, it is important that we are working with the free group,
and not a general group which is generated by two elements. Consider an abstract
(connected) rooted graph with oriented edges. Suppose that every vertex has
exactly two incoming edges labeled by a and b, and two outgoing edges labeled by
a and b. We may then consider this graph as a Schreier graph of the free group.
Indeed, given such a graph Γ one can recover the subgroup K: Given any
element g = w1 · · · wn , of the free group (where w1 · · · wn is a reduced word in
{a, a−1 , b, b−1 }) and a vertex v in the graph, one can “apply the element g to the
vertex v” to get another vertex denoted v.g, as follows:
• Given wi ∈ {a, b} move from v to the adjacent vertex u = v.wi such that
the oriented edge (v, u) is labeled by wi .
• Given wi ∈ {a−1 , b−1 } move from v to the adjacent vertex u = v.wi such
that the oriented edge (u, v) is labeled wi−1 (that is, follow the label wi
using the reverse orientation).
• Given g = w1 · · · wn written as a reduced word in the generators, apply the
generators one at a time, to get v.g = (· · · ((v.w1 ).w2 ) . . .).wn .
Thus, we arrive at a right action of G on the vertex set. Note that this action is
not by graph automorphisms, rather only by permutations of the vertex set. We
say that g preserves v if v.g = v.
To recover the subgroup K ≤ F2 from a graph Γ as above, let K be the set of all
elements of the free group that preserve the root vertex. Moreover, changing the
root - that is, recovering the graph which is obtained be taking the same labeled
graph but when considering a different vertex as the root - yields a conjugation
of K as the corresponding subgroup. Namely, if the original root was v with a
corresponding subgroup K, and if the new root is v.g for some g, then one gets the
corresponding subgroup K g . Hence when ignoring the root, one can think of the
Schreier graph as representing the conjugacy class K G . In particular, if K is the
subgroup associated with a vertex v, and g is such that when rooting the Schreier
graph in v and in v.g we get two isomorphic labeled graphs, then g belongs to the
normalizer NG (K) = {g ∈ G : K g = K}.

ENTROPY OF INTERSECTIONAL IRSS

9

The subgroups of the free group that we will construct will be described by their
Schreier graph. In the following section we explain the connection between Schreier
graphs and Poisson bundles, which is why using Shcreier graphs to construct
subgroups is more adapted to our purpose.
2.3. Random walks on Schreier graphs. Fix a generating probability measure
µ on a group G. Let (Xt )∞
t=1 be i.i.d. random variables taking values in G with
law µ, and let Zt = X1 · · · Xt be the position of the µ-random walk at time t.
Let K be a subgroup. Although this discussion holds for any group, since we
defined Schreier graphs in the context of the free group, assume that G = F2 .
The Markov chain (KZt )t has a natural representation on the Schreier graph
associated with K, by just considering the position on the graph at time t to be
v.Zt where v is a root associated with K. (This is the importance of considering
left-Schreier graphs and right-random walks.) This is a general correspondence,
not special to the free group.
The boundary of the Markov chain KZt where K is fixed, is the boundary
of the random walk on the graph that starts in v, namely, the collection of tail
events of this Markov chain. The Poisson bundle is a bundle of such boundaries,
or alternatively, it is the boundary of the Markov chain KZt where K is a random
subgroup distributed according to λ. This perspective was our intuition behind
the construction in Section 4.
2.4. Random walk entropy. Recall that we assume throughout
that all random
P
walk measures µ have finite entropy, that is H(Z1 ) = − g P[Z1 = g] log(P[Z1 =
g]) < ∞ where Zt as usual is the position of the µ-random walk at time t. The
sequence of numbers H(Zt ) is a sub-additive sequence so the following limit exists
and finite.
Definition 2.1. The random walk entropy of (G, µ) is given by
1
hRW (G, µ) = lim H(Zt ).
t→∞ t
The Furstenberg-entropy of a (G, µ)-stationary space, (X, ν) is given by
Z
X
dg −1 ν
hµ (X, ν) = −
µ(g)
log
dν(x)
dν
X
g∈G
By Jensen’s inequality, hµ (X, ν) ≥ 0 and equality holds if and only if the action
is a measure preserving action. Furthermore, Kaimanovich and Vershik proved
in [20] that hµ (X, ν) ≤ hRW (G, µ) for any stationary action (X, ν), and equality
holds for the Furstenberg-Poisson boundary, namely
(2.1)

hµ (Π(G, µ)) = hRW (G, µ) ,

where Π(G, µ) denotes the Furstenberg-Poisson boundary of (G, µ).

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If hRW (G, µ) = 0 the pair (G, µ) is called Liouville, which is equivalent to
triviality of the Furstenberg-Poisson boundary.
The Furstenberg-entropy realization problem is to find the exact values
in the interval [0, hRW (G, µ)] which are realized as the Furstenberg entropy of
ergodic stationary actions.
In a similar way to the Kaimanovich-Vershik formula (2.1), Bowen proves the
following:
Theorem 2.2 (Bowen, [4]). Let λ be an IRS of G. Then
Z
Z
1
1
hµ (Bµ (λ)) = lim
(2.2)
H(KZt )dλ(K) = inf
H(KZt )dλ(K)
t→∞ t Sub
t t Sub
G
G
Moreover, the (G, µ)-stationary space is ergodic if (and only if ) the IRS λ is
ergodic.
This reduces the problem of finding (G, µ)-stationary actions with many Furstenberg entropy values in [0, hRW (G, µ)], to finding IRSs λ with many different random walk entropy values.
Regarding continuity, recall that the space SubG is equipped with the Chabauty
topology. For finitely generated groups this topology is induced by the natural
metric for which two subgroups are close if they share the exact same elements
from a large ball in G. One concludes that the Shannon entropy, H, as a function
on SubG is a continuous function. The topology on SubG induces a weak* topology
on, P rob (SubG ), the space of IRSs of G. By the continuity of H and the second
equality in (2.2) we get that the Furstenberg-entropy of Poisson bundles of the
form Bµ (λ) is an upper semi-continuous function of the IRS. Since the entropy is
non-negative we conclude:
Corollary 2.3 (Bowen, [4]). If λn → λ in the weak* topology on P rob (SubG ),
such that hµ (Bµ (λ)) = 0 then hµ (Bµ (λn )) → 0.
However, it is important to notice that the Furstenberg-entropy is far from
being a continuous function of the IRS. Indeed, consider the free group, which is
residually finite. There exists a sequence of normal subgroups Nn , such that Nn \F2
is finite for any n, which can be chosen so that, when considered as IRSs λn = δNn ,
we have λn → δ{e} . In that case hµ (Bµ (λn )) = 0 for all n (because H(Nn Zt ) is
bounded in t), but Bµ (δ{e} ) is the Furstenberg-Poisson boundary of the free group,
and its Furstenberg entropy is positive for any µ (follows for example from the
non-amenability of the free group).
We want to point out that this lack of continuity makes it delicate to find
conditions on quotients of the free group to have large entropy. In Section 4
we will deal with this problem by introducing a tool to prove that the entropy
does convergence along a sequence of subgroups which satisfy some geometric
conditions.

