2.1 Stationary spaces

Throughout the paper we assume that $G$ is a discrete countable group and that $\unicode[STIX]{x1D707}$ is a generating probability measure on $G$ .

A measurable action $G\curvearrowright (X,\unicode[STIX]{x1D708})$ where $(X,\unicode[STIX]{x1D708})$ is a standard probability spaces is called $(G,\unicode[STIX]{x1D707})$ -stationary if $\sum _{g\in G}\unicode[STIX]{x1D707}(g)g\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}$ . The basic example of non-invariant $(G,\unicode[STIX]{x1D707})$ -stationary action is the action on the Furstenberg–Poisson boundary, $\unicode[STIX]{x1D6F1}(G,\unicode[STIX]{x1D707})$ (the amount of literature on the Furstenberg–Poisson boundary is vast, and we do not attempt to give a comprehensive overview of this topic; see, for example, [Reference FurstenbergFur73, Reference FurstenbergFur71, Reference Furstenberg and GlasnerFG10, Reference FurmanFur02] and references therein for more on this topic).

The Kakutani fixed point theorem implies that whenever $G$ acts continuously on a compact space $X$ , there is always some $\unicode[STIX]{x1D707}$ -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 as follows. Denote by $\text{Sub}_{G}$ 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, $\text{Sub}_{G}$ is a compact space, with a natural right action by conjugation ( $K^{g}=g^{-1}Kg$ where $K\in \text{Sub}_{G}$ ), which is continuous. An Invariant Random Subgroup (IRS) is a Borel probability measure $\unicode[STIX]{x1D706}$ on $\text{Sub}_{G}$ 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 $\unicode[STIX]{x1D706}$ . For more background regarding IRSs we refer the interested reader to [Reference Abert, Glasner and ViragAGV16, Reference Abért, Glasner and VirágAGV14].

Fix a probability measure $\unicode[STIX]{x1D707}$ on $G$ . Given an IRS $\unicode[STIX]{x1D706}$ of $G$ , one can consider the Poisson bundle over $\unicode[STIX]{x1D706}$ , denoted by $B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706})$ . This is a probability space with a $(G,\unicode[STIX]{x1D707})$ stationary action, and when $\unicode[STIX]{x1D706}$ is an ergodic IRS, the action on $B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706})$ is ergodic.

Poisson bundles were introduced in a greater generality by Kaimanovich in [Reference KaimanovichKai05]. In [Reference BowenBow14] 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 [Reference Hartman and TamuzHT16, Reference BowenBow14, Reference Hartman and TamuzHT15, Reference KaimanovichKai05], 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.

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 $G=\mathbb{F}_{2}$ the free group on two generators, generated by $a$ and $b$ . The generalization to higher rank is straightforward.

Let $K\in \text{Sub}_{G}$ . 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 $K\backslash G$ and the set of edges is $\{(Kg,Kga),(Kg,Kgb):Kg\in K\backslash G\}$ . 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 $\unicode[STIX]{x1D6E4}$ one can recover the subgroup $K$ as follows. Given any element $g=w_{1}\cdots w_{n}$ , of the free group (where $w_{1}\cdots w_{n}$ 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 $w_{i}\in \{a,b\}$ move from $v$ to the adjacent vertex $u=v.w_{i}$ such that the oriented edge $(v,u)$ is labeled by $w_{i}$ .

• ∙ Given $w_{i}\in \{a^{-1},b^{-1}\}$ move from $v$ to the adjacent vertex $u=v.w_{i}$ such that the oriented edge $(u,v)$ is labeled $w_{i}^{-1}$ (that is, follow the label $w_{i}$ using the reverse orientation).

• ∙ Given $g=w_{1}\cdots w_{n}$ written as a reduced word in the generators, apply the generators one at a time, to get $v.g=(\cdots ((v.w_{1}).w_{2})\ldots ).w_{n}$ .

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 $K\leqslant \mathbb{F}_{2}$ from a graph $\unicode[STIX]{x1D6E4}$ 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 $N_{G}(K)=\{g\in G:K^{g}=K\}$ .

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 Schreier graphs to construct subgroups is more adapted to our purpose.

2.3 Random walks on Schreier graphs

Fix a generating probability measure $\unicode[STIX]{x1D707}$ on a group $G$ . Let $(X_{t})_{t=1}^{\infty }$ be independent and identically distributed (i.i.d.) random variables taking values in $G$ with law $\unicode[STIX]{x1D707}$ , and let $Z_{t}=X_{1}\cdots X_{t}$ be the position of the $\unicode[STIX]{x1D707}$ -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 $G=\mathbb{F}_{2}$ .

The Markov chain $(KZ_{t})_{t}$ has a natural representation on the Schreier graph associated with $K$ , by just considering the position on the graph at time $t$ to be $v.Z_{t}$ 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 $KZ_{t}$ 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 $KZ_{t}$ where $K$ is a random subgroup distributed according to $\unicode[STIX]{x1D706}$ . This perspective was our intuition behind the construction in § 4.

2.4 Random walk entropy

Recall that we assume throughout that all random walk measures $\unicode[STIX]{x1D707}$ have finite entropy, that is $H(Z_{1})=-\sum _{g}\mathbb{P}[Z_{1}=g]\log (\mathbb{P}[Z_{1}=g])<\infty$ where $Z_{t}$ as usual is the position of the $\unicode[STIX]{x1D707}$ -random walk at time $t$ . The sequence of numbers $H(Z_{t})$ is a sub-additive sequence so the following limit exists and finite.

Definition 2.1. The random walk entropy of $(G,\unicode[STIX]{x1D707})$ is given by

$$\begin{eqnarray}h_{RW}(G,\unicode[STIX]{x1D707})=\lim _{t\rightarrow \infty }\frac{1}{t}H(Z_{t}).\end{eqnarray}$$

The Furstenberg-entropy of a $(G,\unicode[STIX]{x1D707})$ -stationary space, $(X,\unicode[STIX]{x1D708})$ is given by

$$\begin{eqnarray}h_{\unicode[STIX]{x1D707}}(X,\unicode[STIX]{x1D708})=-\mathop{\sum }_{g\in G}\unicode[STIX]{x1D707}(g)\int _{X}\log \frac{dg^{-1}\unicode[STIX]{x1D708}}{d\unicode[STIX]{x1D708}}\,d\unicode[STIX]{x1D708}(x).\end{eqnarray}$$

By Jensen’s inequality, $h_{\unicode[STIX]{x1D707}}(X,\unicode[STIX]{x1D708})\geqslant 0$ and equality holds if and only if the action is a measure preserving action. Furthermore, Kaimanovich and Vershik proved in [Reference Kaimanovich and VershikKV83] that $h_{\unicode[STIX]{x1D707}}(X,\unicode[STIX]{x1D708})\leqslant h_{RW}(G,\unicode[STIX]{x1D707})$ for any stationary action $(X,\unicode[STIX]{x1D708})$ , and equality holds for the Furstenberg–Poisson boundary, namely

(1) $$\begin{eqnarray}\displaystyle h_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6F1}(G,\unicode[STIX]{x1D707}))=h_{RW}(G,\unicode[STIX]{x1D707}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}(G,\unicode[STIX]{x1D707})$ denotes the Furstenberg–Poisson boundary of $(G,\unicode[STIX]{x1D707})$ .

If $h_{RW}(G,\unicode[STIX]{x1D707})=0$ the pair $(G,\unicode[STIX]{x1D707})$ 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 $[0,h_{RW}(G,\unicode[STIX]{x1D707})]$ which are realized as the Furstenberg entropy of ergodic stationary actions.

In a similar way to the Kaimanovich–Vershik formula (1), Bowen proves the following.

Theorem 2.2 (Bowen, [Reference BowenBow14]).

Let $\unicode[STIX]{x1D706}$ be an IRS of $G$ . Then

(2) $$\begin{eqnarray}\displaystyle h_{\unicode[STIX]{x1D707}}(B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706}))=\lim _{t\rightarrow \infty }\frac{1}{t}\int _{\text{Sub}_{G}}H(KZ_{t})\,d\unicode[STIX]{x1D706}(K)=\inf _{t}\frac{1}{t}\int _{\text{Sub}_{G}}H(KZ_{t})\,d\unicode[STIX]{x1D706}(K). & & \displaystyle\end{eqnarray}$$

Moreover, the $(G,\unicode[STIX]{x1D707})$ -stationary space is ergodic if (and only if) the IRS $\unicode[STIX]{x1D706}$ is ergodic.

This reduces the problem of finding $(G,\unicode[STIX]{x1D707})$ -stationary actions with many Furstenberg-entropy values in $[0,h_{RW}(G,\unicode[STIX]{x1D707})]$ , to finding IRSs $\unicode[STIX]{x1D706}$ with many different random walk entropy values.

Regarding continuity, recall that the space $\text{Sub}_{G}$ 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 as a function on $\text{Sub}_{G}$ , namely $K\mapsto H(KZ_{1})$ , is a continuous function.

The topology on $\text{Sub}_{G}$ induces a weak* topology on, $\text{Prob}(\text{Sub}_{G})$ , the space of IRSs of $G$ . By the continuity of $H$ and the second equality in (2) we get that the Furstenberg-entropy of Poisson bundles of the form $B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706})$ is an upper semi-continuous function of the IRS. Since the entropy is non-negative we conclude the following.

