# Ratio-limit boundaries for random walks on relatively hyperbolic groups

Adam Dor-On  Department of Mathematics
University of Haifa
Haifa
Israel<EMAIL_ADDRESS>,  Matthieu Dussaule  Cogitamus Laboratory
France<EMAIL_ADDRESS>and  Ilya Gekhtman  Department of
Mathematics
Technion - IIT
Haifa
Israel<EMAIL_ADDRESS>
###### Abstract.

We study boundaries arising from limits of ratios of transition probabilities
for random walks on relatively hyperbolic groups. We extend, as well as
determine significant limitations of, a strategy employed by Woess for
computing ratio-limit boundaries for the class of hyperbolic groups. On the
one hand we employ results of the second and third authors to adapt this
strategy to spectrally non-degenerate random walks, and show that the closure
of minimal points in $R$-Martin boundary is the unique smallest invariant
subspace in ratio-limit boundary. On the other hand we show that the general
strategy can fail when the random walk is spectrally degenerate and adapted on
a free product. Using our results, we are able to extend a theorem of the
first author beyond the hyperbolic case and establish the existence of a co-
universal quotient for Toeplitz C*-algebras arising from random walks which
are spectrally non-degenerate on relatively hyperbolic groups. Finally, we
exhibit an example of a relatively hyperbolic group carrying two random walks
such that the ratio limit boundaries are not equivariantly homeomorphic and no
two equivariant quotients of their respective Toeplitz C*-algebras are
equivariantly $*$-isomorphic.

###### 2020 Mathematics Subject Classification:

Primary: 60J50, 20F67, 37A55. Secondary: 37B05, 47L80

The first author was partially supported by EU grant HORIZON-MSCA-SE-2021
Project 101086394, and by an ISF Moked 713510 grant number 2919/19 for his
stay at the Weizmann Institute of Science, Israel.

##  1\. Introduction

### Background

A prominent line of research in geometric group theory is dedicated to finding
appropriate definitions of boundaries on which a group acts, following
examples of Fuchsian groups acting on the hyperbolic plane. Probabilistic
approaches for defining and studying such boundaries were initiated by
Furstenberg [35, 36], going back to ideas of Martin and Poisson in classical
harmonic analysis.

These approaches use random walks to define probabilistic boundaries obtained
by considering different ways a random walker can “converge at infinity”. By a
result of Varopoulos [64] we know that the only groups that support recurrent
random walks are virtually $\mathbb{Z}^{d}$ for $d=0,1,2$. Hence, most random
walks are transient, and it makes sense to define probabilistic
compactifications as the set of points to which a random walk converges.

Throughout this paper we will consider a countable discrete group $\Gamma$
together with a probability measure $\mu$ whose support generates $\Gamma$ as
a semigroup. We will call such a probability measure _admissible_. This
defines a Markov chain on $\Gamma$ given by the transition kernel
$P(g,h)=\mu(g^{-1}h)$ which is called the $\mu$-random walk on $\Gamma$. We
denote by $P^{n}(g,h)=\mu^{*n}(g^{-1}h)$ the associated $n$-step transition
probabilities.

The _weighted Green function_ associated to $\mu$ is defined by

$G(g,h|r)=\sum^{\infty}_{n=0}P^{n}(g,h)r^{n}$

for $r\geq 1$ and $g,h\in\Gamma$. This sum converges when $r<R$ where

$R:=R(\mu):=\big{(}\limsup_{n\rightarrow\infty}\sqrt[n]{P^{n}(e,e)}\big{)}^{-1}$

is the inverse of the spectral radius of the random walk. When $\Gamma$ is
non-amenable, we know that $R>1$ (See [53]), and $G(g,h|R)<\infty$ for all
$g,h$ in $\Gamma$ (see [70, Theorem 7.8]). Thus, it makes sense to define the
_$r$ -Martin kernel_ $K_{r}:\Gamma\times\Gamma\rightarrow(0,\infty)$ at
$r\in[1,R]$ (which is also well-defined as $r\rightarrow R$) by

$K_{r}(g,h):=\frac{G(g,h|r)}{G(e,h|r)}.$

The smallest compactification of $\Gamma$ to which the functions $h\mapsto
K_{r}(g,h)$ extend continuously for all $g\in\Gamma$ is called the $r$-Martin
compactification, and the complement of $\Gamma$ in it is called the
$r$-Martin boundary $\partial_{M,r}\Gamma$.

Martin boundaries are intimately related to discrete harmonic analysis on the
group through the Martin–Poisson integral representation theorem. We will say
that a function $u:\Gamma\rightarrow(0,\infty)$ is $t$-harmonic if

$t\cdot u(x)=\sum_{y\in\Gamma}P(x,y)u(y).$

The set $\mathcal{H}_{1}^{+}(P,t)$ of positive harmonic functions with
$u(e)=1$ is a compact convex set with the topology of pointwise convergence,
and the functions $x\mapsto K_{r}(x,\xi)$ are $r^{-1}$-harmonic whenever
$\xi\in\partial_{M,r}\Gamma$. We will denote by $\partial^{m}_{M,r}\Gamma$ the
points $\xi\in\partial_{M,r}\Gamma$ for which the function $x\mapsto
K_{r}(x,\xi)$ is extreme in $\mathcal{H}_{1}^{+}(P,r^{-1})$. These are often
called _minimal_ points of $\mathcal{H}_{1}^{+}(P,r^{-1})$, and the following
version of Choquet’s theorem shows how to get any function in
$\mathcal{H}_{1}^{+}(P,r^{-1})$ from minimal such.

######  Theorem 1.1 (Poisson–Martin integral representation).

Let $r\in[1,R]$, and $u$ a positive $r^{-1}$-harmonic function with $u(e)=1$.
Then there is a representing probability measure $\nu^{u}$ on
$\partial_{M,r}\Gamma$ such that

$u(x)=\int_{\partial_{M,r}\Gamma}K_{r}(x,\xi)d\nu^{u}(\xi),$

and $\nu^{u}$ is unique among probability measures $\nu$ that have full mass
on $\partial^{m}_{M,r}\Gamma$.

There are several different notions of boundaries for discrete groups in the
literature, and it is useful and interesting to know when they coincide or are
different. For instance, the Martin boundary may a priori depend on both $r$
and $\mu$, and it is useful to know when this is the case. For hyperbolic
groups, it was shown by Ancona [2] that for a large class of random walks on
them, the $1$-Martin boundary can be identified with the Gromov boundary
$\partial\Gamma$ of $\Gamma$. Ancona’s results extend readily to the
$r$-Martin boundary for $r<R$, and Gouezel [40] extended them to $r=R$ when
$\mu$ is symmetric. This opens avenues for using random walks to study
geometric properties of the group. Among several applications, Ancona
inequalities found use in giving a new proof of the Baum-Connes conjecture for
hyperbolic groups [45], and for constructing natural flow with unique KMS
stats on the crossed product C*-algebra $C(\partial\Gamma)\rtimes\Gamma$ [56].

Such emerging interactions between group theory and operator algebra theory
are not mere coincidence. The subject of operator algebras is a branch of
functional analysis sharing deep connections with other areas of pure
Mathematics and Physics. The study focuses on subalgebras of operators on a
complex Hilbert space, that are closed under appropriate topologies. Recent
applications of operator algebraic techniques to representation theory of
infinite discrete groups [52, 13] stem from studying non-commutative
boundaries of non-self-adjoint operator algebras in the sense of Arveson [4,
22]. These works eventually led to the resolution of open problems in group
theory [54], exciting new techniques in stationary dynamics [46] and progress
in ergodic theory of lattices in semisimple Lie groups [9, 6].

One of the main motivations for our work comes from a central question in the
theory of C*-algebras, of the existence of co-universal quotients for various
kinds of Toeplitz C*-algebras. Co-universality is realized in various forms in
C*-algebra theory, and some of its early manifestation are due to Cuntz and
Krieger [21, 20] in their works on C*-algebras associated to topological
Markov chains. The uniqueness of such co-universal quotients is often known in
the literature as _uniqueness theorems_ , including the so-called _gauge-
invariant uniqueness theorem_ as prototypical example. Later developments
include co-universal quotients for Toeplitz-Cuntz-Pimsner C*-algebras [58,
51], as well as more recent works on co-universal quotients for Toeplitz
algebras of product systems [16, 24, 25]. In these works, state-of-the art
techniques for establishing the existence of such co-universal quotient
C*-algebras rely heavily on non-commutative boundary theory.

The class of Toeplitz C*-algebras we are concerned with arise from _subproduct
systems_ , which were initially studied in the work of Shalit and Solel [63].
Such Toeplitz C*-algebras behave very differently from those arising from
product systems. For instance, the question of co-universality does not lend
itself available to non-commutative boundary techniques (see [27, Corollary
3.16]), and the existence of a co-universal quotient usually requires adding
additional symmetries (see [67, Example 2.3]). Thus, completely new techniques
are often necessary in order to prove the existence of a natural co-universal
quotient in various scenarios (see for instance [3, 44]). By using deep
results from the theory of random walks [40, 72], the first author established
the existence of a natural co-universal quotient for Toeplitz C*-algebras
arising from random walks, when the random walks are symmetric, aperiodic, and
on non-elementary hyperbolic groups [23, Corollary 5.2].