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11

The free group is also residually nilpotent, so one can replace the finite quotient
from the discussion above by infinite nilpotent quotients. We arrive at a similar
discontinuity of the entropy function, due to the following classical result. As we
will use it in the sequel, we quote it here for future reference. The original case of
Abelian groups is due to Choquet-Deny [9] and the nilpotent case appears in [27].
Theorem 2.4 (Choquet-Deny-Raugi). Let G be a nilpotent group. For any probability measure µ on G with finite entropy, the pair (G, µ) is Liouville (i.e. hRW (G, µ) =
0).
We mention that the triviality of the Furstenberg-Poisson boundary holds for
general µ without any entropy assumption, although in this paper we consider only
µ with finite entropy. The proof of the Choquet-Deny Theorem follows from the
fact that the center of the group always acts trivially on the Furstenberg-Poisson
boundary and hence Abelian groups are always Liouville. An induction argument
extends this to nilpotent groups as well.
3. Intersectional IRSs
In this section we show how to construct a family of IRSs, given an infinite
conjugacy class.
Let G be a countable discrete group, and let K ≤ G be a subgroup with infinitely
many different conjugates |K G | = ∞. Equivalently, the normalizer NG (K) := {g ∈
G : K g = K} is of infinite index in G.
Recall that the G action on K G is the right action K g = g −1 Kg. Since K ng =
−1 −1
g n Kng = g −1 Kg = K g for any n ∈ NG (K), the K-conjugation depends only
on the coset NG (K)g. That is, conjugation K θ for a right coset θ ∈ NG (K)\G is
well defined.
We use 2NG (K)\G to denote the family of all subsets of NG (K)\G. This is
identified canonically with {0, 1}NG (K)\G . Note that G acts from the right on
subsets in 2NG (K)\G , via Θ.g = {θg : θ ∈ Θ}. Given a non-empty subset
Θ ⊂ NG (K)\G we define the subgroup
\
CoreΘ (K) =
Kθ.
θ∈Θ

Claim 3.1. The map ϕ : 2NG (K)\G → SubG defined by ϕ(Θ) = CoreΘ (K) is
G-equivariant.
In particular, if we denote by λp,K the ϕ-push-forward of the Bernoulli-p product
measure on 2NG (K)\G , then λp,K is an ergodic IRS for any p ∈ (0, 1).
Proof. Note that
g

(CoreΘ (K)) =

\
θ∈Θ

K

θ

g

=

\
θ∈Θ

(K θ )g =

\
θ0 ∈Θ.g

0

K θ = CoreΘ.g (K)

12

YAIR HARTMAN AND ARIEL YADIN

It follows that we can push forward any G-invariant measure on 2NG (K)\G to get
IRSs. For any p ∈ (0, 1) consider the Bernoulli-p product measure on 2NG (K)\G ∼
=
{0, 1}NG (K)\G , namely, each element in NG (K)\G is chosen to be in Θ independently with probability p. These measures are clearly ergodic invariant measures.
It follows that the push-forward measures λp,K are ergodic invariant measures on
SubG .

We continue to determine the limits of λp,K as p → 0 and p → 1. When the
subgroup K is clear from the context, we write λp = λp,K .
Lemma 3.2. Given K ≤ G with |K G | = ∞, there exist two normal subgroups,
p→0
p→1
Core∅ (K), CoreG (K) / G such that λp,K −−→ δCore∅ (K) and λp,K −−→ δCoreG (K) ,
where the convergence is in the weak* topology on P rob (SubG ).
While the definition of CoreG (K) is apparent, it is less obvious how to define
Core∅ (K). We now give an intrinsic description of the normal subgroup Core∅ (K).
3.1. The subgroup Core∅ (K). For any element g ∈ G, let Ωg ⊂ NG (K)\G be
the set of all cosets, such that g belongs to the corresponding conjugation: Ωg =
{θ ∈ NG (K)\G | g ∈ K θ }. For example, for g ∈ K we have that NG (K) ∈ Ωg .
We observe that:
(1) Ωg ∩ Ωh ⊂ Ωgh
(2) Ωg−1 = Ωg
(3) Ωgγ = Ωg γ −1
The first two follow from the fact that K θ is a group and the third since γK θ γ −1 =
−1
K θγ .
Let fg be the complement of Ωg in NG (K)\G. That is, fg = {θ ∈ NG (K)\G :
θ 6∈ Ωg }. Define kgkK = |fg |. The properties above show that
(1) kghkK ≤ kgkK + khkK
(2) kg −1 kK = kgkK
(3) kg γ kK = kgkK
Two normal subgroups of G naturally arise: the subgroup of all elements with
zero norm and the subgroup of all elements with finite norm. By (1) and (2)
above, these are indeed subgroups and normality follows from (3).
The first subgroup is, by definition, the normal core CoreG (K) = {g | kgkK =
0}. We claim that the second subgroup is the appropriate definition for Core∅ (K).
That is, we define Core∅ (K) = {g | kgkK < ∞} and prove that λp,K → δCore∅ (K)
as p → 0.
p→0

Proof of Lemma 3.2. By definition, to show that λp −−→ δCore∅ (K) we need to show
p→0

that for any g ∈ Core∅ (K), we have λp ({F ∈ SubG | g ∈ F }) −−→ 1 and for any
p→0

g∈
/ Core∅ (K), we have λp ({F ∈ SubG | g ∈ F }) −−→ 0.

ENTROPY OF INTERSECTIONAL IRSS

13

Note that for any Θ 6= ∅, by the definition of CoreΘ (K), we have that g ∈
CoreΘ (K) if and only if Θ ∩ fg = ∅.
Now consider Θ to be a random subset according to the Bernoulli-p product measure on 2NG (K)\G . By the above, we have equality of the events {g ∈
CoreΘ (K)} = {fg ∩ Θ = ∅}. The latter (and hence also the former) has probability (1 − p)|fg | = (1 − p)kgkK . This includes the case where |fg | = kgkK = ∞
(whence we have 0 probability for {fg ∩Θ = ∅}). We conclude that for any g ∈ G,
(
0
if kgkK = ∞,
kgkK
λp ({F ∈ SubG | g ∈ F }) = (1 − p)
=
→ 1 as p → 0 if kgkK < ∞.
It follows that λp → δCore∅ (K) as p → 0.
we show that λp → δCoreG (K) as p → 1, where we define CoreG (K) =
T Finally,
g
g K = CoreNG (K)\G (K). Clearly, for any p ∈ (0, 1), we have that CoreG (K) ⊂ F
for λp -almost every F , since CoreG (K) ≤ CoreΘ (K) for any non-empty subset Θ.
On the other hand, fix some g 6∈ CoreG (K). By definition, there exists some
θ ∈ NG (K)\G such that g 6∈ K θ . Whenever Θ is such that θ ∈ Θ, we have that
g 6∈ CoreΘ (K). The probability that θ 6∈ Θ is (1 − p). Hence, for any g with
kgkK > 0,
p→1

λp ({F ∈ SubG | g ∈ F }) ≤ (1 − p) −−→ 0.

We conclude that λp → δCoreG (K) as p → 1.



3.2. Continuity of the entropy along intersectional IRSs. Lemma 3.2 above
shows that the IRSs λp,K interpolate between the two Dirac measures concentrated
on the normal subgroups Core∅ (K) and CoreG (K). In this section we provide a
condition under which this interpolation holds for the random walk entropy as
well.
Fix a random walk µ on G with finite entropy. Let h : [0, 1] → R+ be the
entropy map defined by h(p) = hµ (Bµ (λp )) when p ∈ (0, 1) and h(0), h(1) defined as the µ-random walk entropies on the quotient groups G/Core∅ (K) and
G/CoreG (K) respectively. In particular, the condition that h(0) = 0 is equivalent
to the Liouville property holding for the projected µ-random walk on G/Core∅ (K).
Proposition 3.3. If h(0) = 0 then h is continuous.
Proof. Let
Z
Hn (p) =

H(F Zn )dλp (F ).
SubG

So h(p) = limn→∞ n1 Hn (p). First, we use a standard coupling argument to prove
the for q = p + ε,
(3.1)

0 ≤ Hn (q) − Hn (p) ≤ Hn (ε)

14

YAIR HARTMAN AND ARIEL YADIN

Indeed, let (Uθ )θ∈NG (K)\G be i.i.d. uniform-[0, 1] random variables, one for each
coset of NG (K). We use P, E to denote the probability measure and expectation
with respect to these random variables. Define three subsets
Θp := {θ : Uθ ≤ p}

Θε := {θ : p < Uθ ≤ q}

and Θq := Θp ∪ Θε . We shorthand Cx := CoreΘx (K) for x ∈ {p, ε, q}. Note that
for x ∈ {p, ε, q} the law of Cx is λx . Thus, Hn (x) = E[H(Cx Zn )] (to be clear, the
entropy is with respect to the random walk Zn , and expectation is with respect
to the random subgroup Cx ).
Now, by definition
\
\
\
Cq =
Kθ =
Kθ ∩
K θ = Cp ∩ Cε .
θ∈Θq

θ∈Θp

θ∈Θε

Since the map (F1 g, F2 g) 7→ (F1 ∩ F2 )g is well defined, we have that Cq Zn is
determined by the pair (Cp Zn , Cε Zn ). Also, since Cq ≤ Cp , we have that Cp Zn is
determined by Cq Zn . Thus, P-a.s.,
H(Cp Zn ) ≤ H(Cq Zn ) ≤ H(Cp Zn ) + H(Cε Zn ).