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 $N_{n}$ , such that $N_{n}\backslash \mathbb{F}_{2}$ is finite for any $n$ , which can be chosen so that, when considered as IRSs $\unicode[STIX]{x1D706}_{n}=\unicode[STIX]{x1D6FF}_{N_{n}}$ , we have $\unicode[STIX]{x1D706}_{n}\rightarrow \unicode[STIX]{x1D6FF}_{\{e\}}$ . In that case $h_{\unicode[STIX]{x1D707}}(B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706}_{n}))=0$ for all $n$ (because $H(N_{n}Z_{t})$ is bounded in $t$ ), but $B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6FF}_{\{e\}})$ is the Furstenberg–Poisson boundary of the free group, and its Furstenberg entropy is positive for any $\unicode[STIX]{x1D707}$ (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 § 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.

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 later, we quote it here for future reference. The original case of Abelian groups is due to Choquet and Deny [Reference Choquet and DenyCD60] and the nilpotent case was done by Raugi and appears in [Reference RaugiRau04].

We mention that the triviality of the Furstenberg–Poisson boundary holds for general $\unicode[STIX]{x1D707}$ without any entropy assumption, although in this paper we consider only $\unicode[STIX]{x1D707}$ 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.

3.1 The subgroup $\text{Core}_{\emptyset }(K)$

For any element $g\in G$ , let $\unicode[STIX]{x1D6FA}_{g}\subset N_{G}(K)\backslash G$ be the set of all cosets, such that $g$ belongs to the corresponding conjugation: $\unicode[STIX]{x1D6FA}_{g}=\{\unicode[STIX]{x1D703}\in N_{G}(K)\backslash G\mid g\in K^{\unicode[STIX]{x1D703}}\}$ . For example, for $g\in K$ we have that $N_{G}(K)\in \unicode[STIX]{x1D6FA}_{g}$ .

We observe that:

1. (i) $\unicode[STIX]{x1D6FA}_{g}\cap \unicode[STIX]{x1D6FA}_{h}\subset \unicode[STIX]{x1D6FA}_{gh}$ ;

2. (ii) $\unicode[STIX]{x1D6FA}_{g^{-1}}=\unicode[STIX]{x1D6FA}_{g}$ ;

3. (iii) $\unicode[STIX]{x1D6FA}_{g^{\unicode[STIX]{x1D6FE}}}=\unicode[STIX]{x1D6FA}_{g}\unicode[STIX]{x1D6FE}^{-1}$ .

The first two follow from the fact that $K^{\unicode[STIX]{x1D703}}$ is a group and the third since $\unicode[STIX]{x1D6FE}K^{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D6FE}^{-1}=K^{\unicode[STIX]{x1D703}\unicode[STIX]{x1D6FE}^{-1}}$ .

Let ${\mho}_{g}$ be the complement of $\unicode[STIX]{x1D6FA}_{g}$ in $N_{G}(K)\backslash G$ . That is, ${\mho}_{g}=\{\unicode[STIX]{x1D703}\in N_{G}(K)\backslash G:\unicode[STIX]{x1D703}\not \in \unicode[STIX]{x1D6FA}_{g}\}$ . Define $\Vert g\Vert _{K}=|{\mho}_{g}|$ . The properties above show that:

1. (i) $\Vert gh\Vert _{K}\leqslant \Vert g\Vert _{K}+\Vert h\Vert _{K}$ ;

2. (ii) $\Vert g^{-1}\Vert _{K}=\Vert g\Vert _{K}$ ;

3. (iii) $\Vert g^{\unicode[STIX]{x1D6FE}}\Vert _{K}=\Vert g\Vert _{K}$ .

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 (i) and (ii) above, these are indeed subgroups and normality follows from (iii).

The first subgroup is, by definition, the normal core $\text{Core}_{G}(K)=\{g\mid \Vert g\Vert _{K}=0\}$ . We claim that the second subgroup is the appropriate definition for $\text{Core}_{\emptyset }(K)$ . That is, we define $\text{Core}_{\emptyset }(K)=\{g\mid \Vert g\Vert _{K}<\infty \}$ and prove that $\unicode[STIX]{x1D706}_{p,K}\rightarrow \unicode[STIX]{x1D6FF}_{\text{Core}_{\emptyset }(K)}$ as $p\rightarrow 0$ . Note that $\text{Core}_{\emptyset }(K)$ is precisely the set of all elements $g$ that appear in every conjugate of $K$ except for finitely many.

Proof of Lemma 3.2.

By definition, to show that $\unicode[STIX]{x1D706}_{p}\xrightarrow[{}]{p\rightarrow 0}\unicode[STIX]{x1D6FF}_{\text{Core}_{\emptyset }(K)}$ we need to show that for any $g\in \text{Core}_{\emptyset }(K)$ , we have $\unicode[STIX]{x1D706}_{p}(\{F\in \text{Sub}_{G}\mid g\in F\})\xrightarrow[{}]{p\rightarrow 0}1$ and for any $g\notin \text{Core}_{\emptyset }(K)$ , we have $\unicode[STIX]{x1D706}_{p}(\{F\in \text{Sub}_{G}\mid g\in F\})\xrightarrow[{}]{p\rightarrow 0}0$ .

Note that for any $\unicode[STIX]{x1D6E9}\neq \emptyset$ , by the definition of $\text{Core}_{\unicode[STIX]{x1D6E9}}(K)$ , we have that $g\in \text{Core}_{\unicode[STIX]{x1D6E9}}(K)$ if and only if $\unicode[STIX]{x1D6E9}\cap {\mho}_{g}=\emptyset$ .

Now consider $\unicode[STIX]{x1D6E9}$ to be a random subset according to the Bernoulli- $p$ product measure on $2^{N_{G}(K)\backslash G}$ . By the above, we have equality of the events $\{g\in \text{Core}_{\unicode[STIX]{x1D6E9}}(K)\}=\{{\mho}_{g}\cap \unicode[STIX]{x1D6E9}=\emptyset \}$ . The latter (and hence also the former) has probability $(1-p)^{|{\mho}_{g}|}=(1-p)^{\Vert g\Vert _{K}}$ . This includes the case where $|{\mho}_{g}|=\Vert g\Vert _{K}=\infty$ (whence we have $0$ probability for $\{{\mho}_{g}\cap \unicode[STIX]{x1D6E9}=\emptyset \}$ ). We conclude that for any $g\in G$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{p}(\{F\in \text{Sub}_{G}\mid g\in F\})=(1-p)^{\Vert g\Vert _{K}}=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }\Vert g\Vert _{K}=\infty ,\\ \rightarrow 1\text{ as }p\rightarrow 0\quad & \text{if }\Vert g\Vert _{K}<\infty .\end{array}\right.\end{eqnarray}$$

It follows that $\unicode[STIX]{x1D706}_{p}\rightarrow \unicode[STIX]{x1D6FF}_{\text{Core}_{\emptyset }(K)}$ as $p\rightarrow 0$ .

Finally, we show that $\unicode[STIX]{x1D706}_{p}\rightarrow \unicode[STIX]{x1D6FF}_{\text{Core}_{G}(K)}$ as $p\rightarrow 1$ , where we define $\text{Core}_{G}(K)=\bigcap _{g}K^{g}=\text{Core}_{N_{G}(K)\backslash G}(K)$ . Clearly, for any $p\in (0,1)$ , we have that $\text{Core}_{G}(K)\subset F$ for $\unicode[STIX]{x1D706}_{p}$ -almost every $F$ , since $\text{Core}_{G}(K)\leqslant \text{Core}_{\unicode[STIX]{x1D6E9}}(K)$ for any non-empty subset $\unicode[STIX]{x1D6E9}$ .

On the other hand, fix some $g\not \in \text{Core}_{G}(K)$ . By definition, there exists some $\unicode[STIX]{x1D703}\in N_{G}(K)\backslash G$ such that $g\not \in K^{\unicode[STIX]{x1D703}}$ . Whenever $\unicode[STIX]{x1D6E9}$ is such that $\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E9}$ , we have that $g\not \in \text{Core}_{\unicode[STIX]{x1D6E9}}(K)$ . The probability that $\unicode[STIX]{x1D703}\not \in \unicode[STIX]{x1D6E9}$ is $(1-p)$ . Hence, for any $g$ with $\Vert g\Vert _{K}>0$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{p}(\{F\in \text{Sub}_{G}\mid g\in F\})\leqslant (1-p)\xrightarrow[{}]{p\rightarrow 1}0.\end{eqnarray}$$

We conclude that $\unicode[STIX]{x1D706}_{p}\rightarrow \unicode[STIX]{x1D6FF}_{\text{Core}_{G}(K)}$ as $p\rightarrow 1$ .◻

3.2 Continuity of the entropy along intersectional IRSs

Lemma 3.2 above shows that the IRSs $\unicode[STIX]{x1D706}_{p,K}$ interpolate between the two Dirac measures concentrated on the normal subgroups $\text{Core}_{\emptyset }(K)$ and $\text{Core}_{G}(K)$ . In this section we provide a condition under which this interpolation holds for the random walk entropy as well.