As part of the proof of co-universality in [23], a new probabilistic boundary
called the _ratio-limit boundary_ was discovered from computing quotients of
Toeplitz C*-algebras arising from random walks. The ratio-limit boundary
$\partial_{\rho}\Gamma$ is defined similarly to $r$-Martin boundary, where one
replaces $r$-Martin kernel $K_{r}$ by the ratio limit kernel $H$ given by

$H(x,y)=\lim\frac{P^{n}(x,y)}{P^{n}(e,y)}.$

It is not known whether these limits exist in general, and when they do we say
that the random walk determined by $\mu$ has the _strong ratio-limit property_
(SRLP). It is known that SRLP holds for random walks on nilpotent groups [55],
symmetric random walks on amenable groups [5], symmetric random walks on non-
elementary hyperbolic groups [40], and certain large classes of random walks
on relatively hyperbolic groups [29, 31].

The key insight in [23] was that, under the assumption of SRLP, the question
of existence of a co-universal quotient is equivalent to showing the existence
of a unique closed minimal $\Gamma$-invariant subspace of ratio-limit boundary
$\partial_{\rho}\Gamma$. In a companion paper, Woess [72] studied ratio limit
boundary and showed that when $\Gamma$ is hyperbolic, ratio-limit boundary
coincides with Gromov boundary $\partial\Gamma$. This was used as the last
missing piece for the proof for co-universality in [23], and gives new
applications of Ancona inequalities at $r=R$ beyond probability theory.

Many questions were left open in [23] and [72] and we will address some of
them here in the wider context of _relatively hyperbolic groups_. Relatively
hyperbolic groups are a generalization of hyperbolic groups which exhibit
_some_ hyperbolic behavior while allowing for arbitrary subgroups. This
includes arbitrary free products of finitely generated groups, as well as
fundamental groups of finite volume manifolds with pinched negative scalar
curvature.

The class of relatively hyperbolic groups is a rich source of examples that
show how various properties depend on the underlying random walk and not on
the group itself. For instance, Cartwright [17, 18] disproved a conjecture of
Gerl by exhibiting a relatively hyperbolic group carrying two finitely
supported, admissible and symmetric random walks with distinct local limit
theorems. Also, Woess [68] and the second and third authors [33] proved that
the homeomorphism type of the $R$-Martin boundary may depend on the underlying
random walk on such groups.

Ancona’s results have been extended to relatively hyperbolic groups. In
particular, the third author together with Gerasimov, Potyagailo and Yang [37]
proved a modified version of Ancona’s deviation inequalities for the Green
function at $r=1$, for finitely supported and admissible random walks on
relatively hyperbolic groups, and used it to show that the identity map on
$\Gamma$ extends to a continuous surjection from the $1$-Martin boundary to
the Bowditch boundary of the group, such that the pre-images of conical points
under this surjection are singletons. For relatively hyperbolic groups with
respect to _virtually abelian_ parabolic subgroups, the second and third
authors together with Gerasimov and Potyagailo [30] identified the Martin
boundary precisely. They showed that the preimage of any parabolic point is a
sphere of dimension $d-1$ where $d$ is the rank of the stabilizer subgroup.

Ancona deviation inequalities were extended to the $r$-Martin boundary $r\leq
R$ by the second and third authors [33], and this allowed them to prove that
the Martin boundary covers the Bowditch boundary in this setting as well.
However, for $r=R$, even when the parabolic subgroups are virtually abelian,
the homeomorphism type of the Martin boundary is more delicate and depends on
certain spectral properties of the random walk. More precisely, the measure
$\mu$ is _spectrally degenerate_ along a parabolic subgroup $P$ if the
spectral radius of first return time Markov chain on $P$ induced from the
rescaled measure $R\mu$ has spectral radius $1$ and _spectrally non-
degenerate_ along $P$ otherwise. Roughly, $\mu$ is spectrally degenerate along
$P$ when it is “largely weighted” on $P$ from the point of view of the random
walk (see Definition 4.1). We call $\mu$ spectrally non-degenerate if it is
spectrally non-degenerate along every parabolic subgroup.

### Main results

Our goal in this paper is to study ratio limit behaviour for random walks on
relatively hyperbolic groups, and extend the existence of a co-universal
quotient for Toeplitz C*-algebras to spectrally non-degenerate random walks on
relatively hyperbolic groups.

Our first results is in Section 3 show that if $\Gamma$ is a non-elementary
relatively hyperbolic group, and $\mu$ is a finitely supported, symmetric and
admissible probability measure, then
$\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$ is strongly
proximal. This is made possible by the existence of the $\Gamma$-surjection
from $r$-Martin boundary onto Bowditch boundary established in [37], which
allows us to lift this property to $\overline{\partial_{M,r}^{m}\Gamma}$ as
well. Density of pre-images of conical points in
$\overline{\partial_{M,r}^{m}\Gamma}$, as well as minimality of
$\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$, which were proved
in [37, Corollary 1.6] for $r=1$, but their proof carries mutatis-mutandis for
any $r\in[1,R]$.

Spectral non-degeneracy of the random walk will play an important role in our
paper, and in Section 4 we show that spectrally non-degenerate random walks
are ubiquitous on relatively hyperbolic groups. In fact, in Proposition 4.2 we
show that spectrally non-degenerate adapted random walks exist on arbitraty
non-elementary free products and in Proposition 4.4 we show that symmetric
random walks are automatically spectrally non-degenerate whenever parabolic
subgroups are virtually nilpotent of homogeneous dimension at most $4$.

For $r\in[1,R]$, $s\in\mathbb{N}$ and $x,y\in\Gamma$ we denote

$I^{(s)}(x,y|r):=\sum_{x_{1},...,x_{s}\in\Gamma}G(x,x_{1}|r)G(x_{1},x_{2}|r)...G(x_{s},y|r).$

One of the key reductions used by Woess in the hyperbolic case to identify
ratio-limit boundary with Gromov boundary was to show that

(1.1) $H(x,y)=\lim_{r\rightarrow R}\frac{I^{(1)}(x,y|r)}{I^{(1)}(e,y|r)}.$

In [72, Question 7.1(a)] Woess asked under what general conditions does
equation (1.1) hold. In Section 5 we shed considerable light on this question
in the relatively hyperbolic context. In Corollary 5.1 we provide sufficient
conditions for when $H(x,y)=\lim_{r\rightarrow
R}\frac{I^{(s)}(x,y|r)}{I^{(s)}(e,y|r)}$ where $\Gamma$ is relatively
hyperbolic, and $s\in\mathbb{N}$ is the smallest for which the $s$-derivative
$G^{(s)}(x,y|r)$ diverges as $r\rightarrow R$ for some (all) $x,y\in\Gamma$.
This leads us to Proposition 5.2 where necessary conditions are provided for
$H(x,y)=\lim_{r\rightarrow R}\frac{I^{(s)}(x,y|r)}{I^{(s)}(e,y|r)}$ to hold,
and provides us with the following example.

Example 5.3 _There exists a finitely supported, admissible, symmetric and
adapted random walk on a free products for which the validity of equation (
1.1) fails. In fact, $H(x,y)=\lim_{r\rightarrow
R}\frac{I^{(s)}(x,y|r)}{I^{(s)}(e,y|r)}$ can fail for _any_ $s$ for an adapted
random walk on a free product. _

Examples of this kind, and the necessary conditions in Proposition 5.2, show
the limitation of the original strategy employed by Woess in [72]. It shows
that new techniques are necessary for studying ratio-limit boundaries, even
for adapted random walks on free products.

Despite these aforementioned limitations, the strategy employed by Woess in
[72] can still be carried out for many random walks on relatively hyperbolic
groups, including spectrally non-degenerate random walks. In Section 6 we
employ results of the second author from [28, 29] to show that for a
spectrally non-degenerate random walk on a relatively hyperbolic group
$\Gamma$, for a point $\xi\in\partial_{M,R}\Gamma$ whose image in Bowditch
boundary is conical, we have that $\frac{H(x,y)}{K_{R}(x,y)}\rightarrow 1$ as
$y\rightarrow\xi$. This allows us show in Corollary 6.2 that closure of
minimal points $\overline{\partial^{m}_{M,R}\Gamma}$ embeds inside ratio-limit
boundary $\partial_{\rho}\Gamma$ via a bi-Lipschitz $\Gamma$-equivariant map.
The main result of Section 6 is Theorem 6.4 which shows that
$\Gamma\curvearrowright\partial_{\rho}\Gamma$ has a unique smallest closed
$\Gamma$-invariant subspace, which is $\Gamma$-homeomorphic to
$\overline{\partial^{m}_{M,R}\Gamma}$.