Taking expectation (under E) we obtain (3.1), which then immediately leads to
0 ≤ h(q) − h(p) ≤ h(ε).
Corollary 2.3 asserts that the entropy function is upper-semi continuous. Thus,
lim supε→0 h(ε) ≤ h(0). Thus, if h(0) = 0 we have that limε→0 h(ε) = 0, and that
the entropy function is continuous.

3.3. Applications to entropy realizations. Let µ be a probability measure
on a group G and N / G a normal subgroup. We use µ̄ to denote the projected
measure on the quotient group G/N .
Corollary 3.4. Let µ be a generating finite entropy probability measure on a
discrete group G. Assume that there exists some K ≤ G such that
• hRW (G/Core∅ (K), µ̄) = 0
• hRW (G/CoreG (K), µ̄) = hRW (G, µ)
Then (G, µ) has a full realization.
More generally, if only the first condition is satisfied, we get a realization of the
interval [0, hRW (G/CoreG (K), µ̄)].
Proof. Consider the path of IRSs λp,K defined in Claim 3.1. Let h(p) be defined
as above, before Proposition 3.3. Since h(0) = hRW (G/Core∅ (K), µ̄) = 0, by
Proposition 3.3 we have that h is continuous, and thus we have a realization of
the interval [h(0), h(1)] = [0, hRW (G/CoreG (K), µ̄)].

In the above corollary, the subgroup K may depend on the measure µ. However,
it is possible in some cases to find a subgroup K that provides realization for any
measure µ, and we now discuss alternative conditions for the corollary.

ENTROPY OF INTERSECTIONAL IRSS

15

The conditions in Corollary 3.4 implies that Core∅ (K) is a “large” subgroup and
CoreG (K) is “small” - at least in the sense of the random walk entropy. The best
scenario hence is when Core∅ (K) = G and CoreG (K) = {1}. The prototypical
example is the following:
Example 3.5. Let Sym∗ (X) denote the group of all finitely supported permutations of some infinite countable set X. Fix some x0 ∈ X and define K to be
the subgroup of all the finitely supported permutations of X that stabilize x0 ; i.e.
K = stab(x0 ) = {π ∈ Sym∗ (X) : π(x0 ) = x0 }. Conjugations of K are of the
form stab(x) for x ∈ X.
It follows that CoreG (K) = {e}, because Sym∗ (X) acts transitively on X and
only the trivial permutation stabilizes all points x ∈ X.
Since each element of Sym∗ (X) is a finitely supported permutation, it stabilizes
all but finitely many points in X. Thus, Core∅ (K) = Sym∗ (X).
This is actually a proof of Theorem 1.4: (Sym∗ (X), µ) admits full realization
for any generating, finite entropy, probability measure µ.
Note that Sym∗ (X) is a countable group which is not finitely generated. However, by adding some elements to it, we can get a finitely generated group with a
similar result, as we explain below in Section 3.4.
To continue our discussion, we want to weaken the condition Core∅ (K) = G to
some algebraic condition that will still guarantee that hRW (G/Core∅ (K), µ̄) = 0
for any µ.
Given a subgroup K (or a conjugacy class K G ) we say that K is locally conilpotent in G if G/Core∅ (K) is nilpotent. (The reason for this name will become
more apparent in Section 4.5.) With this definition, Proposition 1.5 follows immediately, as may be seen by the following.
Corollary 3.6. Assume that G admits a subgroup K ≤ G such that G/Core∅ (K)
is a nilpotent group. Then, for any generating finite entropy probability measure
µ, the pair (G, µ) admits a realization of the interval [0, hRW (G/CoreG (K), µ̄)].
If, in addition hRW (G/CoreG (K), µ̄) = hRW (G, µ) (e.g. if CoreG (K) = {e})
then (G, µ) admits full realization.
Proof. When G/Core∅ (K) is nilpotent, the Choquet-Deny Theorem (Theorem 2.4)
tells us that the first condition of Corollary 3.4 is satisfied.

In Section 3.4 we apply this to obtain full realizations for a class of lamplighter
groups as well as to some extensions of the group of finitely supported permutations. However, one should not expect to find a subgroup satisfying the conditions
of Corollary 3.6 in every group. These conditions are quite restrictive, for example
we the following:
Lemma 3.7. If G is a group with a self-normalizing subgroup K that satisfies the
two conditions of Corollaries 3.4 (or 3.6) then G is amenable.

16

YAIR HARTMAN AND ARIEL YADIN

Proof. First, observe that hRW (G/Core∅ (K), µ̄) = 0 implies that Core∅ (K) is
a co-amenable (normal) subgroup in G. Indeed, it is well know that a group,
G/Core∅ (K) in this case, admits a random walk which is Liouville is amenable.
Next, the condition hRW (G/CoreG (K), µ̄) = hRW (G, µ) implies that CoreG (K)
is an amenable group. This follows from Kaimanovich’s amenable extension theorem [18]: Let µ be a random walk on G. The Furstenberg-Poisson boundary
of G/CoreG (K) with the µ-projected random walk is a (G, µ)-boundary, or, in
other words a G-factor of the Furstenberg-Poisson boundary Π(G, µ). Since it has
the same entropy as the random walk entropy, this factor is actually isomorphic
to Π(G, µ). Now Kaimanovich’s theorem asserts that CoreG (K) is an amenable
group.
To conclude that G is amenable we show that Core∅ (K)/CoreG (K) is amenable,
showing that the normal subgroup CoreG (K) is both amenable and co-amenable
in G.
In general, we claim that Core∅ (K)/CoreG (NG (K)) is always an amenable
group. This will complete the proof, because we assumed that NG (K) = K.
Indeed, G acts by permutations on the conjugacy class K G . Note that any
element in Core∅ (K) has finite support as a permutation on K G . Also, it may be
checked that the elements of Core∅ (K) that stabilize every K γ are precisely the
elements of CoreG (NG (K)). Thus, Core∅ (K)/CoreG (NG (K)) is isomorphic to a
subgroup of the amenable group Sym∗ (X).

The subgroups of the free group that we will consider in Section 4 (to prove
Theorem 1.2) are all self-normalizing. Since the free group is non-amenable, we
cannot obtain the conditions of Corollary 3.4 simultaneously in this fashion. For
this reason, we approximate the second condition using a sequence of subgroups.
Corollary 3.8. Let µ be a generating finite entropy probability measure on a
discrete group G. Assume that there exists a sequence of subgroups Kn ≤ G such
that:
(1) hRW (G/Core∅ (Kn ), µ̄) = 0 (e.g. whenever G/Core∅ (Kn ) are nilpotent),
(2) hRW (G/CoreG (Kn ), µ̄) → hRW (G, µ)

Then (G, µ) admits a full realization.

Proof. By the first condition, for any fixed n, the entropy along the path λp,Kn gives
a realization of the interval [0, hRW (G/CoreG (Kn ), µ̄)]. By the second condition,
we conclude that any number in [0, hRW (G, µ)] is realized as the entropy some
IRS of the form λp,Kn .

3.4. Examples: lamplighter groups and permutation groups. We are now
ready to apply our tools to get Theorem 1.3, that is, full realization for lamplighter
groups with Liouville base.

ENTROPY OF INTERSECTIONAL IRSS

17

Proof of Theorem 1.3. Choose the subgroup K = ⊕B\{e} L o {e} where e is the
trivial element in the base group B. It may be simply verified that CoreG (K) = {e}
and ⊕B L o {e} C Core∅ (K).
Whenever (B, µ̄) is Liouville, also (G/Core∅ (K), µ̄) is. This is because B ∼
=
G/(⊕B L o {e}), so G/Core∅ (K) is a quotient group of B. Thus, we have that
hRW (G/Core∅ (K), µ̄) = 0, and we obtain a realization of the full interval
[0, hRW (G/CoreG (K), µ̄)] = [0, hRW (G, µ)].