Fix a random walk $\unicode[STIX]{x1D707}$ on $G$ with finite entropy. Let $h:[0,1]\rightarrow \mathbb{R}_{+}$ be the entropy map defined by $h(p)=h_{\unicode[STIX]{x1D707}}(B_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706}_{p}))$ when $p\in (0,1)$ and $h(0),h(1)$ defined as the $\unicode[STIX]{x1D707}$ -random walk entropies on the quotient groups $G/\text{Core}_{\emptyset }(K)$ and $G/\text{Core}_{G}(K)$ respectively. In particular, the condition that $h(0)=0$ is equivalent to the Liouville property holding for the projected $\unicode[STIX]{x1D707}$ -random walk on $G/\text{Core}_{\emptyset }(K)$ .

Proof. Let

$$\begin{eqnarray}H_{n}(p)=\int _{\text{Sub}_{G}}H(FZ_{n})\,d\unicode[STIX]{x1D706}_{p}(F).\end{eqnarray}$$

So $h(p)=\lim _{n\rightarrow \infty }(1/n)H_{n}(p)$ . First, we use a standard coupling argument to prove that when $q=p+\unicode[STIX]{x1D700}$ , we have

(3) $$\begin{eqnarray}0\leqslant H_{n}(q)-H_{n}(p)\leqslant H_{n}(\unicode[STIX]{x1D700}).\end{eqnarray}$$

Indeed, let $(U_{\unicode[STIX]{x1D703}})_{\unicode[STIX]{x1D703}\in N_{G}(K)\backslash G}$ be i.i.d. uniform- $[0,1]$ random variables, one for each coset of $N_{G}(K)$ . We use $\mathbb{P},\mathbb{E}$ to denote the probability measure and expectation with respect to these random variables. Define three subsets

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{p}:=\{\unicode[STIX]{x1D703}:U_{\unicode[STIX]{x1D703}}\leqslant p\}\quad \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}:=\{\unicode[STIX]{x1D703}:p<U_{\unicode[STIX]{x1D703}}\leqslant q\}\end{eqnarray}$$

and $\unicode[STIX]{x1D6E9}_{q}:=\unicode[STIX]{x1D6E9}_{p}\cup \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}$ . We shorthand $C_{x}:=\text{Core}_{\unicode[STIX]{x1D6E9}_{x}}(K)$ for $x\in \{p,\unicode[STIX]{x1D700},q\}$ . Note that for $x\in \{p,\unicode[STIX]{x1D700},q\}$ the law of $C_{x}$ is $\unicode[STIX]{x1D706}_{x}$ . Thus, $H_{n}(x)=\mathbb{E}[H(C_{x}Z_{n})]$ (to be clear, the entropy is with respect to the random walk $Z_{n}$ , and expectation is with respect to the random subgroup $C_{x}$ ).

Now, by definition

$$\begin{eqnarray}C_{q}=\mathop{\bigcap }_{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E9}_{q}}K^{\unicode[STIX]{x1D703}}=\mathop{\bigcap }_{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E9}_{p}}K^{\unicode[STIX]{x1D703}}\cap \mathop{\bigcap }_{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}}K^{\unicode[STIX]{x1D703}}=C_{p}\cap C_{\unicode[STIX]{x1D700}}.\end{eqnarray}$$

Since the map $(F_{1}g,F_{2}g)\mapsto (F_{1}\cap F_{2})g$ is well defined, we have that $C_{q}Z_{n}$ is determined by the pair $(C_{p}Z_{n},C_{\unicode[STIX]{x1D700}}Z_{n})$ . Also, since $C_{q}\leqslant C_{p}$ , we have that $C_{p}Z_{n}$ is determined by $C_{q}Z_{n}$ . Thus, $\mathbb{P}$ -a.s.,

$$\begin{eqnarray}H(C_{p}Z_{n})\leqslant H(C_{q}Z_{n})\leqslant H(C_{p}Z_{n})+H(C_{\unicode[STIX]{x1D700}}Z_{n}).\end{eqnarray}$$

Taking expectation (under $\mathbb{E}$ ) we obtain (3), which then immediately leads to $0\leqslant h(q)-h(p)\leqslant h(\unicode[STIX]{x1D700})$ .

Corollary 2.3 asserts that the entropy function is upper semi-continuous. Thus, $\limsup _{\unicode[STIX]{x1D700}\rightarrow 0}$ $h(\unicode[STIX]{x1D700})\leqslant h(0)$ . Thus, if $h(0)=0$ we have that $\lim _{\unicode[STIX]{x1D700}\rightarrow 0}h(\unicode[STIX]{x1D700})=0$ , and that the entropy function is continuous.

3.3 Applications to entropy realizations

Let $\unicode[STIX]{x1D707}$ be a probability measure on a group $G$ and $N\lhd G$ a normal subgroup. We use $\bar{\unicode[STIX]{x1D707}}$ to denote the projected measure on the quotient group $G/N$ .

In the above corollary, the subgroup $K$ may depend on the measure $\unicode[STIX]{x1D707}$ . However, it is possible in some cases to find a subgroup $K$ that provides realization for any measure $\unicode[STIX]{x1D707}$ , and we now discuss alternative conditions for the corollary.

The conditions in Corollary 3.4 implies that $\text{Core}_{\emptyset }(K)$ is a ‘large’ subgroup and $\text{Core}_{G}(K)$ is ‘small’, at least in the sense of the random walk entropy. The best scenario hence is when $\text{Core}_{\emptyset }(K)=G$ and $\text{Core}_{G}(K)=\{1\}$ . The prototypical example is the following.

Note that it is not clear that $h_{RW}(\text{Sym}_{\ast }(X),\unicode[STIX]{x1D707})\,>\,0$ , and it may very well be that $(\text{Sym}_{\ast }(X),\unicode[STIX]{x1D707})$ is Liouville for any $\unicode[STIX]{x1D707}$ , making this realization result trivial. However, it is possible to obtain non-trivial examples by modifying the group slightly. $\text{Sym}_{\ast }(X)$ is a countable group which is not finitely generated. By adding some elements to it, we can get a finitely generated group. If this is done properly, as we explain below in § 3.4, one obtains a full realization result for any measure on the group.

To continue our discussion, we want to weaken the condition $\text{Core}_{\emptyset }(K)=G$ to some algebraic condition that will still guarantee that $h_{RW}(G/\text{Core}_{\emptyset }(K),\bar{\unicode[STIX]{x1D707}})=0$ for any $\unicode[STIX]{x1D707}$ .

Given a subgroup $K$ (or a conjugacy class $K^{G}$ ) we say that $K$ is locally co-nilpotent in $G$ if $G/\text{Core}_{\emptyset }(K)$ is nilpotent. (The reason for this name will become more apparent in § 4.5.) With this definition, Proposition 1.5 follows immediately, as may be seen by the following.

In § 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 have the following.

The subgroups of the free group that we will consider in § 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.

3.4 Examples: lamplighter groups and permutation groups

We are now ready to apply our tools to get Theorems 1.3 and 1.4.

Proof of Theorem 1.3.

Choose the subgroup $K=\bigoplus _{B\backslash \{e\}}L\rtimes \{e\}$ where $e$ is the trivial element in the base group $B$ . It may be simply verified that $\text{Core}_{G}(K)=\{e\}$ and $\bigoplus _{B}L\rtimes \{e\}{\vartriangleleft}\text{Core}_{\emptyset }(K)$ .

Whenever $(B,\bar{\unicode[STIX]{x1D707}})$ is Liouville, so is $(G/\text{Core}_{\emptyset }(K),\bar{\unicode[STIX]{x1D707}})$ . This is because $B\cong G/(\bigoplus _{B}L\rtimes \{e\})$ , so $G/\text{Core}_{\emptyset }(K)$ is a quotient group of $B$ . Thus, we have that $h_{RW}(G/\text{Core}_{\emptyset }(K),\bar{\unicode[STIX]{x1D707}})=0$ , and we obtain a realization of the full interval

$$\begin{eqnarray}[0,h_{RW}(G/\text{Core}_{G}(K),\bar{\unicode[STIX]{x1D707}})]=[0,h_{RW}(G,\unicode[STIX]{x1D707})].\square\end{eqnarray}$$

We now turn to discuss extensions of the group of finitely supported permutations. Let $X$ be a countable set, and denote by $\text{Sym}(X)$ the group of all permutations of  $X$ . Recall that $\text{Sym}_{\ast }(X)$ is its subgroup of all finitely supported permutations. While $\text{Sym}_{\ast }(X)$ is not finitely generated, one can add elements from $\text{Sym}(X)$ to get a finitely generated group. A standard example is when adding a ‘shift’ element $\unicode[STIX]{x1D70E}\in \text{Sym}(X)$ , that is, $\unicode[STIX]{x1D70E}$ acts as a free transitive permutation of $X$ . Let $\unicode[STIX]{x1D6F4}=\left\langle \unicode[STIX]{x1D70E}\right\rangle$ be the subgroup generated by $\unicode[STIX]{x1D70E}$ . Formally the element $\unicode[STIX]{x1D70E}$ acts on $\text{Sym}_{\ast }(X)$ by conjugation (recall that both groups are subgroups of $\text{Sym}(X)$ ) and we get a finitely generated group $\text{Sym}_{\ast }(X)\rtimes \unicode[STIX]{x1D6F4}$ .