Theorem 6.4 _Let $\Gamma$ be a non-elementary relatively hyperbolic group and
let $\mu$ be a finitely supported, symmetric, admissible and aperiodic
probability measure on $\Gamma$. Assume further the $\mu$-random walk is
spectrally non-degenerate. Then, there is a bi-Lipschitz $\Gamma$-equivariant
map $i:\overline{\partial^{m}_{M,r}\Gamma}\to\partial_{\rho}\Gamma$. The
action on the ratio-limit boundary
$\Gamma\curvearrowright\partial_{\rho}\Gamma$ is essentially minimal, i.e. it
has a unique smallest closed $\Gamma$-invariant subspace which coincides with
the action $\Gamma\curvearrowright\overline{\partial^{m}_{M,R}\Gamma}$._

We expect that most of our results hold without assuming that the random walk
is spectrally non-degenerate. Although the geometric arguments are really more
difficult to handle, we can still prove that some of our results holds for
certain convergent random walks on the free product
$\mathbb{Z}^{5}*\mathbb{Z}$. This allows us to show that there are two
distinct random walks on $\mathbb{Z}^{5}*\mathbb{Z}$ for which the associated
ratio-limit boundaries are not equivariantly homeomorphic.

Example 7.3 _There exist two finitely supported, symmetric, admissible and
adapted probability measures $\mu$ and $\mu^{\prime}$ on
$\Gamma:=\mathbb{Z}^{5}*\mathbb{Z}$ such that $\partial_{\rho}(\Gamma,\mu)$
and $\partial_{\rho}(\Gamma,\mu^{\prime})$ are not equivariantly
homeomorphic._

Theorem 6.4 is then leveraged in Section 8 to answer the co-universality
question for Toeplitz C*-algebras arising from spectrally non-degenerate
random walks on relatively hyperbolic groups. More precisely, in Theorem 8.4
where we show that the Toeplitz algebra $\mathcal{T}(\Gamma,\mu)$ has a a
unique smallest $\Gamma\times\mathbb{T}$ equivariant quotient which coincides
$C(\overline{\partial^{m}_{M,R}\Gamma}\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$.

Theorem 8.4 _The Toeplitz algebra $\mathcal{T}(\Gamma,\mu)$ has a a unique
smallest $\Gamma\times\mathbb{T}$ equivariant quotient which coincides with
$C(\overline{\partial^{m}_{M,R}\Gamma}\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$._

This provides a positive answer to the first part of [23, Question 5.6] for
spectrally non-degenerate random walks on relatively hyperbolic groups. As an
application of co-universality applied to the random walks in Example 7.3, we
show in Example 8.5 that no two $\Gamma\times\mathbb{T}$-equivariant quotients
of Toeplitz C*-algebras, one of $\mathcal{T}(\Gamma,\mu)$ and one of
$\mathcal{T}(\Gamma,\mu^{\prime})$, are $\Gamma\times\mathbb{T}$-equivariantly
isomorphic.

Our work provides first examples of a finitely generated group for which the
the ratio-limit boundary, as well as the Toeplitz C*-algebra and its
equivariant quotients depend on the finitely supported, symmetric and
admissible random walk. We emphasize that such examples cannot be found among
hyperbolic groups, because in that situation both the Martin boundary and the
ratio limit boundary coincide with the Gromov boundary. This brings new
impetus for the study of ratio limit boundaries on relatively hyperbolic
groups.

### Acknowledgements

The authors are grateful to Alon Dogon, Victor Gerasimov and Leonid Potyagailo
for comments, remarks and suggestions on previous versions of this paper.

##  2\. Random walks and relatively hyperbolic groups.

In this section we discuss necessary material from the the theory of random
walks and the theory of relatively hyperbolic groups. We refer the reader to
[70, 71, 72] for detailed treatments of topological boundary theory for random
walks, and to [11, 12, 60] for some of the basic theory of relatively
hyperbolic groups.

######  Definition 2.1.

Let $\Gamma$ be a discrete group, and $\mu:\Gamma\rightarrow[0,1]$ a
probability measure such that $\operatorname{supp}(\mu)$ generates $\Gamma$ as
a semigroup. We call such a probability measure _admissible_. We will say that
$\mu$ is _finitely supported_ if the set of $g\in\Gamma$ for which $\mu(g)>0$
is finite. The transition kernel $P$ on $\Gamma$ given by
$P(x,y)=\mu(x^{-1}y)$ is called the _random walk_ on $\Gamma$ determined (or
driven) by $\mu$.

The iterates of $P$ are given by $P^{n}(x,y)=\mu^{*n}(x^{-1}y)$ where
$\mu^{*n}$ is the $n$-th convolution power of $\mu$. We denote by $R:=R(P)$
the inverse of the spectral radius of $P$ given by

$R(P):=\big{(}\limsup_{n\rightarrow\infty}\sqrt[n]{P^{n}(x,y)}\big{)}^{-1},$

which is independent of the choice of $x,y\in G$. By a theorem of Kesten [53],
we know that $R>1$ whenever $G$ is non-amenable.

### Martin boundaries and $t$-harmonic functions

One way of measuring the behavior of a random walk at infinity arises from
considering compactifications of $\Gamma$ with respect to Green functions. Let
$\mu$ be a finitely supported and admissible probability measure on $\Gamma$.
The Green function of $\mu$ for $x,y\in G$ is given by

$G(x,y|r):=\sum_{n=0}^{\infty}P^{n}(x,y)r^{n},$

with radius of convergence $R$. The _$r$ -Martin kernel_
$K_{r}:\Gamma\times\Gamma\rightarrow(0,\infty)$ at $r\in[1,R]$ is then given
by

$K_{r}(x,y):=\frac{G(x,y|r)}{G(e,y|r)},$

and is well-defined even for $r=R$ by [71, Lemma 3.66]. Then, for any
$r\in[1,R]$, the compactification of $\Gamma$ (see for instance [71, Theorem
7.13]) with respect to the $r$-Martin kernel functions $\\{y\mapsto
K_{r}(x,y)\\}_{x\in\Gamma}$ defines the so-called _$r$ -Martin
compactification_ $\Delta_{M,r}\Gamma$, so that $y\mapsto K_{r}(x,y)$ extend
to a continuous functions on $\Delta_{M,r}\Gamma$.

In fact, since for every $x\in\Gamma$ we have $C_{x}>0$ so that
$K_{r}(x,y)\leq C_{x}$, the $r$-Martin compactification $\Delta_{M,r}\Gamma$
can also be obtained by completing the bounded metric $d_{M,r}$ on $\Gamma$
given by

(2.1)
$d_{M,r}(y,y^{\prime})=\sum_{x\in\Gamma}\frac{|K_{r}(x,y)-K_{r}(x,y^{\prime})|+|1_{y}-1_{y^{\prime}}|}{2^{\phi(x)}C_{x}},$

where $1_{y}$ is the characteristic function of $y$, and
$\phi:\Gamma\rightarrow\mathbb{N}$ is some bijection.

The action of $\Gamma$ on itself by left multiplication extends to an action
by homeomorphisms $\Gamma\curvearrowright\Delta_{M,r}\Gamma$, and the _$r$
-Martin boundary_ is then the closed $\Gamma$-invariant subspace
$\partial_{M,r}\Gamma=\Delta_{M,r}\Gamma\setminus\Gamma$. A sequence
$g_{n}\in\Gamma$ converges to a point $\xi\in\partial_{M,r}\Gamma$ if $g_{n}$
is outside any finite set and $\lim_{n}K_{r}(x,g_{n})=K_{r}(x,\xi)$.

The $r$-Martin compactification is intimately related to $r^{-1}$-harmonic
functions. We refer to [70, Section 24] for the theory of harmonic functions,
which is done for general Markov chains (and not just random walks). Most of
the literature deals with the case when $r=1$. To obtain results for the case
when $r\in(1,R]$, one often applies a _Doob transform_ or _$u$ -process_ to
$P$ via some positive $r^{-1}$-harmonic function $u$ to reduce to the case
when $r=1$ (see [70, Page 261]).

We say that a positive function $u:\Gamma\rightarrow(0,\infty)$ is _$t$
-harmonic_ if we have $t\cdot u(x)=\sum_{y\in\Gamma}P(x,y)u(y)$ for every
$x\in\Gamma$. We say that $u$ is _normalized_ if $u(e)=1$, and we denote by
$\mathcal{H}^{+}(P,t)$ the set of all positive normalized $t$-harmonic
functions for $P$, which we imbue with the topology of pointwise convergence.

The key attribute of $\partial_{M,r}\Gamma$ is that every normalized positive
$r$-harmonic function $u$ on $\Gamma$ has a Poisson-Martin integral
representation

(2.2) $\displaystyle
u(x)=\int_{\partial_{M,r}\Gamma}K_{r}(x,\xi)d\nu^{u}(\xi),$

where $\nu^{u}$ is a Borel probability measure on $\partial_{M,r}\Gamma$. We
call a function $u$ in $\mathcal{H}^{+}(P,t)$ _minimal_ if it cannot be
written as a convex combination of two distinct functions in
$\mathcal{H}^{+}(P,t)$. The _minimal $r$-Martin boundary_
$\partial_{M,r}^{m}\Gamma$ is the Borel set of points
$\xi\in\partial_{M,r}\Gamma$ for which $x\mapsto K(x,\xi)$ is minimal, or
equivalently those points $\xi$ for which $x\mapsto K(x,\xi)$ has a unique
representing measure, which must hence be a Dirac measure $\delta_{\xi}$. By
[70, Corollary 24.18] we know that the measure $\nu^{u}$ in equation (2.2)
would be the unique representing measure for $u$ if we require that
$\nu^{u}(\partial_{M,r}^{m}\Gamma)=1$.