We now turn to discuss extensions of the group of finitely supported permutations. Let X be a countable set, and denote by Sym(X) the group of all permutations of X. Recall that Sym∗ (X) is its subgroup of all finitely supported permutations. While Sym∗ (X) is not finitely generated, one can add elements from
Sym(X) to get a finitely generated group. A standard example is when adding a
“shift” element σ ∈ Sym(X), that is, σ acts as a free transitive permutation of
X. Let Σ = hσi be the subgroup generated by σ. Formally the element σ acts on
Sym∗ (X) by conjugation (recall that both groups are subgroups of Sym(X)) and
we get a finitely generated group Sym∗ (X) o Σ.
One can replace Σ by any other subgroup of Sym(X). The reason that we want
to change Σ by a “larger” group is to have a non-Liouville group Sym∗ (X) o Σ.
Theorem 3.9. Let Σ ≤ Sym(X) and consider G = Sym∗ (X) o Σ. Let µ be a
generating finite entropy probability measure on G, such that the the projected
measure µ̄ on Σ ∼
= G/Sym∗ (X) o {e} is Liouville. Then (G, µ) has full realization.
In particular, if Σ is a nilpotent group, (G, µ) admits full realization for any µ.
Proof. Fix some point x0 ∈ X and let K = stab(x0 ) o {e} where stab(x0 ) ≤
Sym∗ (X) is the stabilizer of x0 . It is easy to verify that CoreG (K) = {e} and that
Core∅ (K) = Sym∗ (X) o {e}, hence by the assumption hRW (G/Core∅ (K), µ̄) = 0
and by Corollary 3.4 we get a full realization.

Remark 3.10. Depending of Σ and µ, it might be that (G, µ) is Liouville where
G = Sym∗ (X) o Σ. In that case, this result of full realization is trivial as the only
realizable value is 0.
However, it is easy to find nilpotent Σ such that G is finitely generated and that
(G, µ) is not Liouville (say, for any generating, symmetric, finitely supported µ).
4. Full realization for the free group
We now turn to discuss our realization results for G = Fr the free group of rank
r for any r ≥ 2. It suffices to prove that the free group satisfies the conditions of
Corollary 3.8. We construct the subgroups Kn by describing their Schreier graphs.
For simplicity of the presentation we describe the result only for the free group on
two generators, F2 = ha, bi. The generalization to higher rank is straightforward.

18

YAIR HARTMAN AND ARIEL YADIN

4.1. Schreier graphs. Recall our notations for Schreier graphs from Section 2.2.
It will be convenient to use |v| to denote the graph distance of a vertex v to the
root vertex in a fixed Schreier graph.
4.2. Fixing. In this section we describe a condition on Schreier graphs that we call
fixing. For the normal core of the associated subgroup, this condition will ensure
that the random walk entropy of the quotient approximates the full random walk
entropy as required by Condition 2 in Corollary 3.8 (see Proposition 4.7 below).
Let Λ be a Schreier graph with root o.
For a vertex v in Λ, the shadow of v, denoted shd(v) is the set of vertices u in
Λ such that any path from u to the root o must pass through v.
For an integer n define the finite subtree (F2 )≤n to be the induced (labeled)
subgraph on {g ∈ F2 : |g| ≤ n}. We say that Λ is n-tree-like if the ball of
radius n about o in Λ is isomorphic to (F2 )≤n , and if every vertex at distance n
from o has a non-empty shadow. Informally, an n-tree-like graph is one that is
made up by gluing disjoint graphs to the leaves of a depth-n regular tree.
Lemma 4.1. Let C ≤ K ≤ F2 be subgroups. Let ΛC , ΛK be the Schreier graphs
of C, K respectively.
If ΛK is n-tree-like, then ΛC is also n-tree-like.
Proof. This follows from the fact that C ≤ K implies that ΛC is a cover of ΛK
(see e.g. [21, Lemma 20]).

For an n-tree-like Λ, since any vertex is connected to the root, every vertex at
distance greater than n from the root o is in the shadow of some vertex at distance
precisely n. For a vertex v in Λ with |v| > n, and k ≤ n we define the k-prefix
of v in Λ to be pref k (v) = pref k,Λ (v) = u for the unique u such that both |u| = k
and v ∈ shd(u). If |v| ≤ k then for concreteness we define pref k (v) = v.
Since n-tree-like graphs look like F2 up to distance n, the behavior of pref k on
n-tree-like graphs where n ≥ k is the same as on F2 . This is the content of the
next elementary lemma, whose proof is left as an exercise.
Lemma 4.2. Let Λ be a n-tree-like Schreier graph. Let ϕ : Λ → F2 be a function
mapping the ball of radius n in Λ isomorphically to (F2 )≤n .
Then, for any vertex |v| ≤ n in Λ we have that
pref k,Λ (v) = ϕ−1 pref k,F2 (ϕ(v)).

We use (Zt )t to denote the µ-random walk on F2 and (Z̃t )t the projected walk on
a Schreier graph Λ with root o. (So Z̃t = KZt where K is the associated subgroup
to the root o in the Schreier graph Λ.)
Another notion we require is the k-prefix at infinity. Define the random
variable pref k (Z̃∞ ) = pref k,Λ (Z̃∞ ), taking values in the set of vertices of distance
k from o in Λ, as follows: If the sequence pref k (Z̃t )t stabilizes at u, that is, if there

ENTROPY OF INTERSECTIONAL IRSS

19

exists t0 such that for all t > t0 we have pref k (Z̃t ) = pref k (Z̃t−1 ) = u, then define
pref k (Z̃∞ ) := u. Otherwise, define pref k (Z̃∞ ) = o. Note that for an n-tree-like
graph with n > 0 we have that pref k (Z̃∞ ) = o (almost surely) if and only if the
walk (Z̃t )t returns to the ball of radius k infinitely many times.
Definition 4.3. Let 0 < k < n be natural numbers and α ∈ (0, 1). We say that
Λ is (k, n, α)-fixing if Λ is n-tree-like and for any |v| ≥ n,
P[∀ t , pref k (Z̃t ) = pref k (Z̃0 ) | Z̃0 = v] ≥ α.
That is, with probability at least α, for any |v| ≥ n, the projected random
walk started at v never leaves shd(pref k (v)). Hence, the random walk started at
depth n in the graph fixes the k-prefix with probability at least α. Note that this
definition depends on the random walk µ.
A good example of a fixing graph is the Cayley graph of the free group itself.
Lemma 4.4. Let µ be a finitely supported generating probability measure on F2 .
Then, there exists α = α(µ) > 0 such that the Cayley graph of F2 is (k, k + 1, α)fixing for any k > 0.
Proof. It is well known that a µ-random walk (Zt )t on F2 is transient (for example,
this follows from non-amenability of F2 ). Set r = max{|g| : µ(g) > 0} be the
maximal jump possible by the µ-random walk. In order to change the k-prefix, the
walk must return to distance at most k + r from the origin. That is, if |Zt | > k + r
for all t > t0 , we have that pref k (Zt ) = pref k (Zt0 ) for all t > t0 .
By forcing finitely many initial steps (depending on r, µ), there exists t0 and
α > 0 such that P[A | |Z0 | = k + 1] ≥ α where
A = {∀0 < t ≤ t0 , pref k (Zt ) = pref k (Z0 ) and |Zt0 | > k + r}.
By perhaps making α smaller, using transience, we also have that
P[∀ t > 0 , |Zt | ≥ |Z0 |] ≥ α.
Combining these two estimates, with the Markov property at time t0 ,
P[∀t, pref k (Zt ) = pref k (Z0 ) | |Z0 | = k + 1] ≥ P[A and {∀ t > t0 , |Zt | ≥ |Zt0 |}]

≥ P[A] · P[∀ t > 0 , |Zt | ≥ |Z0 |] ≥ α2 .

This is the definition of F2 being (k, k + 1, α)-fixing.



A very useful property of fixing graphs is that when α is close enough to 1, there
exists a finite random time - the first time the random walk leaves the tree area on which we can already guess with a high accuracy the k-prefix at infinity of the
random walk. A precise formulation is the following.