One can replace $\unicode[STIX]{x1D6F4}$ by any other subgroup of $\text{Sym}(X)$ . The reason that we want to replace $\unicode[STIX]{x1D6F4}$ by a ‘larger’ group is to have a non-Liouville group $\text{Sym}_{\ast }(X)\rtimes \unicode[STIX]{x1D6F4}$ .

The following is a generalization of Theorem 1.4.

4.1 Schreier graphs

Recall our notations for Schreier graphs from § 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 $\unicode[STIX]{x1D6EC}$ be a Schreier graph with root $o$ .

For a vertex $v$ in $\unicode[STIX]{x1D6EC}$ , the shadow of $v$ , denoted $\text{shd}(v)$ is the set of vertices $u$ in $\unicode[STIX]{x1D6EC}$ such that any path from $u$ to the root $o$ must pass through $v$ .

For an integer $n$ define the finite subtree $(\mathbb{F}_{2})_{{\leqslant}n}$ to be the induced (labeled) subgraph on $\{g\in \mathbb{F}_{2}:|g|\leqslant n\}$ . We say that $\unicode[STIX]{x1D6EC}$ is $n$ -tree-like if the ball of radius $n$ about $o$ in $\unicode[STIX]{x1D6EC}$ is isomorphic to $(\mathbb{F}_{2})_{{\leqslant}n}$ , 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.

For an $n$ -tree-like $\unicode[STIX]{x1D6EC}$ , since every 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 $\unicode[STIX]{x1D6EC}$ with $|v|>n$ , and $k\leqslant n$ we define the $k$ -prefix of $v$ in $\unicode[STIX]{x1D6EC}$ to be $\text{pref}_{k}(v)=\text{pref}_{k,\unicode[STIX]{x1D6EC}}(v)=u$ for the unique $u$ such that both $|u|=k$ and $v\in \text{shd}(u)$ . If $|v|\leqslant k$ then for concreteness we define $\text{pref}_{k}(v)=v$ .

Since $n$ -tree-like graphs look like $\mathbb{F}_{2}$ up to distance $n$ , the behavior of $\text{pref}_{k}$ on $n$ -tree-like graphs where $n\geqslant k$ is the same as on $\mathbb{F}_{2}$ . This is the content of the next elementary lemma, whose proof is left as an exercise.

Lemma 4.2. Let $\unicode[STIX]{x1D6EC}$ be a $n$ -tree-like Schreier graph. Let $\unicode[STIX]{x1D711}:\unicode[STIX]{x1D6EC}\rightarrow \mathbb{F}_{2}$ be a function mapping the ball of radius $n$ in $\unicode[STIX]{x1D6EC}$ isomorphically to $(\mathbb{F}_{2})_{{\leqslant}n}$ .

Then, for any vertex $|v|\leqslant n$ in $\unicode[STIX]{x1D6EC}$ we have that

$$\begin{eqnarray}\text{pref}_{k,\unicode[STIX]{x1D6EC}}(v)=\unicode[STIX]{x1D711}^{-1}\,\text{pref}_{k,\mathbb{F}_{2}}(\unicode[STIX]{x1D711}(v)).\end{eqnarray}$$

We use $(Z_{t})_{t}$ to denote the $\unicode[STIX]{x1D707}$ -random walk on $\mathbb{F}_{2}$ and $(\tilde{Z}_{t})_{t}$ the projected walk on a Schreier graph $\unicode[STIX]{x1D6EC}$ with root $o$ . (So $\tilde{Z}_{t}=KZ_{t}$ where $K$ is the associated subgroup to the root $o$ in the Schreier graph $\unicode[STIX]{x1D6EC}$ .)

Another notion we require is the $k$ -prefix at infinity. Define the random variable $\text{pref}_{k}(\tilde{Z}_{\infty })=\text{pref}_{k,\unicode[STIX]{x1D6EC}}(\tilde{Z}_{\infty })$ , taking values in the set of vertices of distance $k$ from $o$ in $\unicode[STIX]{x1D6EC}$ , as follows. If the sequence $\text{pref}_{k}(\tilde{Z}_{t})_{t}$ stabilizes at $u$ , that is, if there exists $t_{0}$ such that for all $t>t_{0}$ we have $\text{pref}_{k}(\tilde{Z}_{t})=\text{pref}_{k}(\tilde{Z}_{t-1})=u$ , then define $\text{pref}_{k}(\tilde{Z}_{\infty }):=u$ . Otherwise, define $\text{pref}_{k}(\tilde{Z}_{\infty })=o$ . Note that for an $n$ -tree-like graph with $n>0$ we have that $\text{pref}_{k}(\tilde{Z}_{\infty })=o$ (almost surely) if and only if the walk $(\tilde{Z}_{t})_{t}$ returns to the ball of radius $k$ infinitely many times.

Definition 4.3. Let $0<k<n$ be natural numbers and $\unicode[STIX]{x1D6FC}\in (0,1)$ . We say that $\unicode[STIX]{x1D6EC}$ is $(k,n,\unicode[STIX]{x1D6FC})$ -fixing if $\unicode[STIX]{x1D6EC}$ is $n$ -tree-like and for any $|v|\geqslant n$ ,

$$\begin{eqnarray}\mathbb{P}[\forall t,\text{pref}_{k}(\tilde{Z}_{t})=\text{pref}_{k}(\tilde{Z}_{0})\mid \tilde{Z}_{0}=v]\geqslant \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

That is, with probability at least $\unicode[STIX]{x1D6FC}$ , for any $|v|\geqslant n$ , the projected random walk started at $v$ never leaves $\text{shd}(\text{pref}_{k}(v))$ . Hence, the random walk started at depth $n$ in the graph fixes the $k$ -prefix with probability at least $\unicode[STIX]{x1D6FC}$ . Note that this definition depends on the random walk $\unicode[STIX]{x1D707}$ .

A good example of a fixing graph is the Cayley graph of the free group itself.

Proof. It is well known that a $\unicode[STIX]{x1D707}$ -random walk $(Z_{t})_{t}$ on $\mathbb{F}_{2}$ is transient (for example, this follows from non-amenability of $\mathbb{F}_{2}$ ). Let $r=\max \{|g|:\unicode[STIX]{x1D707}(g)>0\}$ be the maximal jump possible by the $\unicode[STIX]{x1D707}$ -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 $|Z_{t}|>k+r$ for all $t>t_{0}$ , we have that $\text{pref}_{k}(Z_{t})=\text{pref}_{k}(Z_{t_{0}})$ for all $t>t_{0}$ .

By forcing finitely many initial steps (depending on $r,\unicode[STIX]{x1D707}$ ), there exists $t_{0}$ and $\unicode[STIX]{x1D6FC}>0$ such that $\mathbb{P}[A\mid |Z_{0}|=k+1]\geqslant \unicode[STIX]{x1D6FC}$ where

$$\begin{eqnarray}A=\{\forall 0<t\leqslant t_{0},\text{pref}_{k}(Z_{t})=\text{pref}_{k}(Z_{0})\text{ and }|Z_{t_{0}}|>k+r\}.\end{eqnarray}$$

By perhaps making $\unicode[STIX]{x1D6FC}$ smaller, using transience, we also have that

$$\begin{eqnarray}\mathbb{P}[\forall t>0,|Z_{t}|\geqslant |Z_{0}|]\geqslant \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Combining these two estimates, with the Markov property at time $t_{0}$ ,

$$\begin{eqnarray}\displaystyle \mathbb{P}[\forall t,\text{pref}_{k}(Z_{t})=\text{pref}_{k}(Z_{0})\mid |Z_{0}|=k+1] & {\geqslant} & \displaystyle \mathbb{P}[A\text{ and }\{\forall t>t_{0},|Z_{t}|\geqslant |Z_{t_{0}}|\}]\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \mathbb{P}[A]\cdot \mathbb{P}[\forall t>0,|Z_{t}|\geqslant |Z_{0}|]\geqslant \unicode[STIX]{x1D6FC}^{2}.\nonumber\end{eqnarray}$$

This is the definition of $\mathbb{F}_{2}$ being $(k,k+1,\unicode[STIX]{x1D6FC})$ -fixing.◻

A very useful property of fixing graphs is that when $\unicode[STIX]{x1D6FC}$ 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.

Lemma 4.5. Let $\unicode[STIX]{x1D6EC}$ be a $(k,n,\unicode[STIX]{x1D6FC})$ -fixing Schreier graph. Let $(Z_{t})_{t}$ be a $\unicode[STIX]{x1D707}$ -random walk on $\mathbb{F}_{2}$ and $(\tilde{Z}_{t})_{t}$ the projected walk on $\unicode[STIX]{x1D6EC}$ . Define the stopping time

$$\begin{eqnarray}T=\inf \{t:|\tilde{Z}_{t}|\geqslant n\}.\end{eqnarray}$$

Then,

$$\begin{eqnarray}|H(Z_{1}\mid \text{pref}_{k}(\tilde{Z}_{T}))-H(Z_{1}\mid \text{pref}_{k}(\tilde{Z}_{\infty }))|\leqslant \unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC})+2(\log 4)(1-\unicode[STIX]{x1D6FC})k,\end{eqnarray}$$

where $0<\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC})\rightarrow 0$ as $\unicode[STIX]{x1D6FC}\rightarrow 1$ .