When $\mu$ is finitely supported and admissible, by [70, Lemma 24.16] the
positive normalized functions $x\mapsto K_{r}(x,\xi)$ are $r^{-1}$-harmonic
for $\xi\in\partial_{M,r}\Gamma$, so that whenever
$\nu\in\operatorname{Prob}(\partial_{M,r}\Gamma)$, the function

$u_{\nu}(x)=\int_{\partial_{M,r}\Gamma}K_{r}(x,\xi)d\nu(\xi)$

is in $\mathcal{H}^{+}(P,r^{-1})$. Let us imbue
$\operatorname{Prob}(\partial_{M,r}\Gamma)$ with the weak* topology, which is
the smallest topology such that for every continuous function
$f:\partial_{M,r}\Gamma\to\mathbb{R}$ the map $\nu\mapsto\int f(\xi)d\nu(\xi)$
is continuous. Note that this is often called the weak topology on
$\operatorname{Prob}(\partial_{M,r}\Gamma)$ in literature on probability.
Define a map
$\varphi:\operatorname{Prob}(\partial_{M,r}\Gamma)\rightarrow\mathcal{H}^{+}(P,r^{-1})$
given by $\varphi(\nu)=u_{\nu}$. Since for every $x\in\Gamma$ the function
$K_{r}(x,\xi)$ is continuous on $\partial_{M,r}\Gamma$, it is easy to see that
$\varphi$ is continuous. That is, if $\nu_{\alpha}\rightarrow\nu$ in the weak*
topology, then $u_{\nu_{\alpha}}$ converges to $u_{\nu}$ pointwise. Since
every $u\in\mathcal{H}^{+}(P,r^{-1})$ has some representing measure $\nu^{u}$,
we see that $\varphi$ is a continuous surjective map.

### Ratio-limit boundaries

Suppose now that the random walk defined by $\mu$ on $\Gamma$ is aperiodic in
the sense that there exists $n_{0}$ such that $\mu^{*n}(e)>0$ for any $n\geq
n_{0}$, and let $\omega$ be a non-principle ultrafilter on $\mathbb{N}$. By
analogy with $R$-Martin kernel functions, for $x,y\in\Gamma$ we may define the
_ratio-limit kernel_ $H:\Gamma\times\Gamma\rightarrow(0,\infty)$ by

(2.3) $H(x,y):=\lim_{m\rightarrow\omega}\frac{P^{m}(x,y)}{P^{m}(e,y)}.$

By a result of Gerl [38, 39], we know that
$\lim_{m\rightarrow\infty}\frac{P^{m+1}(x,z)}{P^{m}(x,z)}=R^{-1}$, so that the
function $x\mapsto H(x,y)$ is $R^{-1}$-harmonic for every $y\in\Gamma$.

The compactification of $\Gamma$ with respect to the ratio-limit kernel
functions $\\{y\mapsto H(x,y)\\}_{x\in\Gamma}$ defines the ratio-limit
compactification $\Delta_{\rho}\Gamma$. Just like Martin compactifications, so
too can the ratio-limit compactification $\Delta_{\rho}\Gamma$ be obtained by
completing a bounded metric. Indeed, for every $x\in\Gamma$ we have $D_{x}>0$
so that $H(x,y)\leq D_{x}$, and the compactification $\Delta_{\rho}\Gamma$ is
obtained by completing the metric $d_{\rho}$ on $\Gamma$ given by

(2.4)
$d_{\rho}(y,y^{\prime})=\sum_{x\in\Gamma}\frac{|H(x,y)-H(x,y^{\prime})|+|1_{y}-1_{y^{\prime}}|}{2^{\phi(x)}D_{x}},$

where $1_{y}$ is the characteristic function of $y$, and
$\phi:\Gamma\rightarrow\mathbb{N}$ is some bijection.

Again the left multiplication of $\Gamma$ on itself extends to an action
$\Gamma\curvearrowright\Delta_{\rho}\Gamma$ by homeomorphisms, and the closed
$\Gamma$-invariant subspace
$\partial_{\rho}\Gamma:=\Delta_{\rho}\Gamma\setminus\Gamma$ is called the
_ratio-limit boundary_. A sequence $g_{n}\in\Gamma$ converges to a point
$\xi\in\partial_{\rho}\Gamma$ if $g_{n}$ is outside any finite set and
$\lim_{n}H(x,g_{n})=H(x,\xi)$.

We will say that the random walk determined by $\mu$ has the _strong ratio-
limit property_ (SRLP) if the following limits exist for every $x,y\in\Gamma$.

$H(x,y):=\lim_{m\rightarrow\infty}\frac{P^{m}(x,y)}{P^{m}(e,y)}.$

######  Remark 2.2.

Note that when the random walk determined by $\mu$ has SRLP, all of the above
defined notions are independent of the non-principle ultrafilter $\omega$. It
is still unknown to us whether the limits always exist as
$m\rightarrow\infty$, and whether the definition of ratio-limit boundary is
independent of the choice of non-principle ultrafilter.

Since $x\mapsto H(x,y)$ is $R^{-1}$-harmonic for every $y\in\Gamma$ and the
random walk we consider is finitely supported and admissible, it follows that
$H(x,\eta)$ is $R^{-1}$-harmonic for every $\eta\in\partial_{\rho}\Gamma$.
Hence, by Martin–Poisson representation theorem, for every
$\eta\in\Delta_{\rho}\Gamma$ there is a probability measure
$\nu^{\eta}\in\operatorname{Prob}(\partial_{M,R}\Gamma)$ such that

$H(x,\eta)=\int_{\partial_{M,R}\Gamma}K_{R}(x,\xi)d\nu^{\eta},$

which is unique if we require that $\nu^{\eta}(\partial_{M,R}^{m}\Gamma)=1$.

Since distinct elements of $\Delta_{\rho}\Gamma$ may give rise to the same
ratio-limit kernels, it makes sense to discuss a “separated” version of the
ratio-limit compactification and boundary.

######  Definition 2.3.

Suppose $\Gamma$ is a discrete group, and $\mu$ an aperiodic probability
measure on $\Gamma$. Let $\omega$ be a non-principle ultrafilter on
$\mathbb{N}$. We define the ratio-limit radical as the subset

$R_{\mu}^{\omega}:=R_{\mu}:=\\{\ g\in\Gamma\ |\ H(x,g)=H(x,e),\ \forall
x\in\Gamma\ \\}.$

In other words, $R_{\mu}$ is the largest subset of elements $g\in\Gamma$ of
all $R^{-1}$-harmonic functions $x\mapsto H(x,g)$ that coincide with the
$R^{-1}$-harmonic function $x\mapsto H(x,e)$. When $\mu$ is symmetric, $g\in
R_{\mu}$ if and only if for every $y\in\Gamma$ we have $H(g,y)=H(g,e)$. In
this case, the next result improves upon [23, Proposition 3.2], showing that
$R_{\mu}$ is always a _normal_ subgroup.

######  Proposition 2.4.

Let $\Gamma$ be discrete, $\mu$ an aperiodic probability measure on $\Gamma$,
and $\omega$ is an non-principle ultrafilter. Then $R_{\mu}$ is a subgroup. If
moreover $\mu$ is symmetric, then $R_{\mu}$ is normal.

###### Proof.

First note that we have the cocycle identity $H(gh,z)=H(h,g^{-1}z)H(g,z)$ for
$h,g,z\in\Gamma$. Hence, if $g,h\in R_{\mu}$ we get that

$H(x,gh)=\frac{H(g^{-1}x,h)}{H(g^{-1},h)}=\frac{H(g^{-1}x,e)}{H(g^{-1},e)}=H(x,g)=H(x,e)$

and

$H(x,g^{-1})=\frac{H(gx,e)}{H(g,e)}=\frac{H(gx,g)}{H(g,g)}=H(x,g^{-1}g)=H(x,e).$

Thus, $R_{\mu}$ is a subgroup. Next, if $\mu$ is symmetric, we have another
cocycle identity given by $H(gh,z)=H(g,zh^{-1})H(h,z)$ for $h,g,z\in\Gamma$.
Hence, if $g\in R_{\mu}$ and $h\in\Gamma$, we also have

$H(x,h^{-1}gh)=\frac{H(hx,gh)}{H(h,gh)}=\frac{H(hxh^{-1},g)/H(h^{-1},g)}{H(hh^{-1},g)/H(h^{-1},g)}=H(hxh^{-1},g)=$
$H(hxh^{-1},e)=H(xh^{-1},h^{-1})H(h,e)=H(x,e)H(h^{-1},h^{-1})H(h,e).$

But now, using homogeneity we get that $H(h^{-1},h^{-1})H(h,e)=1$, and we
obtain $H(x,h^{-1}gh)=H(x,e)$. Hence, $h^{-1}gh\in R_{\mu}$, and we see that
$R_{\mu}$ is normal. ∎

Next, note that the function $(x,yR_{\mu})\mapsto H(x,y)$ is well-defined on
$\Gamma\times\Gamma/R_{\mu}$, and we may use them to define the _reduced
ratio-limit compactification_ $\Delta_{\rho}^{r}\Gamma$. This is obtained by
completing the space $\Gamma/R_{\mu}$ with respect to the metric
$d_{\rho}^{r}$ on $\Gamma/R_{\mu}$ given by

$d_{\rho}^{r}(yR_{\mu},y^{\prime}R_{\mu})=\sum_{x\in\Gamma}\frac{|H(x,y)-H(x,y^{\prime})|}{2^{\phi(x)}D_{x}},$

where $\phi:\Gamma\rightarrow\mathbb{N}$ is some bijection. Left
multiplication of $\Gamma$ on $\Gamma/R_{\mu}$ then extends to an action
$\Gamma\curvearrowright\Delta_{\rho}^{r}\Gamma$ by homeomorphisms, and the
closed $\Gamma$-invariant subspace given by
$\partial_{\rho}^{r}\Gamma:=\Delta_{\rho}^{r}\Gamma\setminus[\Gamma/R_{\mu}]$
is called the _reduced ratio-limit boundary_.