20

YAIR HARTMAN AND ARIEL YADIN

Lemma 4.5. Let Λ be a (k, n, α)-fixing Schreier graph. Let (Zt )t be a µ-random
walk on F2 and (Z̃t )t the projected walk on Λ. Define the stopping time
T = inf{t : |Z̃t | ≥ n}.
Then,
H(Z1 | pref k (Z̃T )) − H(Z1 | pref k (Z̃∞ )) ≤ ε(α) + 2(log 4)(1 − α)k,
where 0 < ε(α) → 0 as α → 1.
Proof. First note that a general entropy inequality is
H(X | A) − H(X | B) ≤ H(A | B) + H(B | A),
which for us provides the inequality
H(Z1 | pref k (Z̃T )) − H(Z1 | pref k (Z̃∞ ))

≤ H(pref k (Z̃T ) | pref k (Z̃∞ )) + H(pref k (Z̃∞ ) | pref k (Z̃T )).

We now bound this last quantity using Fano’s inequality (see e.g. [11, Section
2.10]). Taking α̃ = P[pref k (Z̃T ) = pref k (Z̃∞ )], since the support of both pref k (Z̃∞ ), pref k (Z̃T )
is of size 4 · 3k−1 < 4k ,
H(pref k (Z̃∞ ) | pref k (Z̃T )) ≤ H(α̃, 1 − α̃) + (1 − α̃) log 4k ,

H(pref k (Z̃T ) | pref k (Z̃∞ )) ≤ H(α̃, 1 − α̃) + (1 − α̃) log 4k .

where here we use the usual notation H(α̃, 1 − α̃) to denote the Shannon entropy
of the probability vector (α̃, 1 − α̃).
Since p 7→ H(p, 1 − p) decreases for p ≥ 12 , it suffices to prove that α̃ =
P[pref k (Z̃T ) = pref k (Z̃∞ )] ≥ α.
But now, the very definition of (k, n, α)-fixing implies that for any relevant u,
P[pref k (Z̃∞ ) = u | pref k (ZT ) = u]

≥ P[∀ t > 0 , pref k (Z̃T +t ) ∈ shd(pref k (u)) | pref k (ZT ) = u] ≥ α,

where we have used the strong Markov property at the stopping time T . Averaging
over the relevant u implies that α̃ ≥ α.

The following is the main technical estimate of this subsection.
Lemma 4.6. Fix a generating probability measure µ on F2 with finite support.
Let (Γj )j be a sequence of Schreier graphs such that Γj is (kj , nj , αj )-fixing. Let
Kj be the subgroup corresponding to Γj .
Suppose that kj → ∞, nj − kj → ∞ and (1 − αj )kj → 0 as j → ∞. Then,
lim sup hRW (F2 /CoreF2 (Kj ), µ̄) = hRW (F2 , µ).
j→∞

ENTROPY OF INTERSECTIONAL IRSS

21

Proof. Let (Zt )t be a µ-random walk on F2 .
Take j large enough so that nj −kj is much larger that r = max{|g| : µ(g) > 0}.
To simplify the presentation we use Γj = Γ, K = Kj , C = Cj = CoreF2 (Kj ) and
k = kj , n = nj , α = αj , omitting the subscript j when it is clear from the context.
Let Λ = Λj beTthe Schreier graph corresponding
T to C = Cj .
Let TC = t σ(CZt , CZt+1 , . . .) and T = t σ(Zt , Zt+1 , . . .) denote that tail
σ-algebras of the random walk on Λ and on F2 respectively. Kaimanovich and
Vershik (eq. 13 on p. 465 in [20]) show that
hRW (F2 /C, µ̄) = H(CZ1 ) − H(CZ1 | TC )
and
hRW (G, µ) = H(Z1 ) − H(Z1 | T ).

Now, since C ≤ K, and since Γ is n-tree-like, also Λ is (Lemma 4.1). Since Λ is
n-tree-like, CZ1 and Z1 determine one another (Lemma 4.2), implying H(CZ1 ) =
H(Z1 ) and H(CZ1 | TC ) = H(Z1 | TC ). Since TC ⊂ T , the inequality H(Z1 | TC ) ≥
H(Z1 | T ) is immediate. Hence, we only need to show that lim inf j H(Z1 | TCj ) ≤
H(Z1 | T ).
Set T = inf{t : |CZt | ≥ n} as in Lemma 4.5. Since Λ is n-tree-like,
pref k (CZT −1 ) and pref k (ZT −1 ) determine one another (Lemma 4.2). Also, since
n − k is much larger than r, and specifically much larger than any jump the
walk can make in one step, we know that pref k (CZT ) = pref k (CZT −1 ) and
pref k (ZT ) = pref k (ZT −1 ). All in all, H(Z1 | pref k (CZT )) = H(Z1 | pref k (ZT )).
Now, set ε = ε(α, k) = 2H(α, 1 − α) + 2(log 4)(1 − α)k. Since pref k (CZ∞ ) is
measurable with respect to TC , by using Lemma 4.5 twice, once for Λ and once
for the tree, we get the bound
H(Z1 | TC ) ≤ H(Z1 | pref k (CZ∞ )) ≤ H(Z1 | pref k (CZT )) + ε

= H(Z1 | pref k (ZT )) + ε ≤ H(Z1 | pref k (Z∞ )) + 2ε.

Now, under the assumptions of the lemma, ε → 0 as j → ∞. Also, since
kj → ∞ we have that
H(Z1 | pref kj (Z∞ )) → H(Z1 | T ).
Thus, taking a limit on the above provides us with
lim sup H(Z1 | TCj ) ≤ lim sup H(Z1 | pref kj (Z∞ )) = H(Z1 | T ).
j→∞

j→∞

Hence, as mentioned, we conclude that lim supj H(Z1 | TCj ) = H(Z1 | T ), and
since H(Z1 ) = H(CZ1 ) we get that
lim sup hRW (F2 /CoreF2 (Kj ), µ̄) = hRW (F2 , µ).
j→∞



22

YAIR HARTMAN AND ARIEL YADIN

To complete the relevance of fixing to convergence of the random walk entropies,
we conclude with the following.
Proposition 4.7. Fix a generating probability measure µ on F2 with finite support.
Let (Γj )j be a sequence of Schreier graphs such that Γj is (kj , nj , αj )-fixing. Let
Kj be the subgroup corresponding to Γj .
Suppose that kj → ∞ and αj → 1 as j → ∞. Then,
lim sup hRW (F2 /CoreF2 (Kj ), µ̄) = hRW (F2 , µ).
j→∞

Proof. Since we are only interested in lim sup we may pass to a subsequence
and show that the parameters along this subsequence satisfy the assumptions
of Lemma 4.6. Notice that the assumption kj → ∞ implies in particular that
nj → ∞.
Note that by definition, if Λ is (k, n, α)-fixing, then it is also (k 0 , n0 , α0 )-fixing
for any k 0 ≤ k, n0 ≥ n, α0 ≤ α.
For any m choose jm large enough so that for all i ≥ jm we have both αi >
1 − m−2 and ki > 2m. The subsequence of graphs (Γjm )m satisfies that Γjm is
(m, njm , 1− m12 )-fixing. Since njm −m ≥ kjm −m > m, Lemma 4.6 is applicable. 
4.3. Gluing graphs. Let Λ be a Schreier graph of the free group F2 = ha, bi and
let S = {a, a−1 , b, b−1 }. We say that the pair (Λ, ) is s-marked if  is a (oriented)
edge in Λ labeled by s ∈ S.
For s ∈ S we define Ns to be the following graph: the vertices are non-negative
the integers N. The edges are given by (x + 1, x), each labeled by s, and self-loops
labeled by ξ ∈ {a, b} \ {s, s−1 } at each vertex. In order to be a Schreier graph,
this graph is missing one outgoing edge from 0 labeled by s.
b

b

a
0

b

a
1

b

a
2

b

a
3

4

Figure 1. Part of the labeled graph Na .
Given a s-marked pair (Λ, ) and an integer n > 0, we construct a Schreier
graph Γn (Λ, ):
The Cayley graph of F2 is the 4-regular tree. For an integer n and a label
(generator) s recall the finite subtree (F2 )≤n which is the induced subgraph on
{g ∈ F2 : |g| ≤ n}, and let (F2 )n be the set of vertices {g : |g| = n}. For g ∈
(F2 )n there is exactly one outgoing edge incident to g in (F2 )≤n . If this edge is
labeled by ξ ∈ S we say that g is a ξ-leaf.