Proof. First note that a general entropy inequality is

$$\begin{eqnarray}\displaystyle H(X\mid A)-H(X\mid B)\leqslant H(A\mid B)+H(B\mid A), & & \displaystyle \nonumber\end{eqnarray}$$

which for us provides the inequality

$$\begin{eqnarray}\displaystyle & & \displaystyle |H(Z_{1}\mid \text{pref}_{k}(\tilde{Z}_{T}))-H(Z_{1}\mid \text{pref}_{k}(\tilde{Z}_{\infty }))|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant H(\text{pref}_{k}(\tilde{Z}_{T})\mid \text{pref}_{k}(\tilde{Z}_{\infty }))+H(\text{pref}_{k}(\tilde{Z}_{\infty })\mid \text{pref}_{k}(\tilde{Z}_{T})).\nonumber\end{eqnarray}$$

We now bound this last quantity using Fano’s inequality (see, for example, [Reference Cover and ThomasCT12, § 2.10]). Taking  $\tilde{\unicode[STIX]{x1D6FC}}=\mathbb{P}[\text{pref}_{k}(\tilde{Z}_{T})=\text{pref}_{k}(\tilde{Z}_{\infty })]$ , since the support of both $\text{pref}_{k}(\tilde{Z}_{\infty }),\text{pref}_{k}(\tilde{Z}_{T})$ is of size $4\cdot 3^{k-1}<4^{k}$ ,

$$\begin{eqnarray}\displaystyle H(\text{pref}_{k}(\tilde{Z}_{\infty })\mid \text{pref}_{k}(\tilde{Z}_{T})) & {\leqslant} & \displaystyle H(\tilde{\unicode[STIX]{x1D6FC}},1-\tilde{\unicode[STIX]{x1D6FC}})+(1-\tilde{\unicode[STIX]{x1D6FC}})\log 4^{k},\nonumber\\ \displaystyle H(\text{pref}_{k}(\tilde{Z}_{T})\mid \text{pref}_{k}(\tilde{Z}_{\infty })) & {\leqslant} & \displaystyle H(\tilde{\unicode[STIX]{x1D6FC}},1-\tilde{\unicode[STIX]{x1D6FC}})+(1-\tilde{\unicode[STIX]{x1D6FC}})\log 4^{k},\nonumber\end{eqnarray}$$

where here we use the usual notation $H(\tilde{\unicode[STIX]{x1D6FC}},1-\tilde{\unicode[STIX]{x1D6FC}})$ to denote the Shannon entropy of the probability vector $(\tilde{\unicode[STIX]{x1D6FC}},1-\tilde{\unicode[STIX]{x1D6FC}})$ .

Since $p\mapsto H(p,1-p)$ decreases for $p\geqslant {\textstyle \frac{1}{2}}$ , it suffices to prove that $\tilde{\unicode[STIX]{x1D6FC}}=\mathbb{P}[\text{pref}_{k}(\tilde{Z}_{T})=\text{pref}_{k}(\tilde{Z}_{\infty })]\geqslant \unicode[STIX]{x1D6FC}$ .

But now, the very definition of $(k,n,\unicode[STIX]{x1D6FC})$ -fixing implies that for any relevant $u$ ,

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{P}[\text{pref}_{k}(\tilde{Z}_{\infty })=u\mid \text{pref}_{k}(Z_{T})=u]\nonumber\\ \displaystyle & & \displaystyle \quad \geqslant \mathbb{P}[\forall t>0,\text{pref}_{k}(\tilde{Z}_{T+t})\in \text{shd}(\text{pref}_{k}(u))\mid \text{pref}_{k}(Z_{T})=u]\geqslant \unicode[STIX]{x1D6FC},\nonumber\end{eqnarray}$$

where we have used the strong Markov property at the stopping time $T$ . Averaging over the relevant $u$ implies that $\tilde{\unicode[STIX]{x1D6FC}}\geqslant \unicode[STIX]{x1D6FC}$ .◻

The following is the main technical estimate of this subsection.

Lemma 4.6. Fix a generating probability measure $\unicode[STIX]{x1D707}$ on $\mathbb{F}_{2}$ with finite support. Let $(\unicode[STIX]{x1D6E4}_{j})_{j}$ be a sequence of Schreier graphs such that $\unicode[STIX]{x1D6E4}_{j}$ is $(k_{j},n_{j},\unicode[STIX]{x1D6FC}_{j})$ -fixing. Let $K_{j}$ be the subgroup corresponding to $\unicode[STIX]{x1D6E4}_{j}$ .

Suppose that $k_{j}\rightarrow \infty$ , $n_{j}-k_{j}\rightarrow \infty$ and $(1-\unicode[STIX]{x1D6FC}_{j})k_{j}\rightarrow 0$ as $j\rightarrow \infty$ . Then,

$$\begin{eqnarray}\limsup _{j\rightarrow \infty }h_{RW}(\mathbb{F}_{2}/\text{Core}_{\mathbb{F}_{2}}(K_{j}),\bar{\unicode[STIX]{x1D707}})=h_{RW}(\mathbb{F}_{2},\unicode[STIX]{x1D707}).\end{eqnarray}$$

Proof. Let $(Z_{t})_{t}$ be a $\unicode[STIX]{x1D707}$ -random walk on $\mathbb{F}_{2}$ .

Take $j$ large enough so that $n_{j}-k_{j}$ is much larger that $r=\max \{|g|:\unicode[STIX]{x1D707}(g)>0\}$ . To simplify the presentation we use $\unicode[STIX]{x1D6E4}_{j}=\unicode[STIX]{x1D6E4},K=K_{j},C=C_{j}=\text{Core}_{\mathbb{F}_{2}}(K_{j})$ and $k=k_{j},n=n_{j},\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{j}$ , omitting the subscript $j$ when it is clear from the context. Let $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6EC}_{j}$ be the Schreier graph corresponding to $C=C_{j}$ .

Let ${\mathcal{T}}_{C}=\bigcap _{t}\unicode[STIX]{x1D70E}(CZ_{t},CZ_{t+1},\ldots )$ and ${\mathcal{T}}=\bigcap _{t}\unicode[STIX]{x1D70E}(Z_{t},Z_{t+1},\ldots )$ denote that tail $\unicode[STIX]{x1D70E}$ -algebras of the random walk on $\unicode[STIX]{x1D6EC}$ and on $\mathbb{F}_{2}$ respectively. Kaimanovich and Vershik (see [Reference Kaimanovich and VershikKV83, (13) on p. 465]) show that

$$\begin{eqnarray}h_{RW}(\mathbb{F}_{2}/C,\bar{\unicode[STIX]{x1D707}})=H(CZ_{1})-H(CZ_{1}\mid {\mathcal{T}}_{C})\end{eqnarray}$$

and

$$\begin{eqnarray}h_{RW}(G,\unicode[STIX]{x1D707})=H(Z_{1})-H(Z_{1}\mid {\mathcal{T}}).\end{eqnarray}$$

Now, since $C\leqslant K$ , and since $\unicode[STIX]{x1D6E4}$ is $n$ -tree-like, also $\unicode[STIX]{x1D6EC}$ is (Lemma 4.1). Since $\unicode[STIX]{x1D6EC}$ is $n$ -tree-like, $CZ_{1}$ and $Z_{1}$ determine one another (Lemma 4.2), implying $H(CZ_{1})=H(Z_{1})$ and $H(CZ_{1}\mid {\mathcal{T}}_{C})=H(Z_{1}\mid {\mathcal{T}}_{C})$ . Since ${\mathcal{T}}_{C}\subset {\mathcal{T}}$ , the inequality $H(Z_{1}\mid {\mathcal{T}}_{C})\geqslant H(Z_{1}\mid {\mathcal{T}})$ is immediate. Hence, we only need to show that $\liminf _{j}H(Z_{1}\mid {\mathcal{T}}_{C_{j}})\leqslant H(Z_{1}\mid {\mathcal{T}})$ .

Set $T=\inf \{t:|CZ_{t}|\geqslant n\}$ as in Lemma 4.5. Since $\unicode[STIX]{x1D6EC}$ is $n$ -tree-like, $\text{pref}_{k}(CZ_{T-1})$ and $\text{pref}_{k}(Z_{T-1})$ 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 $\text{pref}_{k}(CZ_{T})=\text{pref}_{k}(CZ_{T-1})$ and $\text{pref}_{k}(Z_{T})=\text{pref}_{k}(Z_{T-1})$ . All in all, $H(Z_{1}\mid \text{pref}_{k}(CZ_{T}))=H(Z_{1}\mid \text{pref}_{k}(Z_{T}))$ .