By [72, Lemma 6.3] there is a $\Gamma$-equivariant continuous surjection
$\Delta_{\rho}\Gamma\rightarrow\Delta_{\rho}^{r}\Gamma$ which is injective on
$\partial_{\rho}\Gamma$. Thus, $R^{-1}$-harmonic functions arising from
compactifying via ratio-limit kernels, are parameterized by elements of
$\Delta_{\rho}^{r}\Gamma$. Since in this paper we will be mostly concerned
with non-amenable groups, the following corollary shows that we may always
assume that $\partial_{\rho}^{r}\Gamma$ is non-empty.

######  Corollary 2.5.

Let $\Gamma$ be discrete, $\mu$ a _symmetric_ aperiodic probability measure on
$\Gamma$, and $\omega$ a non-principle ultrafilter. Then
$\partial^{r}_{\rho}\Gamma=\emptyset$ if and only if $\Gamma$ is amenable.

###### Proof.

Suppose that $\partial^{r}_{\rho}\Gamma=\emptyset$. Then we must have that
$\Gamma/R_{\mu}$ is finite. Since for each $g\in R_{\mu}$ we have
$H(g,g)=H(g,e)$, and as $\mu$ is symmetric, we get that

$\lim_{n}\frac{P^{n}(e,g)}{P^{n}(e,e)}=\lim_{n}\frac{P^{n}(e,e)}{P^{n}(e,g)},$

which must hence be equal to $1$. The set of elements $g\in\Gamma$ satisfying
$\lim_{n}\frac{P^{n}(e,g)}{P^{n}(e,e)}=1$ was shown to be an amenable subgroup
by [34, Theorem 4.1 & Theorem 4.2]. Thus, we deduce that $R_{\mu}$ is
amenable. Since $R_{\mu}$ is amenable and $\Gamma/R_{\mu}$ is finite, we get
that $\Gamma$ is amenable.

Conversely, if $\Gamma$ is amenable and $\mu$ is symmetric, a result of Avez
[5] shows that $H(x,y)=1$ for all $x,y\in\Gamma$. Thus, $\Gamma=R_{\mu}$, and
hence $\partial_{\rho}^{r}\Gamma=\emptyset$. ∎

### Relatively hyperbolic group and their intrinsic geometry

Let $s$ be an isometry on a hyperbolic space $X$. We will say that $s$ is
hyperbolic if it has two fixed points $x^{\pm}_{s}$ on $\partial X$. Such an
isometry then has North-South dynamics on $\partial X$, in the sense that
$s^{\pm n}x\underset{n\rightarrow\infty}{\to}x^{\pm}_{s}$ for any
$x\in\partial X\setminus x^{\mp}_{s}$.

Suppose now that $\Gamma$ is a non-elementary, finitely generated group of
isometries of some proper, geodesic hyperbolic space $X$. We denote by
$\Lambda_{\Gamma}$ the closed set of limit points of $\Gamma x$ in the Gromov
boundary $\partial X$ for some element $x\in X$ (whose definition does not
depend on $x$), or equivalently the unique minimal closed $\Gamma$ invariant
subset of $\partial X$. Let $s\in\Gamma$ be hyperbolic as an isometry on $X$.
It is well-known that the set of all attractors and repellers of hyperbolic
elements

$\\{\ x_{s}^{\pm}\ |\ s\in\Gamma,\ \text{hyperbolic}\ \\}$

is dense in $\Lambda_{\Gamma}$, and that
$\Gamma\curvearrowright\Lambda_{\Gamma}$ is minimal (see for instance [60,
Proposition 4.7]).

Let $\Gamma$ be a finitely generated discrete group and $\Omega$ a finite
collection of subgroups. We construct the coned-off (sometimes called
”relative”) Cayley graph $\operatorname{Gr}(\Gamma;\Omega)$ as follows.
Consider the usual Cayley graph $\operatorname{Gr}(\Gamma)$, and for each
$P\in\Omega$ and coset $gP$ we add a vertex $gP$ and an edge of length
$\frac{1}{2}$ from $h$ to $gP$ for any $h\in gP$. The resulting graph is
$\operatorname{Gr}(\Gamma;\Omega)$, and the shortest path metric makes
$\operatorname{Gr}(\Gamma;\Omega)$ into a geodesic metric space. We will call
the shortest path metric on $\operatorname{Gr}(\Gamma)$ the _word metric_ and
the shortest path metric on $\operatorname{Gr}(\Gamma;\Omega)$ the _relative
metric_. A geodesic in $\operatorname{Gr}(\Gamma)$ will be called a _word
geodesic_ , and a geodesic in $\operatorname{Gr}(\Gamma,\Omega)$ will be
called a _relative geodesic_.

######  Definition 2.6.

Let $\Gamma$ be a finitely generated discrete group and $\Omega$ a finite
collection of subgroups. We say that $\Gamma$ is _hyperbolic relative to
$\Omega$_ if

  1. (1)

$\operatorname{Gr}(\Gamma;\Omega)$ is $\delta$ hyperbolic for some $\delta>0$

  2. (2)

For each $L\in\mathbb{N}$, any edge $e$ in $\operatorname{Gr}(\Gamma;\Omega)$
belongs to at most finitely many simple cycles of length $L$.

We let $\partial\operatorname{Gr}(\Gamma,\Omega)$ be the Gromov boundary of
$\operatorname{Gr}(\Gamma,\Omega)$ and we let
$V_{\infty}(\operatorname{Gr}(\Gamma,\Omega))$ be the set of vertices with
infinite degree in $\operatorname{Gr}(\Gamma,\Omega)$. We also set
$\partial_{B}(\Gamma;\Omega)=\partial\operatorname{Gr}(\Gamma,\Omega)\cup
V_{\infty}(\operatorname{Gr}(\Gamma,\Omega))$. In [12, Section 8], Bowditch
defines a topology on $\Gamma\cup\partial_{B}(\Gamma;\Omega)$ such that the
induced topology on
$\operatorname{Gr}(\Gamma,\Omega)\cup\partial\operatorname{Gr}(\Gamma,\Omega)$
is the visual topology on the Gromov completion of a hyperbolic space [12,
Proposition 8.5] and such that $V_{\infty}(\operatorname{Gr}(\Gamma,\Omega))$
is dense in $\partial_{B}(\Gamma;\Omega)$. The space
$\partial_{B}(\Gamma;\Omega)$ is compact [12, Proposition 8.6] and is called
the Bowditch boundary of $\Gamma$.

The action of $\Gamma$ on $\partial_{B}(\Gamma;\Omega)$ is minimal and
geometrically finite. That is, it is a convergence action (i.e. the action on
the space of discrete triples is properly discontinuous) and every point in
$\partial_{B}(\Gamma;\Omega)$ is either conical or bounded parabolic, where
$\xi\in\partial_{B}(\Gamma;\Omega)$ is

  1. (1)

a _conical point_ if there is a sequence $(g_{n})$ in $\Gamma$ and two
distinct points $\xi_{1},\xi_{2}\in\partial_{B}(\Gamma;\Omega)$ such that for
any $\xi\neq\zeta\in\partial_{B}(\Gamma;\Omega)$, the sequences $(g_{n}\xi)$
and $(g_{n}\zeta)$ converge to $\xi_{1}$ and $\xi_{2}$ respectively, or;

  2. (2)

a _bounded parabolic point_ if the stabilizer $\Gamma_{\xi}$ is infinite and
acts cocompactly on $\partial_{B}(\Gamma;\Omega)\setminus\\{\xi\\}$.

Relative hyperbolicity and Bowditch boundary can also be identified abstractly
as follows. Equivalently, we say that $\Gamma$ is relatively hyperbolic if
there is some proper, geodesic hyperbolic space $X$ on which $\Gamma$ acts by
isometries on $X$, geometrically finitely on $\partial X$, and the maximal
parabolic subgroups (i.e. infinite stabilizers of points of $\partial X)$ are
conjugate to elements of $\Omega$. An infinite order element $g$ in a
relatively hyperbolic group $\Gamma$ is then called hyperbolic if it is a
hyperbolic isometry on a space $X$ as above. The limit set $\Lambda_{\Gamma}$
in the Gromov boundary of such a space is then unique up to
$\Gamma$-equivariant homeomorphism [12, Theorem 9.4], and coincides with the
Bowditch boundary $\partial_{B}(\Gamma;\Omega)$ [12, Proposition 9.1]. Under
this identification, $\partial\operatorname{Gr}(\Gamma,\Omega)$ is the set of
conical limit points and $V_{\infty}(\operatorname{Gr}(\Gamma,\Omega))$ is the
set of bounded parabolic limit points. It follows from this definition that
$x_{s}^{\pm}$ are conical for every hyperbolic element $s\in\Gamma$, and that
there are only countably many bounded parabolic points in
$\partial_{B}(\Gamma;\Omega)$.