ENTROPY OF INTERSECTIONAL IRSS

23

w

x

y

z

Figure 2. The subgraph (F2 )≤3 . Edges labeled a are dotted and
labeled b are solid. The vertices x, y, z, w are a, a−1 , b, b−1 -leaves
respectively.
For a ξ-leaf g, in order to complete (F2 )≤n into a Schreier graph, we need
to specify 3 more outgoing edges, with the labels ξ −1 and the two labels from
S \ {ξ, ξ −1 }.
Now, recall our s-marked pair (Λ, ). Let g be a ξ-leaf for ξ 6∈ {s, s−1 }. Let
(Λg , g ) be a copy of (Λ, ), and suppose that g = (xg , yg ). Connect Λg to g by
deleting the edge g from Λg and adding the directed edges (xg , g), (g, yg ) labeling
them both s. Also, let Ng be a copy of Nξ above, and connect this copy to g by a
directed edge (0g , g) labeled by ξ (here 0g is the copy of 0 in Ng ). This takes care
of ξ-leaves for ξ 6∈ {s, s−1 }.
If ξ ∈ {s, s−1 }, the construction is simpler: For any ξ-leaf γ (where ξ ∈ {s, s−1 }),
let Nγ be a copy of Nξ (with 0γ the copy of 0 as before), and connect Nγ by an
edge (0γ , γ) labeled by ξ. Finally add an additional self-loop with the missing
labels at γ (i.e. the labels in S \ {s, s−1 } each in one direction along the loop).
By adding all the above copies to all the leaves in (F2 )≤n , we obtain a Schreier
graph, which we denote Γn (Λ, ). See Figure 3 for a visualization (which in this
case is probably more revealing than the written description). It is immediate
from the construction that Γn (Λ, ) is n-tree-like.
4.4. Marked pairs and transience. Recall the definition of a transient random
walk (see e.g. [22, 26]): We say that a graph Λ is µ-transient if for any vertex v
in Λ there is some positive probability that the µ-random walk started at v will
never return to v.

24

YAIR HARTMAN AND ARIEL YADIN

Ng
0g
g

γ

yg

xg

0γ



Nγ
Λg
Figure 3. A visiulization of gluing (Λ, ) to (F2 )≤3 . The dotted
edges correspond to the label a, and the full edges to b. The vertex g
is a a-leaf. Thus, a copy of Λ, denoted Λg is connected by removing
the edge  (dashed) and adding to edges (xg , g), (g, yg ) labeled b
(solid). A copy of Na , denoted Ng , is connected via an edge (0g , g)
labeled a (dotted). Also, to the b-leaf γ, a copy of Nb , denoted Nγ is
added via an edge (0γ , γ) labeled b (full), and an additional self-loop
labeled a (dotted) is added at γ.

We now connect the transience of Λ above to that of fixing from the previous
subsection.
Proposition 4.8. Let µ be a finitely supported generating probability measure on
F2 . Let (Λ, ) be a µ-transient s-marked pair.
Then, for any ε > 0 and k, there exist n > k such that the graph Γn (Λ, ) is
(k, n, 1 − ε)-fixing.
The proof of this proposition is quite technical. For the benefit of the reader,
before proving the proposition we first provide a rough sketch, outlining the main
idea of the proof.

ENTROPY OF INTERSECTIONAL IRSS

25

Proof Sketch. Consider the graph Γn (Λ, ). This is essentially a finite tree with
copies of Λ and copies of N glued to the (appropriate) leaves. At any vertex v with
|v| > k, there is a fixed positive probability α of escaping to the leaves without
changing the k-prefix (because F2 is (k, k + 1, α)-fixing).
Once near the leaves, with some fixed probability δ the walk reaches a copy
of Λ without going back and changing the k-prefix. Note that although there
are copies of the recurrent Nγ (e.g. for symmetric random walks) glued to some
leaves, nonetheless, since k is much smaller than n, there is at least one copy of
the transient Λ in the shadow of v.
Finally, once in the copy of Λ, because of transience, the walk has some positive
probability β to escape to infinity without ever leaving Λ, and thus without ever
changing the k-prefix.
All together, with probability at least αδβ the walk will never change the kprefix.
If this event fails, the walk may jump back some distance, but not too much
since we assume that µ has finite support. So there is some r for which the walk
(deterministically) cannot retreat more than distance r, even conditioned on the
above event failing.
Thus, starting at |v| > k + `r for large enough `, there are at least ` attempts to
escape to infinity without changing the k-prefix, each with a conditional probability of success at least αδβ. By taking ` large enough, we can make the probability
of changing the k-prefix as small as desired.
We now proceed with the actual proof.
Proof of Proposition 4.8. Step I. Let (Z̃t )t denote the projected µ-random walk
on the Schreier graph Λ. Assume that  = (x, y), and note that since Λ is µtransient, then
P[∀ t , Z̃t 6= x | Z̃0 = y] + P[∀ t , Z̃t 6= y | Z̃0 = x] > 0.

By possibly changing  to (y, x), without loss of generality we may assume that
β := P[∀ t , Z̃t 6= x | Z̃0 = y] > 0.

Let ` > 0 be large enough, to be chosen below. Set n = k + (` + 2)r + 1. We
now change notation and use (Z̃t )t to denote the projected walk on the Schreier
graph Γn (Λ, ). Denote r = max{|g| : µ(g) > 0}, the maximal possible jump of
the µ-random walk. A key fact we will use throughout is that: in order to change
the k-prefix, the walk must at some point reach a vertex u with |u| ≤ k + r.
Let τ = inf{t : n − r ≤ |Z̃t | ≤ n}. If we start from |Z̃0 | ≤ n this is a.s. finite,
since the jumps cannot be larger than r. Now, by Lemma 4.4, F2 is (k, k + 1, η)fixing for all k and some η = η(µ) > 0. Since Γn (Λ, ) is n-tree-like, if v is a vertex
in Γn (Λ, ) with k < |v| < n, then
(4.1)

P[pref k (Z̃τ ) = pref k (v) | Z̃0 = v] ≥ η

26

YAIR HARTMAN AND ARIEL YADIN

For s ∈ S let ts be the smallest number such that µts (s) > 0. Let tS = max{ts :
s ∈ S}. Let δ = min{µts (s) : s ∈ S}. Let v, u be two adjacent vertices in
Γn (Λ, ), and assume that the label of (v, u) is s ∈ S. Note that the definitions
above ensure that
P[Z̃ts = u | Z̃0 = v] ≥ δ.

Thus, when |v| > k + tS r, with probability at least δ we can move from v to
u in at most tS steps without changing the k-prefix. This holds specifically for
|v| ≥ n − r > k + tS r by our choice of n (as long as ` ≥ tS ).
Consider the graph Γn (Λ, ). Recall that it contains many copies of Λ (glued to
the appropriate leaves). Let Λ1 , . . . , Λm be the list of these copies, and denote by
j = (xj , yj ) the corresponding copies of  in each. Define Y = {y1 , . . . , ym }.
Now, define stopping times
Um = inf{t : |Z̃t | < m}

T = inf{t : Z̃t ∈ Y }

and

(where inf ∅ = ∞). Any vertex v in Γn (Λ, ) of depth |v| < n must have some
copy of (Λ, ) in its shadow shd(v). So there exists some path of length at most
n−|v|+1 from v into some copy of (Λ, ), ending in some vertex in Y . If |v| ≥ n−r
then we can use the strong Markov property at the first time the walk gets to some
vertex u with |u| < n. Since this u must have |u| ≥ n − r, we obtain that for any
|v| ≥ n − r,
P[T < Un−r | Z̃0 = v] ≥ δ r+1 .
By µ-transience of Λ, we have that starting from any yj ∈ Y , with probability at
least β the walk never crosses (yj , xj ), and so never leaves Λj (and thus always
stays at distance at least n from the root). We conclude that for any |v| ≥ n − r,
P[pref k (Z̃∞ ) = pref k (Z̃0 ) | Z̃0 = v] ≥ P[Un−r = ∞ | Z̃0 = v]

≥ P[T < Un−r | Z̃0 = v] · inf P[∀t , Z̃t ∈ Λj | Z̃0 = yj ]
yj ∈Y

≥ δ r+1 β.
(where we have used that k + r < n − r). Combining this with (4.1), using the
strong Markov property at time τ , we obtain that for any |v| > k,
(4.2)

P[pref k (Z̃∞ ) = pref k (v) | Z̃0 = v] ≥ ηδ r+1 β.