Now, set $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC},k)=2H(\unicode[STIX]{x1D6FC},1-\unicode[STIX]{x1D6FC})+2(\log 4)(1-\unicode[STIX]{x1D6FC})k$ . Since $\text{pref}_{k}(CZ_{\infty })$ is measurable with respect to ${\mathcal{T}}_{C}$ , by using Lemma 4.5 twice, once for $\unicode[STIX]{x1D6EC}$ and once for the tree, we get the bound

$$\begin{eqnarray}\displaystyle H(Z_{1}\mid {\mathcal{T}}_{C}) & {\leqslant} & \displaystyle H(Z_{1}\mid \text{pref}_{k}(CZ_{\infty }))\leqslant H(Z_{1}\mid \text{pref}_{k}(CZ_{T}))+\unicode[STIX]{x1D700}\nonumber\\ \displaystyle & = & \displaystyle H(Z_{1}\mid \text{pref}_{k}(Z_{T}))+\unicode[STIX]{x1D700}\leqslant H(Z_{1}\mid \text{pref}_{k}(Z_{\infty }))+2\unicode[STIX]{x1D700}.\nonumber\end{eqnarray}$$

Now, under the assumptions of the lemma, $\unicode[STIX]{x1D700}\rightarrow 0$ as $j\rightarrow \infty$ . Also, since $k_{j}\rightarrow \infty$ we have that

$$\begin{eqnarray}H(Z_{1}\mid \text{pref}_{k_{j}}(Z_{\infty }))\rightarrow H(Z_{1}\mid {\mathcal{T}}).\end{eqnarray}$$

Thus, taking a limit on the above provides us with

$$\begin{eqnarray}\limsup _{j\rightarrow \infty }H(Z_{1}\mid {\mathcal{T}}_{C_{j}})\leqslant \limsup _{j\rightarrow \infty }H(Z_{1}\mid \text{pref}_{k_{j}}(Z_{\infty }))=H(Z_{1}\mid {\mathcal{T}}).\end{eqnarray}$$

Hence, as mentioned, we conclude that $\limsup _{j}H(Z_{1}\mid {\mathcal{T}}_{C_{j}})=H(Z_{1}\mid {\mathcal{T}})$ , and since $H(Z_{1})=H(CZ_{1})$ we get that

$$\begin{eqnarray}\limsup _{j\rightarrow \infty }h_{RW}(\mathbb{F}_{2}/\text{Core}_{\mathbb{F}_{2}}(K_{j}),\bar{\unicode[STIX]{x1D707}})=h_{RW}(\mathbb{F}_{2},\unicode[STIX]{x1D707}).\square\end{eqnarray}$$

To complete the relevance of fixing to convergence of the random walk entropies, we finish with the following.

Proposition 4.7. Fix a generating probability measure $\unicode[STIX]{x1D707}$ on $\mathbb{F}_{2}$ with finite support. Let $(\unicode[STIX]{x1D6E4}_{j})_{j}$ be a sequence of Schreier graphs such that $\unicode[STIX]{x1D6E4}_{j}$ is $(k_{j},n_{j},\unicode[STIX]{x1D6FC}_{j})$ -fixing. Let $K_{j}$ be the subgroup corresponding to $\unicode[STIX]{x1D6E4}_{j}$ .

Suppose that $k_{j}\rightarrow \infty$ and $\unicode[STIX]{x1D6FC}_{j}\rightarrow 1$ as $j\rightarrow \infty$ . Then,

$$\begin{eqnarray}\limsup _{j\rightarrow \infty }h_{RW}(\mathbb{F}_{2}/\text{Core}_{\mathbb{F}_{2}}(K_{j}),\bar{\unicode[STIX]{x1D707}})=h_{RW}(\mathbb{F}_{2},\unicode[STIX]{x1D707}).\end{eqnarray}$$

4.3 Gluing graphs

For this section it is instructive to consult Figures 1, 2 and 3. Let $\unicode[STIX]{x1D6EC}$ be a Schreier graph of the free group $\mathbb{F}_{2}=\langle a,b\rangle$ and let $S=\{a,a^{-1},b,b^{-1}\}$ . We say that the pair $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ is $s$ -marked if $\unicode[STIX]{x1D716}$ is a (oriented) edge in $\unicode[STIX]{x1D6EC}$ labeled by $s\in S$ .

For $s\in S$ we define ${\mathcal{N}}_{s}$ to be the following graph. The vertices are non-negative the integers $\mathbb{N}$ . The edges are given by $(x+1,x)$ , each labeled by $s$ , and self-loops labeled by $\unicode[STIX]{x1D709}\in \{a,b\}\setminus \{s,s^{-1}\}$ at each vertex. In order to be a Schreier graph, this graph is missing one outgoing edge from $0$ labeled by $s$ .

Given a $s$ -marked pair $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ and an integer $n>0$ , we construct a Schreier graph $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ as follows.

The Cayley graph of $\mathbb{F}_{2}$ is the $4$ -regular tree. Recall that for an integer $n$ we defined the finite subtree $(\mathbb{F}_{2})_{{\leqslant}n}$ which is the induced subgraph on $\{g\in \mathbb{F}_{2}:|g|\leqslant n\}$ , and let $(\mathbb{F}_{2})_{n}$ be the set of vertices $\{g:|g|=n\}$ . For $g\in (\mathbb{F}_{2})_{n}$ there is exactly one outgoing edge incident to $g$ in $(\mathbb{F}_{2})_{{\leqslant}n}$ . If this edge is labeled by $\unicode[STIX]{x1D709}\in S$ we say that $g$ is a $\unicode[STIX]{x1D709}$ -leaf.

Figure 1. Part of the labeled graph ${\mathcal{N}}_{a}$ .

Figure 2. The subgraph $(\mathbb{F}_{2})_{{\leqslant}3}$ . 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 $\unicode[STIX]{x1D709}$ -leaf $g$ , in order to complete $(\mathbb{F}_{2})_{{\leqslant}n}$ into a Schreier graph, we need to specify three more outgoing edges, with the labels $\unicode[STIX]{x1D709}^{-1}$ and the two labels from $S\setminus \{\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}^{-1}\}$ .

Now, recall our $s$ -marked pair $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ . Let $g$ be a $\unicode[STIX]{x1D709}$ -leaf for $\unicode[STIX]{x1D709}\not \in \{s,s^{-1}\}$ . Let $(\unicode[STIX]{x1D6EC}_{g},\unicode[STIX]{x1D716}_{g})$ be a copy of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ , and suppose that $\unicode[STIX]{x1D716}_{g}=(x_{g},y_{g})$ . Connect $\unicode[STIX]{x1D6EC}_{g}$ to $g$ by deleting the edge $\unicode[STIX]{x1D716}_{g}$ from $\unicode[STIX]{x1D6EC}_{g}$ and adding the directed edges $(x_{g},g),(g,y_{g})$ labeling them both $s$ . Also, let ${\mathcal{N}}_{g}$ be a copy of ${\mathcal{N}}_{\unicode[STIX]{x1D709}}$ above, and connect this copy to $g$ by a directed edge $(0_{g},g)$ labeled by $\unicode[STIX]{x1D709}$ (here $0_{g}$ is the copy of $0$ in ${\mathcal{N}}_{g}$ ). This takes care of $\unicode[STIX]{x1D709}$ -leaves for $\unicode[STIX]{x1D709}\not \in \{s,s^{-1}\}$ .

If $\unicode[STIX]{x1D709}\in \{s,s^{-1}\}$ , the construction is simpler. For any $\unicode[STIX]{x1D709}$ -leaf $\unicode[STIX]{x1D6FE}$ (where $\unicode[STIX]{x1D709}\in \{s,s^{-1}\}$ ), let ${\mathcal{N}}_{\unicode[STIX]{x1D6FE}}$ be a copy of ${\mathcal{N}}_{\unicode[STIX]{x1D709}}$ (with $0_{\unicode[STIX]{x1D6FE}}$ the copy of $0$ as before), and connect ${\mathcal{N}}_{\unicode[STIX]{x1D6FE}}$ by an edge $(0_{\unicode[STIX]{x1D6FE}},\unicode[STIX]{x1D6FE})$ labeled by  $\unicode[STIX]{x1D709}$ . Finally add an additional self-loop with the missing labels at $\unicode[STIX]{x1D6FE}$ (i.e. the labels in $S\setminus \{s,s^{-1}\}$ each in one direction along the loop).

By adding all the above copies to all the leaves in $(\mathbb{F}_{2})_{{\leqslant}n}$ , we obtain a Schreier graph, which we denote $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ . 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 $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ is $n$ -tree-like.

Figure 3. A visualization of gluing $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ to $(\mathbb{F}_{2})_{{\leqslant}3}$ . The dotted edges correspond to the label $a$ , and the full edges to $b$ . The vertex $g$ is an $a$ -leaf. Thus, a copy of $\unicode[STIX]{x1D6EC}$ , denoted $\unicode[STIX]{x1D6EC}_{g}$ is connected by removing the edge $\unicode[STIX]{x1D716}$ (dashed) and adding to edges $(x_{g},g),(g,y_{g})$ labeled $b$ (solid). A copy of ${\mathcal{N}}_{a}$ , denoted ${\mathcal{N}}_{g}$ , is connected via an edge $(0_{g},g)$ labeled $a$ (dotted). Also, to the $b$ -leaf $\unicode[STIX]{x1D6FE}$ , a copy of ${\mathcal{N}}_{b}$ , denoted ${\mathcal{N}}_{\unicode[STIX]{x1D6FE}}$ is added via an edge $(0_{\unicode[STIX]{x1D6FE}},\unicode[STIX]{x1D6FE})$ labeled $b$ (full), and an additional self-loop labeled $a$ (dotted) is added at $\unicode[STIX]{x1D6FE}$ .