Let $\alpha$ be a word geodesic of a relatively hyperbolic group $\Gamma$. A
point $p\in\alpha$ is said to be an $(\epsilon_{1},\epsilon_{2})$ transition
point if the length $\epsilon_{2}$ sub-segment of $\alpha$ centered at $p$ is
not contained in an $\epsilon_{1}$ neighborhood of a single coset of a
parabolic subgroup. When $\Gamma$ is relatively hyperbolic, by [48,
Proposition 8.13] we know that relative geodesics are uniformly close to
geodesics in $\operatorname{Gr}(\Gamma)$ with respect to the Hausdorff
distance. More precisely, there exists $C>0$ such that the following holds for
any two $x,y\in\Gamma$. Inside $\operatorname{Gr}(\Gamma)$, the set of points
in $\Gamma$ on a relative geodesic from $x$ to $y$ is within Hausdorff
distance $C$ of the set of transition points on a geodesic from $x$ to $y$.

By [73, Lemma 2.20] we know there exist $\epsilon_{1},\epsilon_{2}>0$ such
that for every conical $x\in\partial_{B}(\Gamma;\Omega)$ and any word geodesic
ray $\gamma$ converging to $x$ there is a sequence of
$(\epsilon_{1},\epsilon_{2})$ transition points $p_{n}\in\gamma$ converging to
$x$. Moreover, by [32, Lemma 2.4] (which itself is a simple corollary of [50,
Lemma 1]) there exists $D>0$ such that for any triangle whose sides are word
geodesic segments or rays with vertices in
$\Gamma\cup\partial_{B}(\Gamma;\Omega)$, any $(\epsilon_{1},\epsilon_{2})$
transition point on one side is within $D$ of an $(\epsilon_{1},\epsilon_{2})$
transition point on one of the other two. We will need the following lemma for
what is to follow.

######  Lemma 2.7.

There are $\epsilon_{1},\epsilon_{2}>0$ such that for all $g\in\Gamma$ there
is a conical $x\in\partial_{B}\Gamma$ such that some word geodesic $[e,x]$
contains an $(\epsilon_{1},\epsilon_{2})$ transition point within
$2\epsilon_{2}$ of $g$.

###### Proof.

Let $\Omega_{D,\epsilon_{1},\epsilon_{2}}(g)$ denote the set of
$x\in\partial_{B}(\Gamma;\Omega)$ such that word some geodesic $[e,x]$
contains an $(\epsilon_{1},\epsilon_{2})$ transition point within
$2\epsilon_{2}$ of $g$ and some point within $D$ of $g$. By [73, Lemma 5.2],
there exist constants $D^{\prime},\epsilon_{1},\epsilon_{2}>0$ for which
$\Omega_{D^{\prime},\epsilon_{1},\epsilon_{2}}(g)$ has positive Patterson-
Sullivan measure. Moreover by [73, Theorem 1.7] the Patterson-Sullivan has no
atoms, and must hence give full weight to conical points. Hence, the
intersection of $\Omega_{D^{\prime},\epsilon_{1},\epsilon_{2}}(g)$ with the
set of conical points still has positive Patterson-Sullivan measure, and the
result follows. ∎

### Deviation inequalities of Green functions along geodesics

The following versions of Ancona inequalities for relative hyperbolic groups
were first proved in [37, Corollary 9.2] when $r=1$ and then in [33, Theorem
3.6] uniformly for $r\leq R$. We will call them the _weak relative Ancona
inequalities_ to contrast with a stronger version which will be described
below. We remark that the term ”relative” is used to insist on the fact that
in relatively hyperbolic groups, these deviation inequalities do not hold
along word geodesics, but only along transition points on them, or
equivalently along relative geodesics.

######  Proposition 2.8.

Let $\Gamma$ be a relatively hyperbolic group and $\mu$ a finitely supported,
admissible and symmetric probability measure on $\Gamma$. Then for any
$\epsilon_{1},\epsilon_{2},D>0$ there is a $C>0$ such that for any $1\leq
r\leq R$ and any $x,y,z\in\Gamma$ with $y$ within $D$ of an
$(\epsilon_{1},\epsilon_{2})$ transition point on a word geodesic $[x,z]$ we
have

(2.5) $G(x,z|r)\leq CG(x,y|r)G(y,z|r)$

Equivalently, by [48, Proposition 8.13], for any $D>0$ there is a $C>0$ such
that (2.5) holds for any $x,y,z$ with $y$ within word distance $D$ of an
_relative_ geodesic from $x$ to $z$.

We will also need the following deviation inequality for random walks on
relatively hyperbolic groups, which we call the _strong relative Ancona
inequality_. These were proved by the second and third named authors [33,
Theorem 3.14] and [29, Theorem 2.15]. For hyperbolic groups, these were first
proved for by Izumi-Neshveyev-Okayasu [49] for $r=1$ and by Gouezël [40]
uniformly for $r\leq R$.

We say that two relative geodesics $[x,y]$ and $[x^{\prime},y^{\prime}]$
$c$-fellow travel for a time at least $n$ if the $n$ first points of both
these relative geodesics are within $c$ of each other in the word metric on
the Cayley graph $\operatorname{Gr}(\Gamma)$.

######  Proposition 2.9.

Let $\Gamma$ be a relatively hyperbolic group and $\mu$ a finitely supported,
admissible and symmetric probability measure on $\Gamma$. Then for any $c>0$
there is a $C>0$ and $0<\alpha<1$ such that if $x,x^{\prime},y,y^{\prime}$ are
four points such that relative geodesics $[x,y]$ and $[x^{\prime},y^{\prime}]$
$c$-fellow travel for a time at least $n$ we have

(2.6)
$\left|\frac{G(x,y|r)G(x^{\prime},y^{\prime}|r)}{G(x,y|r)G(x^{\prime},y^{\prime}|r)}-1\right|\leq
C\alpha^{n}$

Finally, a pivotal connection between Bowditch boundary and $r$-Martin
boundary is given as follows. By [37, Theorem 1.5, Corollary 1.7 & Corollary
7.10] we know that for any $1\leq r\leq R$ there is a $\Gamma$-equivariant
surjection
$\pi:\Delta_{M,r}\Gamma\rightarrow\Gamma\cup\partial_{B}(\Gamma;\Omega)$ such
that $\pi|_{\Gamma}={\operatorname{id}}_{\Gamma}$ and such that for any point
$x\in\partial_{B}(\Gamma;\Omega)$ the pre-image $\pi^{-1}(x)$ contains a
minimal point in $\partial_{M,r}\Gamma$, and if
$x\in\partial_{B}(\Gamma;\Omega)$ is conical, then the pre-image $\pi^{-1}(x)$
is a singleton. Although these results are formally stated for the case of
$r=1$, the proofs go through verbatim for $r$-Martin kernels with any
$r\in[1,R]$. This is because weak relative Ancona inequalities hold for $r=R$
in [33, Theorem 3.6] when $\mu$ is symmetric, and the maximum principle for
Martin kernels also holds for $r$-Martin kernels for any $r\in[1,R]$.

##  3\. Minimal points, minimal actions, and strong proximality

In this section we show that for $\Gamma$ relatively hyperbolic, and $\mu$ a
finitely supported, admissible and symmetric measure on $\Gamma$, the closure
of minimal points $\overline{\partial_{M,R}^{m}\Gamma}$ is a minimal and
strongly proximal subspace of $R$-Martin boundary. It is a deep open problem
to determine whether minimal points of $\partial_{M,R}\Gamma$ are dense when
$\Gamma$ is relatively hyperbolic, and we provide examples for when we
automatically have $\overline{\partial_{M,R}^{m}\Gamma}=\partial_{M,R}\Gamma$.
We emphasize here that aside from strong proximality, the analogous statements
for $r=1$ were proved in [37, Corollary 1.6] and the arguments carry over to
the case where $r\in(1,R]$ (using the Ancona inequalities for $r=R$ from [33,
Theorem 3.6]). We provide a self-contained proof of them here for the benefit
of the reader.

In what follows, for two functions $u,w$ on $\Gamma$ we will denote
$u\asymp_{A}w$ if for the constant $A>0$ we have
$\frac{1}{A}\leq\frac{u(g)}{w(g)}\leq A$ for all $g\in\Gamma$. Suppose that
$\mu$ is as above, and recall that $d_{M,R}$ is the Green metric from Section
2. Then, since $P$ is $\Gamma$ invariant and irreducible, we get a local
Harnack inequality [72, Equation (4.3)] (See also [70, (25.1)]). That is, for
every $g\in\Gamma$ there is a constant $A>0$, depending only on the word
distance between $g$ and $h$, such that
$G(g,\cdot|r)=G(\cdot,g|r)\asymp_{A}G(\cdot,h|r)=G(h,\cdot|r)$ for any
$r\in[1,R]$.