Step II. Now, define the events
Aj = {pref k (Z̃∞ ) 6= pref k (Z̃Un−jr )}.
When Z̃0 = v for |v| ≥ n = k + (` + 2)r + 1 the event {pref k (Z̃∞ ) 6= pref k (Z̃0 )}
implies that
U|v|−r < U|v|−2r < · · · < U|v|−`r < Uk+r < ∞.

ENTROPY OF INTERSECTIONAL IRSS

27

But, at time U|v|−jr we have that
|Z̃U|v|−jr | ≥ |Z̃U|v|−jr −1 | − r ≥ |v| − (j + 1)r > k + r.

Thus, we have by (4.2), using the strong Markov property at time U|v|−jr ,
implying that

P[Aj+1 | Z̃0 , . . . , Z̃U|v|−jr ] ≤ 1 − ηδ r+1 β,

P[Aj+1 | (A1 )c ∩ · · · ∩ (Aj )c ] ≤ 1 − ηδ r+1 β.
Thus, as long as |v| ≥ n we have that

P[pref k (Z̃∞ ) 6= pref k (v) | Z̃0 = v] ≤ P[A1 ∪ · · · ∪ A` | Z̃0 = v] ≤ (1 − ηδ r+1 β)` .

Choosing ` such that (1 − ηδ r β)` < ε, we obtain that for any |v| ≥ n
P[pref k (Z̃∞ ) 6= pref k (v) | Z̃0 = v] < ε.

That is, the graph Γn (Λ, ) is (k, n, 1 − ε)-fixing.



Remark 4.9. Proposition 4.8 is the only place we require the measure µ to have
finite support. Indeed, we believe that the proposition should hold with weaker
assumptions on µ, perhaps even only assuming that µ is generating and has finite
entropy. If this is the case, we could obtain full realization results for the free
group for such measures.
4.5. Local properties. Recall from Section 3.1 that given a subgroup K ≤ F2
we define kgkK as the number of cosets of the normalizer b ∈ NG (K)\G for which
g 6∈ K b . Recall also that Core∅ (K) = {g : kgkK < ∞}.
In this subsection we will provide conditions on K under which Core∅ (K) contains a given subgroup of F2 . This will be useful in determining properties of
F2 /Core∅ (K) (specifically, whether F2 /Core∅ (K) is nilpotent).
First, some notation: Let g ∈ F2 . Let g = w1 · · · w|g| be a reduced word with
wi ∈ S for all i. Let Λ be a Schreier graph with corresponding subgroup K ≤ F2 .
Fix a vertex v in Λ. Let v0 (g) = v and inductively define vn+1 (g) to be the unique
vertex of Λ such that (vn (g), vn+1 (g)) is labeled by wn+1 . In other words, if v = Kγ
then vi (g) = Kγw1 · · · wi .
Recall that g ∈ K if and only if for v = o the root of Λ. Hence, for any g ∈ K,
we have v|g| (g) = v0 (g). From this we can deduce that when v = Kγ, then
v|g| (g) = v0 (g) ⇐⇒ γgγ −1 ∈ K ⇐⇒ g ∈ K γ .
Definition 4.10. Let Γ1 , . . . , Γn be some Schreier graphs with roots ρ1 , . . . , ρn .
Let BΛ (v, r) be the ball of radius r around v in the graph Λ (and similarly for
Γj ). We say that Λ (or sometimes K) is locally-(Γ1 , . . . , Γn ) if for any r > 0
there exists R > 0 such that for all vertices v in Λ with |v| > R we have that
the ball BΛ (v, r) is isomorphic (as a labeled graph) to at least one of the balls
BΓ1 (ρ1 , r), . . . , BΓn (ρn , r).

28

YAIR HARTMAN AND ARIEL YADIN

That is, a subgroup K is locally-(Γ1 , . . . , Γn ) if, after ignoring some finite area,
locally we see one of the graphs Γ1 , . . . , Γn .
The main purpose of this subsection is to prove:
Proposition 4.11. Let K ≤ F2 . Let G1 , . . . , Gn ≤ F2 be subgroups with corresponding Schreier graphs Γ1 , . . . , Γn .
If K is locally-(Γ1 , . . . , Γn ) then G1 ∩ · · · ∩ Gn ≤ Core∅ (K).
Proof. Let g ∈ G1 ∩ · · · ∩ Gn . We need to prove that ||g||K < ∞.
Let Λ be the Schreier graph of K. Let r = |g|. Let R be large enough so that for
|v| > R in Λ the ball BΛ (v, r) is isomorphic to one of BΓj (ρj , r) (where as before,
ρj is the root vertex of Γj ).
Let |v| > R and let j be such that BΛ (v, r) is isomorphic to BΓj (ρj , r). Set
Γ = Γj , ρ = ρj .
Consider the paths ρ = ρ0 (g), . . . , ρr (g) in Γ and v = v0 (g), . . . , vr (g) in Λ. Since
these each sit in BΓ (ρ, r) and BΛ (v, r) respectively, we have that ρr (g) = ρ0 (g) if
and only if vr (g) = v0 (g). Thus, g ∈ Gj implies that g ∈ K γ where v = Kγ.
In conclusion, we have shown that if γ is such that |Kγ| > R then g ∈ K γ .
Hence, there are only finitely many γ ∈ F2 such that g 6∈ K γ , which implies that
||g||K < ∞.

4.6. Full realization. In this subsection we prove Theorem 1.2.
In light of Corollary 3.8 and Proposition 4.7, in order to prove full realization
for F2 , we need to find a sequence of subgroups Kn with Schreier graphs Γn such
that the following properties hold:
• Γn is (kn , kn0 , αn )-fixing, with kn → ∞ and αn → 1.
• The subgroups Core∅ (Kn ) are co-nilpotent in F2 .
To show the first property, we will use the gluing construction from Proposition
4.8, by finding a suitable marked pair (Λ, ) that is µ-transient.
Lemma 4.12. Let N be a normal co-nilpotent subgroup in F2 , and let Λ be its
associated Schreier graph. Choose some edge  in Λ labeled with s ∈ S and consider
(Λ, ) as a marked pair. For any n > 0 let Kn ≤ F2 be the subgroup corresponding
to the root of the Shcreier graph Γn (Λ, ).
Then, for any n > 0, the normal subgroup Core∅ (Kn ) is co-nilpotent in F2 .
Proof. Let G0 = F2 , Gj+1 = [Gj , F2 ] be the descending central series of F2 . Since
N is co-nilpotent, there exists m such that Gm C N .
By definition of Γn (Λ, ), outside a ball of radius n in Γn (Λ, ), we only have
glued copies of Λ or of Ns for s ∈ S = {a, a−1 , b, b−1 } (recall Subsection 4.3).
Let Zs denote the Schreier graph with vertices in Z, edges (x + 1, x) labeled by
s and a self loop with label ξ 6∈ {s, s−1 } at each vertex. Let ϕs : F2 → Z be the
homomorphism defined via s 7→ −1, s−1 7→ 1 and ξ 7→ 0 for ξ 6∈ {s, s−1 }. Then,

ENTROPY OF INTERSECTIONAL IRSS

29

the subgroup corresponding to Zs is ker ϕs . Since F2 / ker ϕs is abelian, we have
that Gm C [F2 , F2 ] C ker ϕs .
It is now immediate that Kn is locally-(Λ, Za , Za−1 , Zb , Zb−1 ). By Proposition
4.11, this implies that
\
Gm ≤ N ∩
ker ϕs ≤ Core∅ (Kn ).
s∈S

Thus, Core∅ (Kn ) is co-nilpotent.