4.4 Marked pairs and transience

Recall the definition of a transient random walk (see, for example, [Reference PetePet, Reference Lyons and PeresLP16]). We say that a graph $\unicode[STIX]{x1D6EC}$ is $\unicode[STIX]{x1D707}$ -transient if for any vertex $v$ in $\unicode[STIX]{x1D6EC}$ there is some positive probability that the $\unicode[STIX]{x1D707}$ -random walk started at $v$ will never return to $v$ .

We now connect the transience of $\unicode[STIX]{x1D6EC}$ above to that of fixing from the previous subsection.

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.

Proof Sketch

Consider the graph $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ . This is essentially a finite tree with copies of $\unicode[STIX]{x1D6EC}$ and copies of $\mathbb{N}$ glued to the (appropriate) leaves. At any vertex $v$ with $|v|>k$ , there is a fixed positive probability $\unicode[STIX]{x1D6FC}$ of escaping to the leaves without changing the $k$ -prefix (because $\mathbb{F}_{2}$ is $(k,k+1,\unicode[STIX]{x1D6FC})$ -fixing).

Once near the leaves, with some fixed probability $\unicode[STIX]{x1D6FF}$ the walk reaches a copy of $\unicode[STIX]{x1D6EC}$ without going back and changing the $k$ -prefix. Note that although there are copies of the recurrent  ${\mathcal{N}}_{\unicode[STIX]{x1D6FE}}$ (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 $\unicode[STIX]{x1D6EC}$ in the shadow of $v$ .

Finally, once in the copy of $\unicode[STIX]{x1D6EC}$ , because of transience, the walk has some positive probability $\unicode[STIX]{x1D6FD}$ to escape to infinity without ever leaving $\unicode[STIX]{x1D6EC}$ , and thus without ever changing the $k$ -prefix.

All together, with probability at least $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}$ the walk will never change the $k$ -prefix.

If this event fails, the walk may jump back some distance, but not too much since we assume that $\unicode[STIX]{x1D707}$ 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 $|v|>k+\ell r$ for large enough $\ell$ , there are at least $\ell$ attempts to escape to infinity without changing the $k$ -prefix, each with a conditional probability of success at least $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}$ . By taking $\ell$ 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 $(\tilde{Z}_{t})_{t}$ denote the projected $\unicode[STIX]{x1D707}$ -random walk on the Schreier graph $\unicode[STIX]{x1D6EC}$ . Assume that $\unicode[STIX]{x1D716}=(x,y)$ , and note that since $\unicode[STIX]{x1D6EC}$ is $\unicode[STIX]{x1D707}$ -transient, then

$$\begin{eqnarray}\mathbb{P}[\forall t,\tilde{Z}_{t}\neq x\mid \tilde{Z}_{0}=y]+\mathbb{P}[\forall t,\tilde{Z}_{t}\neq y\mid \tilde{Z}_{0}=x]>0.\end{eqnarray}$$

By possibly changing $\unicode[STIX]{x1D716}$ to $(y,x)$ , without loss of generality we may assume that

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}:=\mathbb{P}[\forall t,\tilde{Z}_{t}\neq x\mid \tilde{Z}_{0}=y]>0.\end{eqnarray}$$

Let $\ell >0$ be large enough, to be chosen below. Set $n=k+(\ell +2)r+1$ . We now change notation and use $(\tilde{Z}_{t})_{t}$ to denote the projected walk on the Schreier graph $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ . Denote $r=\max \{|g|:\unicode[STIX]{x1D707}(g)>0\}$ , the maximal possible jump of the $\unicode[STIX]{x1D707}$ -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 $|u|\leqslant k+r$ .

Let $\unicode[STIX]{x1D70F}=\inf \{t:n-r\leqslant |\tilde{Z}_{t}|\leqslant n\}$ . If we start from $|\tilde{Z}_{0}|\leqslant n$ this is a.s. finite, since the jumps cannot be larger than $r$ . Now, by Lemma 4.4, $\mathbb{F}_{2}$ is $(k,k+1,\unicode[STIX]{x1D702})$ -fixing for all $k$ and some $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D707})>0$ . Since $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ is $n$ -tree-like, if $v$ is a vertex in $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ with $k<|v|<n$ , then

(4) $$\begin{eqnarray}\displaystyle \mathbb{P}[\text{pref}_{k}(\tilde{Z}_{\unicode[STIX]{x1D70F}})=\text{pref}_{k}(v)\mid \tilde{Z}_{0}=v]\geqslant \unicode[STIX]{x1D702}. & & \displaystyle\end{eqnarray}$$

For $s\in S$ let $t_{s}$ be the smallest number such that $\unicode[STIX]{x1D707}^{t_{s}}(s)>0$ . Let $t_{S}=\max \{t_{s}:s\in S\}$ . Let $\unicode[STIX]{x1D6FF}=\min \{\unicode[STIX]{x1D707}^{t_{s}}(s):s\in S\}$ . Let $v,u$ be two adjacent vertices in $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ , and assume that the label of $(v,u)$ is $s\in S$ . Note that the definitions above ensure that

$$\begin{eqnarray}\mathbb{P}[\tilde{Z}_{t_{s}}=u\mid \tilde{Z}_{0}=v]\geqslant \unicode[STIX]{x1D6FF}.\end{eqnarray}$$

Thus, when $|v|>k+t_{S}r$ , with probability at least $\unicode[STIX]{x1D6FF}$ we can move from $v$ to $u$ in at most $t_{S}$ steps without changing the $k$ -prefix. This holds specifically for $|v|\geqslant n-r>k+t_{S}r$ by our choice of $n$ (as long as $\ell \geqslant t_{S}$ ).

Consider the graph $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ . Recall that it contains many copies of $\unicode[STIX]{x1D6EC}$ (glued to the appropriate leaves). Let $\unicode[STIX]{x1D6EC}_{1},\ldots ,\unicode[STIX]{x1D6EC}_{m}$ be the list of these copies, and denote by $\unicode[STIX]{x1D716}_{j}=(x_{j},y_{j})$ the corresponding copies of $\unicode[STIX]{x1D716}$ in each. Define $Y=\{y_{1},\ldots ,y_{m}\}$ .

Now, define stopping times

$$\begin{eqnarray}U_{m}=\inf \{t:|\tilde{Z}_{t}|<m\}\quad \text{and}\quad T=\inf \{t:\tilde{Z}_{t}\in Y\}\end{eqnarray}$$

(where $\inf \emptyset =\infty$ ). Any vertex $v$ in $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ of depth $|v|<n$ must have some copy of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ in its shadow $\text{shd}(v)$ . So there exists some path of length at most $n-|v|+1$ from $v$ into some copy of $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ , ending in some vertex in $Y$ . If $|v|\geqslant n-r$ , 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 $|u|\geqslant n-r$ , we obtain that for any $|v|\geqslant n-r$ ,

$$\begin{eqnarray}\displaystyle \mathbb{P}[T<U_{n-r}\mid \tilde{Z}_{0}=v]\geqslant \unicode[STIX]{x1D6FF}^{r+1}. & & \displaystyle \nonumber\end{eqnarray}$$

By $\unicode[STIX]{x1D707}$ -transience of $\unicode[STIX]{x1D6EC}$ , we have that starting from any $y_{j}\in Y$ , with probability at least $\unicode[STIX]{x1D6FD}$ the walk never crosses $(y_{j},x_{j})$ , and so never leaves $\unicode[STIX]{x1D6EC}_{j}$ (and thus always stays at distance at least $n$ from the root). We conclude that for any $|v|\geqslant n-r$ ,

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{P}[\text{pref}_{k}(\tilde{Z}_{\infty })=\text{pref}_{k}(\tilde{Z}_{0})\mid \tilde{Z}_{0}=v]\nonumber\\ \displaystyle & & \displaystyle \geqslant \mathbb{P}[U_{n-r}=\infty \mid \tilde{Z}_{0}=v]\nonumber\\ \displaystyle & & \displaystyle \geqslant \mathbb{P}[T<U_{n-r}\mid \tilde{Z}_{0}=v]\cdot \inf _{y_{j}\in Y}\mathbb{P}[\forall t,\tilde{Z}_{t}\in \unicode[STIX]{x1D6EC}_{j}\mid \tilde{Z}_{0}=y_{j}]\nonumber\\ \displaystyle & & \displaystyle \geqslant \unicode[STIX]{x1D6FF}^{r+1}\unicode[STIX]{x1D6FD}\nonumber\end{eqnarray}$$

(where we have used that $k+r<n-r$ ). Combining this with (4), using the strong Markov property at time $\unicode[STIX]{x1D70F}$ , we obtain that for any $|v|>k$ ,