######  Proposition 3.1.

(c.f. [37, Proposition 7.12]) Let $\Gamma$ be a non-elementary hyperbolic
relative to a finite collection of subgroups $\Omega$. Let $\mu$ be a finitely
supported, admissible and symmetric probability measure on $\Gamma$, and take
$r\in[1,R]$. Then, there exists a constant $C>0$ such that for any
$\xi\in\partial_{M,r}\Gamma$ there is a sequence of conical points $\zeta_{n}$
in the Bowditch boundary such that the sequence $(\pi^{-1}(\zeta_{n}))$
converges to $\alpha\in\partial_{M,r}\Gamma$ such that for all $g\in\Gamma$ we
have

$C\geq\frac{K_{r}(g,\alpha)}{K_{r}(g,\xi)}\geq\frac{1}{C}$

###### Proof.

Let $\xi\in\partial_{M,r}\Gamma$ and $x=\pi(\xi)$. Let $(g_{n})$ be a sequence
in $\Gamma$ which converges to $\xi$. By Lemma 2.7 there exist
$\epsilon_{1},\epsilon_{2}>0$ and a sequence $(\zeta_{n})$ of conical points
of the Bowditch boundary for which some geodesic ray $[e,\zeta_{n}]$ contains
an $(\epsilon_{1},\epsilon_{2})$ transition point within $2\epsilon_{2}$ of
$g_{n}$. Note that since $\zeta_{n}$ are conical, any geodesic ray with
endpoint $\zeta_{n}$ also converges to $\pi^{-1}(\zeta_{n})$ inside the
$R$-Martin boundary.

Now, by [32, Lemma 2.4] there is a constant $D>0$ such that for any
$g\in\Gamma$ and large enough $n$, any geodesic $[g,\zeta_{n}]$ has an
$(\epsilon_{1},\epsilon_{2})$ transition point within $D$ of $g_{n}$. By the
relative Ancona inequality (in the form of [37, Corollary 9.2]) there exists
$B>0$ independent of $n$ such that for a transition point
$q_{n}\in[e,\zeta_{n}]$ within $D$ of $g_{n}$ we have
$G(e,g_{n}|r)\asymp_{B}G(e,q_{n}|r)G(q_{n},g_{n}|r)$ for a transition point
$p_{n}\in[g,\zeta_{n}]$ within $D$ of $g_{n}$ we have
$G(g,g_{n}|r)\asymp_{B}G(g,p_{n}|r)G(p_{n},g_{n}|r)$. Let
$p_{n}\in[g,\zeta_{n}],q_{n}\in[e,\zeta_{n}]$ be unbounded sequences of
$(\epsilon_{1},\epsilon_{2})$ transition points with word distance (between
$p_{n}$ and $q_{n}$) at most $D$ and

$K_{r}(e,q_{n})/K_{r}(e,\pi^{-1}(\zeta_{n}))\in[(1+1/n)^{-1},1+1/n],$
$K_{r}(g,p_{n})/K_{r}(g,\pi^{-1}(\zeta_{n}))\in[(1+1/n)^{-1},1+1/n].$

By Harnack inequalities there is a constant $A>0$ such that
$G(p_{n},\cdot|r)=G(\cdot,p_{n}|r)\asymp_{A}G(\cdot,q_{n}|r)=G(q_{n},\cdot|r)$,
so we get that

$K_{r}(g,p_{n})=\frac{G(g,p_{n}|r)}{G(e,p_{n}|r)}\asymp_{A}\frac{G(g,p_{n}|r)}{G(e,q_{n}|r)}$
$\asymp_{B^{2}}\frac{G(g,g_{n}|r)G(q_{n},g_{n}|r)}{G(e,g_{n}|r)G(p_{n},g_{n}|r)}\asymp_{A}K_{r}(g,g_{n}).$

Hence, we get
$K_{r}(g,\pi^{-1}(\zeta_{n}))\asymp_{A^{2}B^{2}(1+\frac{1}{n})}K_{r}(g,g_{n})\to
K_{r}(g,\xi)$. Thus, if we take $C=A^{2}B^{2}$, any limit point $\alpha$ of
the sequence $(\pi^{-1}(\zeta_{n}))$ will satisfy

$C\geq K_{r}(g,\alpha)/K_{r}(g,\xi)\geq\frac{1}{C}.\qed$

######  Corollary 3.2.

Let $\Gamma$ be a non-elementary hyperbolic relative to a finite collection of
subgroups $\Omega$. Let $\mu$ be a finitely supported, admissible and
symmetric probability measure on $\Gamma$, and $r\in[1,R]$. Then any minimal
point in the $r$-Martin boundary is a limit of a sequence of preimages of
conical points. Hence, conical points are dense in
$\overline{\partial_{M,r}^{m}\Gamma}$.

###### Proof.

Suppose $\xi\in\partial_{M,r}\Gamma$ is minimal, and let $\alpha$ be a limit
point of pre-images of conical points $\pi^{-1}(\zeta_{n})$ such that $0\leq
K_{r}(g,\alpha)\leq CK_{r}(g,\xi)$. Minimality of $\xi$ implies that
$K_{r}(g,\alpha)=K_{r}(g,\xi)$, since these functions are normalized at $g=e$.
Thus, we get that $\alpha=\xi$. ∎

The following proposition shows that the action
$\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$ is strongly
proximal. The same result holds for $\partial_{M,r}\Gamma$ when its minimal
points are dense. Recall that $\Gamma\curvearrowright X$ with $X$ compact
Hausdorff is _strongly proximal_ if whenever $\nu\in\operatorname{Prob}(X)$
then the closure $\overline{\Gamma\nu}$ of its $\Gamma$ orbit contains a Dirac
mass $\delta_{x}$ for $x\in X$. By essentially the same proof as in [37,
Corollary 1.6] we know that
$\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$ is minimal for any
$r\in[1,R]$, and we provide our own proof of this for the benefit of the
reader.

######  Proposition 3.3.

Let $\Gamma$ be a non-elementary hyperbolic relative to a finite collection of
subgroups $\Omega$. Let $\mu$ be a finitely supported, admissible and
symmetric probability measure on $\Gamma$, and $r\in[1,R]$. Then the action
$\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$ is minimal and
strongly proximal.

###### Proof.

Let $s\in\Gamma$ be hyperbolic for a geometrically finite action
$\Gamma\curvearrowright X$. Since $x_{s}^{\pm}$ are conical, there pre-images
are minimal points in the Martin boundary and there are unique $s$-fixed
points $\xi_{s}^{\pm}$ in $\overline{\partial_{M,r}^{m}\Gamma}$ such that
$\pi(\xi_{s}^{\pm})=x_{s}^{\pm}$. Suppose that
$\xi\in\overline{\partial_{M,r}^{m}\Gamma}\setminus\\{\xi_{s}^{-}\\}$. We show
that $s^{n}\xi\rightarrow\xi_{s}^{+}$. Indeed, by contradiction assume that
$s^{n}\xi$ does not converge to $\xi_{+}$. Then, by compactness there is a
subequence of $(s^{n}\xi)$ which converges to $\xi^{\prime}\neq\xi_{+}$. Since
$\pi$ is $\Gamma$-equivariant, we get that $s^{n}\pi(\xi)$ converges to
$x^{+}=\pi(\xi^{+})=\pi(\xi^{\prime})$, contradicting the fact that the pre-
image of the conical point $x^{+}$ is a singleton.

Thus, every hyperbolic element $s\in\Gamma$ has an attractor and repeller
$\xi_{s}^{\pm}$ in $\overline{\partial_{M,r}^{m}\Gamma}$. The set of
$\xi\in\overline{\partial_{M,r}^{m}\Gamma}$, with $\pi(\xi)$ conical, is dense
in $\overline{\partial_{M,r}^{m}\Gamma}$, and the set $\\{\ x_{s}^{\pm}\ |\
s\in\Gamma,\ \text{hyperbolic}\ \\}$ is dense inside conical points of
$\partial_{B}(\Gamma;\Omega)$. Consequently, we get that the set of attractors
and repellers $\\{\ \xi_{s}^{\pm}\ |\ s\in\Gamma,\ \text{hyperbolic}\ \\}$ is
dense in $\overline{\partial_{M,r}^{m}\Gamma}$.

Now, as $\Gamma$ is non-elementary, there are two hyperbolic elements
$s,t\in\Gamma$ such that the set $\\{\xi_{s}^{\pm},\xi_{t}^{\pm}\\}$ has at
least three elements. We assume without loss of generality that
$\xi_{s}^{\pm}\neq\xi_{t}^{-}$. We will first show that
$\overline{\partial_{M,r}^{m}\Gamma}$ is strongly proximal. Fix
$\tau\in\operatorname{Prob}(\overline{\partial_{M,r}^{m}\Gamma})$. Then, since
$s\xi_{s}^{-}=\xi_{s}^{-}$, by the dominated convergence theorem we get that

$s^{n}\tau\rightarrow\tau(\\{\xi_{s}^{-}\\})\delta_{\xi_{s}^{-}}+(1-\tau(\\{\xi_{s}^{-}\\}))\delta_{\xi_{s}^{+}}.$

Also, since $\xi_{s}^{\pm}\neq\xi_{t}^{-}$ the same manipulation shows that
$t^{n}\xi_{s}^{-}\rightarrow\xi_{t}^{+}$, so that setting
$\nu=\tau(\\{\xi_{s}^{-}\\})\delta_{\xi_{s}^{-}}+(1-\tau(\\{\xi_{s}^{-}\\}))\delta_{\xi_{s}^{+}}$
we get that $t^{m}\nu\rightarrow\delta_{\xi_{t}^{+}}$. Thus, the closure of
the $\Gamma$-orbit of $\tau$ contains a Dirac measure, so that
$\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$ is strongly
proximal.