We are now ready to prove our main result, Theorem 1.2.
Proof of Theorem 1.2. By Corollary 3.8, Proposition 4.7 and Lemma 4.12 it suffices to find N C F2 such that F2 /N is nilpotent and transient. Since any 2generated nilpotent group is a quotient of F2 , it suffices to find a 2-generated
nilpotent group, whose Cayley graph is transient, and then take N to be the corresponding kernel of the canonical projection from F2 onto our nilpotent group.
It is well known that recurrent subgroups must have at most quadratic volume growth (this follows for instance from combining the Coulhon-Sallof-Coste
inequality [10] and the method of evolving sets [23], see e.g. [26, Chapters 5 &
8]). Thus, it suffices to find a 2-generated nilpotent group that has volume growth
larger than quadratic. This is not difficult, and there are many possibilities.
For a concrete example, we may consider take H3 (Z), the Heisenberghgroupi (over
1 a c
Z). This is the group whose elements are 3 × 3 matrices of the form 0 1 b with
0 0 1
a, b, c ∈ Z. It is well known that H3 (Z) is two-step nilpotent,
with
the
commutator
h
i
1 0 c
subgroup [H3 (Z), H3 (Z)] containing matrices of the form 0 1 0 . It follows that
0 0 1
[[H3 (Z), H3 (Z)], H3 (Z)] = {1} and H3 (Z)/[H3 (Z), H3 (Z)] ∼
= Z2 . This structure
shows that H3 (Z) has volume growth at least like a degree 3 polynomial (in fact
the volume growth is like r4 ), and thus must be transient. h
i
h
i
1 1 0
1 0 0
Finally, H3 (Z) is generated as a group by two elements 0 1 0 and 0 1 1 , so
0 0 1
0 0 1
can be seen as a quotient of F2 , i.e. H3 (Z) ∼
= F2 /N for some N . This N will satisfy
our requirements, and this finishes the proof for F2 .
Let us note that for Fr with r > 2 the construction is even simpler: Since Zr
is abelian and transient for any r > 2, we can just take N to be the commutator
subgroup of Fr .

4.7. Proof of Theorem 1.6.
Proof of Theorem 1.6. Let N be a normal co-nilpotent subgroup in F2 , and let Λ
be its associated Schreier graph. Choose some edge  in Λ labeled with s ∈ S and
consider (Λ, ) as a marked pair. For any n > 0 let Kn ≤ F2 be the subgroup
corresponding to the root of the Shcreier graph Γn (Λ, ), where Γn (Λ, ) is defined
in Section 4.3.

30

YAIR HARTMAN AND ARIEL YADIN

We have already seen in the proof of Lemma 4.12 that Core∅ (Kn ) is co-nilpotent
in F2 , so the entropy function is continuous and the interval of values [0, c] is
realizable for c = hRW (G/CoreG (Kn ), µ̄)], by Corollary 3.6.
So we only need to prove that hRW (G/CoreG (Kn ), µ̄) > 0 for some n. (Of course
we proved that when µ has finite support this converges to the random walk
entropy, but here we are dealing with any finite entropy generating probability
measure µ, not necessarily with finite support.)
By the well known Kaimanovich-Vershik entropy criterion [20], if there exists a
non-constant bounded harmonic function on the Schreier graph Γn (Λ, ), then the
entropy hRW (G/CoreG (Kn ), µ̄) is positive.
If the graph (Λ, ) is transient, then considering the random walk on Γn (Λ, ),
for each glued copy of (Λ, ) the random walk has positive probability to eventually
end up in that copy. In other words, there exists 0 < α < 1 such that for any
|v| = n we have
P[∃ t0 : ∀ t > t0 Z̃t ∈ shd(v)] ≥ α.

Thus, for a fixed |v| = n, the function

h(g) = hv (g) := Pg [∃ t0 : ∀ t > t0 Z̃t ∈ shd(v) | Z̃0 = g]
is a non-constant bounded harmonic function on G.
Finding a transient Schreier graph (Λ, ) with corresponding subgroup N C F2
which is co-nilpotent, can be done exactly as in the proof of Theorem 1.2.

References
[1] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean
Raimbault, and Iddo Samet, On the growth of L2 -invariants for sequences of lattices in Lie
groups, arXiv:1210.2961 (2012).
[2] Miklos Abert, Yair Glasner, and Balint Virag, The measurable Kesten theorem,
arXiv:1111.2080 (2011).
[3] Miklós Abért, Yair Glasner, and Bálint Virág, Kestenś theorem for invariant random subgroups, Duke Mathematical Journal 163 (2014), no. 3, 465–488.
[4] Lewis Bowen, Random walks on random coset spaces with applications to Furstenberg entropy., Inventiones mathematicae 196 (2014), no. 2.
[5]
, Invariant random subgroups of the free group, Groups, Geometry, and Dynamics 9
(2015), no. 3, 891–916.
[6] Lewis Bowen, Rostislav Grigorchuk, and Rostyslav Kravchenko, Invariant random subgroups
of lamplighter groups, Israel Journal of Mathematics 207 (2015), no. 2, 763–782.
[7] Lewis Bowen, Yair Hartman, and Omer Tamuz, Generic stationary measures and actions,
Transactions of the American Mathematical Society (2017).
[8] Peter Burton, Martino Lupini, and Omer Tamuz, Weak equivalence of stationary actions
and the entropy realization problem, arXiv preprint arXiv:1603.05013 (2016).
[9] Gustave Choquet and Jacques Deny, Sur lequation de convolution mu= mu-star-sigma,
Comptes rendus hebdomadaires des seances de l’academie des sciences 250 (1960), no. 5,
799–801.

ENTROPY OF INTERSECTIONAL IRSS

31

[10] Thierry Coulhon and Laurent Saloff-Coste, Isopérimétrie pour les groupes et les variétés,
Rev. Mat. Iberoamericana 9 (1993), no. 2, 293–314.
[11] Thomas M Cover and Joy A Thomas, Elements of information theory, John Wiley & Sons,
2012.
[12] Alex Furman, Random walks on groups and random transformations, Handbook of dynamical systems 1 (2002), 931–1014.
[13] H. Furstenberg, Random walks and discrete subgroups of Lie groups, Advances in Probability
and Related Topics 1 (1971), 1–63.
[14] H Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Harmonic
analysis on homogeneous spaces 26 (1973), 193–229.
[15] H. Furstenberg and E. Glasner, Stationary dynamical systems, Dynamical numbers—
interplay between dynamical systems and number theory, 2010, pp. 1–28. MR2762131
[16] Yair Hartman and Omer Tamuz, Furstenberg entropy realizations for virtually free groups
and lamplighter groups, Journal d’Analyse Mathématique 126 (2015), no. 1, 227–257.
, Stabilizer rigidity in irreducible group actions, Israel Journal of Mathematics 216
[17]
(2016), no. 2, 679–705.
[18] Vadim A Kaimanovich, The poisson boundary of amenable extensions, Monatshefte für
Mathematik 136 (2002), no. 1, 9–15.
, Amenability and the liouville property, Israel Journal of Mathematics 149 (2005),
[19]
no. 1, 45–85.
[20] Vadim A Kaimanovich and Anatoly M Vershik, Random walks on discrete groups: boundary
and entropy, The annals of probability (1983), 457–490.
[21] Paul-Henry Leemann, Schreir graphs: Transitivity and coverings, International Journal of
Algebra and Computation 26 (2016), no. 01, 69–93.
[22] Russell Lyons and Yuval Peres, Probability on trees and networks, Vol. 42, Cambridge University Press, 2016.
[23] Ben Morris and Yuval Peres, Evolving sets, mixing and heat kernel bounds, Probability
Theory and Related Fields 133 (2005), no. 2, 245–266.
[24] Amos Nevo, The spectral theory of amenable actions and invariants of discrete groups,
Geometriae Dedicata 100 (2003), no. 1, 187–218.
[25] Amos Nevo and Robert J Zimmer, Rigidity of Furstenberg entropy for semisimple Lie group
actions, Annales scientifiques de l’ecole normale supérieure, 2000, pp. 321–343.
[26] Gabor Pete, Probability and geometry on groups. available at: http://math.bme.hu/
~gabor/PGG.pdf.
[27] Albert Raugi, A general Choquet–Deny theorem for nilpotent groups, Annales de l’IHP
probabilités et statistiques 40 (2004), no. 6, 677–683.
[28] Anatolii Moiseevich Vershik, Totally nonfree actions and the infinite symmetric group, Mosc.
Math. J 12 (2012), no. 1, 193–212.
YH: Northwestern University, Evanston, IL
E-mail address: hartman@math.northwestern.edu
AY: Ben-Gurion University of the Negev, Be’er Sheva ISRAEL
E-mail address: yadina@bgu.ac.il