(5) $$\begin{eqnarray}\displaystyle \mathbb{P}[\text{pref}_{k}(\tilde{Z}_{\infty })=\text{pref}_{k}(v)\mid \tilde{Z}_{0}=v]\geqslant \unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}^{r+1}\unicode[STIX]{x1D6FD}. & & \displaystyle\end{eqnarray}$$

Step II. Now, define the events

$$\begin{eqnarray}A_{j}=\{\text{pref}_{k}(\tilde{Z}_{\infty })\neq \text{pref}_{k}(\tilde{Z}_{U_{n-jr}})\}.\end{eqnarray}$$

When $\tilde{Z}_{0}=v$ for $|v|\geqslant n=k+(\ell +2)r+1$ the event $\{\text{pref}_{k}(\tilde{Z}_{\infty })\neq \text{pref}_{k}(\tilde{Z}_{0})\}$ implies that

$$\begin{eqnarray}U_{|v|-r}<U_{|v|-2r}<\cdots <U_{|v|-\ell r}<U_{k+r}<\infty .\end{eqnarray}$$

But, at time $U_{|v|-jr}$ we have that

$$\begin{eqnarray}|\tilde{Z}_{U_{|v|-jr}}|\geqslant |\tilde{Z}_{U_{|v|-jr}-1}|-r\geqslant |v|-(j+1)r>k+r.\end{eqnarray}$$

Thus, we have by (5), using the strong Markov property at time $U_{|v|-jr}$ ,

$$\begin{eqnarray}\mathbb{P}[A_{j+1}\mid \tilde{Z}_{0},\ldots ,\tilde{Z}_{U_{|v|-jr}}]\leqslant 1-\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}^{r+1}\unicode[STIX]{x1D6FD},\end{eqnarray}$$

implying that

$$\begin{eqnarray}\mathbb{P}[A_{j+1}\mid (A_{1})^{c}\cap \cdots \cap (A_{j})^{c}]\leqslant 1-\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}^{r+1}\unicode[STIX]{x1D6FD}.\end{eqnarray}$$

Thus, as long as $|v|\geqslant n$ we have that

$$\begin{eqnarray}\displaystyle \mathbb{P}[\text{pref}_{k}(\tilde{Z}_{\infty })\neq \text{pref}_{k}(v)\mid \tilde{Z}_{0}=v]\leqslant \mathbb{P}[A_{1}\cup \cdots \cup A_{\ell }\mid \tilde{Z}_{0}=v]\leqslant (1-\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}^{r+1}\unicode[STIX]{x1D6FD})^{\ell }. & & \displaystyle \nonumber\end{eqnarray}$$

Choosing $\ell$ such that $(1-\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}^{r}\unicode[STIX]{x1D6FD})^{\ell }<\unicode[STIX]{x1D700}$ , we obtain that for any $|v|\geqslant n$

$$\begin{eqnarray}\mathbb{P}[\text{pref}_{k}(\tilde{Z}_{\infty })\neq \text{pref}_{k}(v)\mid \tilde{Z}_{0}=v]<\unicode[STIX]{x1D700}.\end{eqnarray}$$

That is, the graph $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ is $(k,n,1-\unicode[STIX]{x1D700})$ -fixing.◻

Remark 4.9. Proposition 4.8 is the only place we require the measure $\unicode[STIX]{x1D707}$ to have finite support. Indeed, we believe that the proposition should hold with weaker assumptions on $\unicode[STIX]{x1D707}$ , perhaps even only assuming that $\unicode[STIX]{x1D707}$ 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 § 3.1 that given a subgroup $K\leqslant \mathbb{F}_{2}$ we define $\Vert g\Vert _{K}$ as the number of cosets of the normalizer $b\in N_{G}(K)\backslash G$ for which $g\not \in K^{b}$ . Recall also that $\text{Core}_{\emptyset }(K)=\{g:\Vert g\Vert _{K}<\infty \}$ .

In this subsection we will provide conditions on $K$ under which $\text{Core}_{\emptyset }(K)$ contains a given subgroup of $\mathbb{F}_{2}$ . This will be useful in determining properties of $\mathbb{F}_{2}/\text{Core}_{\emptyset }(K)$ (specifically, whether $\mathbb{F}_{2}/\text{Core}_{\emptyset }(K)$ is nilpotent).

First, some notation. Let $g\in \mathbb{F}_{2}$ . Let $g=w_{1}\cdots w_{|g|}$ be a reduced word with $w_{i}\in S$ for all $i$ . Let $\unicode[STIX]{x1D6EC}$ be a Schreier graph with corresponding subgroup $K\leqslant \mathbb{F}_{2}$ . Fix a vertex $v$ in $\unicode[STIX]{x1D6EC}$ . Let $v_{0}(g)=v$ and inductively define $v_{n+1}(g)$ to be the unique vertex of $\unicode[STIX]{x1D6EC}$ such that $(v_{n}(g),v_{n+1}(g))$ is labeled by $w_{n+1}$ . In other words, if $v=K\unicode[STIX]{x1D6FE}$ then $v_{i}(g)=K\unicode[STIX]{x1D6FE}w_{1}\cdots w_{i}$ .

Recall that $g\in K$ if and only if for $v=o$ the root of $\unicode[STIX]{x1D6EC}$ . Hence, for any $g\in K$ , we have $v_{|g|}(g)=v_{0}(g)$ . From this we can deduce that when $v=K\unicode[STIX]{x1D6FE}$ , then

$$\begin{eqnarray}v_{|g|}(g)=v_{0}(g)\;\Longleftrightarrow \;\unicode[STIX]{x1D6FE}g\unicode[STIX]{x1D6FE}^{-1}\in K\;\Longleftrightarrow \;g\in K^{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

That is, a subgroup $K$ is locally- $(\unicode[STIX]{x1D6E4}_{1},\ldots ,\unicode[STIX]{x1D6E4}_{n})$ if, after ignoring some finite area, locally we see one of the graphs $\unicode[STIX]{x1D6E4}_{1},\ldots ,\unicode[STIX]{x1D6E4}_{n}$ .

The main purpose of this subsection is to prove the following.

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 $\mathbb{F}_{2}$ , we need to find a sequence of subgroups $K_{n}$ with Schreier graphs $\unicode[STIX]{x1D6E4}_{n}$ such that the following properties hold.

• ∙ The Schreier graph $\unicode[STIX]{x1D6E4}_{n}$ is $(k_{n},k_{n}^{\prime },\unicode[STIX]{x1D6FC}_{n})$ -fixing, with $k_{n}\rightarrow \infty$ and $\unicode[STIX]{x1D6FC}_{n}\rightarrow 1$ .

• ∙ The subgroups $\text{Core}_{\emptyset }(K_{n})$ are co-nilpotent in $\mathbb{F}_{2}$ .

To show the first property, we will use the gluing construction from Proposition 4.8, by finding a suitable marked pair $(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ that is $\unicode[STIX]{x1D707}$ -transient.

Proof. Let $G_{0}=\mathbb{F}_{2},G_{j+1}=[G_{j},\mathbb{F}_{2}]$ be the descending central series of $\mathbb{F}_{2}$ . Since $N$ is co-nilpotent, there exists $m$ such that $G_{m}{\vartriangleleft}N$ .

By definition of $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ , outside a ball of radius $n$ in $\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D716})$ , we only have glued copies of $\unicode[STIX]{x1D6EC}$ or of ${\mathcal{N}}_{s}$ for $s\in S=\{a,a^{-1},b,b^{-1}\}$ (recall § 4.3).

Let ${\mathcal{Z}}_{s}$ denote the Schreier graph with vertices in $\mathbb{Z}$ , edges $(x+1,x)$ labeled by $s$ and a self loop with label $\unicode[STIX]{x1D709}\not \in \{s,s^{-1}\}$ at each vertex. Let $\unicode[STIX]{x1D711}_{s}:\mathbb{F}_{2}\rightarrow \mathbb{Z}$ be the homomorphism defined via $s\mapsto -1,s^{-1}\mapsto 1$ and $\unicode[STIX]{x1D709}\mapsto 0$ for $\unicode[STIX]{x1D709}\not \in \{s,s^{-1}\}$ . Then, the subgroup corresponding to ${\mathcal{Z}}_{s}$ is $\ker \unicode[STIX]{x1D711}_{s}$ . Since $\mathbb{F}_{2}/\ker \unicode[STIX]{x1D711}_{s}$ is abelian, we have that $G_{m}{\vartriangleleft}[\mathbb{F}_{2},\mathbb{F}_{2}]{\vartriangleleft}\ker \unicode[STIX]{x1D711}_{s}$ .

It is now immediate that $K_{n}$ is locally- $(\unicode[STIX]{x1D6EC},{\mathcal{Z}}_{a},{\mathcal{Z}}_{a^{-1}},{\mathcal{Z}}_{b},{\mathcal{Z}}_{b^{-1}})$ . By Proposition 4.11, this implies that

$$\begin{eqnarray}G_{m}\leqslant N\cap \mathop{\bigcap }_{s\in S}\ker \unicode[STIX]{x1D711}_{s}\leqslant \text{Core}_{\emptyset }(K_{n}).\end{eqnarray}$$

Thus, $\text{Core}_{\emptyset }(K_{n})$ is co-nilpotent.◻

We are now ready to prove our main result, Theorem 1.2.