Next, we show that $\Gamma\curvearrowright\overline{\partial_{M,r}^{m}\Gamma}$
is minimal. If we restrict the above argument to Dirac measures, we have shown
above that for any element $\xi\in\overline{\partial_{M,r}^{m}\Gamma}$ and
hyperbolic element $s\in\Gamma$, the closure of the orbit $\Gamma\xi$ contains
$\xi_{s}^{\pm}$. Since $\overline{\partial_{M,r}^{m}\Gamma}$ is the closure of
$\\{\ \xi_{s}^{\pm}\ |\ s\in\Gamma,\ \text{hyperbolic}\ \\}$, we are done. ∎

There are many natural examples where minimal points are automatically dense
in $\partial_{M,r}\Gamma$ , so that the action
$\Gamma\curvearrowright\partial_{M,r}\Gamma=\overline{\partial_{M,r}^{m}\Gamma}$
is minimal and strongly proximal. For instance,

  1. (1)

when $\Gamma=\Gamma_{1}*...*\Gamma_{n}$ is a free product of infinite groups
$\Gamma_{1},...,\Gamma_{n}$ for $n\geq 2$, and $\mu$ is a finitely supported,
admissible, adapted and symmetric generating measure on this free product, by
[70, Theorem 26.1] the preimages of conical points are dense in the Martin
boundary, and hence so are minimal points in $\partial_{M,r}\Gamma$.

  2. (2)

when $\Gamma$ is a non-elementary hyperbolic, all points in
$\partial_{M,r}\Gamma$ are automatically minimal, because they are all
conical.

  3. (3)

More generally, when $\Gamma$ is non-elementary hyperbolic relative to a
finite collection of _virtually abelian_ subgroups, by [33, Theorem 1.4], all
points in $\partial_{M,r}\Gamma$ are minimal.

##  4\. Spectral non-degeneracy

In this section we discuss spectral non-degeneracy, which is one of our
standing assumptions for random walks on relatively hyperbolic groups. It
roughly means that the $R$-induced first-return time random walk on a
parabolic subgroup has spectral radius greater than one. Our goal is to show
that spectrally non-degenerate random walks are ubiquitous, by showing that
they exist on a variety of relatively hyperbolic groups.

Let $\Gamma$ be be hyperbolic relative to a finite collection of subgroups
$\Omega$ group and let $\mu$ be a finitely supported, admissible and symmetric
probability measure on $\Gamma$. For a parabolic subgroup $H\in\Omega$, we
consider the first return kernel $P_{r,H}$ to $H$ associated with the measure
$r\mu$. That is, for $x,y$ in $H$, we set

$P_{r,H}(x,y)=\sum_{n\geq 1}\sum_{z_{1},...,z_{n-1}\notin
H}r^{n}\mu(x^{-1}z_{1})...\mu(z_{n-1}^{-1}y).$

We denote by $G_{r,H}$ the Green function associated with $P_{r,H}$ and by
$R_{H}(r)$ the inverse of the spectral radius of $P_{r,H}$. By [33, Lemma
4.4], for every $r\leq R$, for every $x,y\in H$,

(4.1) $G(x,y|r)=G_{r,H}(x,y|1).$

In particular, since $G(x,y|R)$ is finite, for every $r\leq R$, $R_{H}(r)\geq
1$.

######  Definition 4.1.

Let $\Gamma$ be be hyperbolic relative to a finite collection of subgroups
$\Omega$ group and let $\mu$ be a finitely supported, admissible and symmetric
probability measure on $\Gamma$. We say that the random walk determined by
$\mu$ is _spectrally degenerate_ along a parabolic subgroup $H\in\Omega$ if
$R_{H}(R)=1$. We say that the random walk determined by $\mu$ is _spectrally
non-degenerate_ if it is not spectrally degenerate along any $H\in\Omega$.

Spectral non-degeneracy is of particular importance in the study of random
walks on relatively hyperbolic groups. For instance, it is a determining
property characterizing the homeomorphism type of the $R$-Martin boundary as
shown in [33], as well as a key property for establishing local limit theorems
[29]. Regardless of whether or not $\mu$ is spectrally non-degenerate, it
follows from equation (4.1) and [33, Lemma 6.2, Proposition 6.3],
respectively, that

(4.2) $G_{R,H}(e,e|1)<\infty\ \text{and}\
\frac{d}{dt}_{|t=1}G_{R,H}(e,e|t)<\infty.$

We now restrict our attention to free products, which are particular cases of
relatively hyperbolic groups. Following the terminology of [70], a probability
measure $\mu$ on $\Gamma_{1}*\Gamma_{2}$ is called adapted if it can be
written as

$\mu=\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2},$

where $\alpha_{i}\geq 0$, $\alpha_{1}+\alpha_{2}=1$ and $\mu_{i}$ are
probability measures on $\Gamma_{i}$ for $i=1,2$.

We assume that each $\mu_{i}$ are finitely supported, admissible and symmetric
on $\Gamma_{i}$ for $i=1,2$. We let $R_{i}$ be the inverse of the spectral
radius of $\mu_{i}$ and $G_{i}$ be the Green function associated with
$\mu_{i}$. For each $i=1,2$, by [70, Proposition 9.18] there exists a
continuous function $\zeta_{i}$ of $r$ such that for $x,y\in\Gamma_{i}$, for
$r\leq R$,

(4.3)
$\frac{G(x,y|r)}{G(e,e|r)}=\frac{G_{i}(x,y|\zeta_{i}(r))}{G_{i}(e,e|\zeta_{i}(r))}.$

Moreover, $\zeta_{i}(R)\leq R_{i}$ and $\zeta_{i}(r)<R_{i}$ if $r<R$. We also
define

$\theta=RG(e,e|R),\ \theta_{i}=R_{i}G_{i}(e,e|R_{i})$

and

(4.4)
$\overline{\theta}=\min\big{\\{}\frac{\theta_{1}}{\alpha_{1}},\frac{\theta_{2}}{\alpha_{2}}\big{\\}}.$

We define implicitly the function $\Phi$ as

(4.5) $G(e,e|r)=\Phi(rG(e,e|r)),$

which is analytic on an open neighborhood of the interval $(0,\theta)$ and we
consider the function $\Psi$ defined for $s<\theta$ as

(4.6) $\Psi(s)=\Phi(s)-t\Phi^{\prime}(s).$

Since $G(e,e|R)$ is finite, the functions $\Phi$ and $\Psi$ are well-defined
on $(0,\theta]$.

We define similarly the functions $\Phi_{i}$ and $\Psi_{i}$ associated with
the probability measures $\mu_{i}$. By [70, Theorem 9.19] we get that

(4.7) $\begin{split}\Phi(s)=\Phi_{1}(\alpha_{1}s)+\Phi_{2}(\alpha_{2}s)-1\\\
\Psi(s)=\Psi_{1}(\alpha_{1}s)+\Psi_{2}(\alpha_{2}s)-1,\end{split}$

This shows that both functions $\Phi$ and $\Psi$ can be extended to the
interval $(0,\overline{\theta}]$. The situation where
$\Psi(\overline{\theta})<0$ is called the ”typical case” in [70]. This is
equivalent to the fact that $\zeta_{i}(R)<R_{i}$ for $i=1,2$. By [29,
Proposition 2.9], this corresponds to the case of a spectrally non-degenerate
random walk. Thus, in order to prove the following Proposition, we need to
construct an example of an adapted random walk where
$\Psi(\overline{\theta})<0$.

######  Proposition 4.2.

Let $\Gamma=\Gamma_{1}*\Gamma_{2}$ be a non-elementary free product of two
groups $\Gamma_{1},\Gamma_{2}$. Then, there exists a symmetric adapted
probability measure $\mu$ with finite support generating $\Gamma$, which is
spectrally non-degenerate.

###### Proof.

We have two cases to consider. Either one of the free factors $\Gamma_{i}$ is
finite or both are infinite.

Let us first assume that $\Gamma_{1}$ is finite and by contradiction that
$\mu$ is spectrally degenerate along $\Gamma_{1}$. Then, $P_{R,\Gamma_{1}}$ is
a $\Gamma_{1}$-invariant transition kernel whose spectral radius is 1. Since
$\Gamma_{1}$ is finite, the irreducible transition kernel $P_{R,\Gamma_{1}}$
must be recurrent, so in particular we get that $G_{R,\Gamma_{1}}(e,e|1)$ is
infinite, yielding a contradiction with equation (4.2). Thus, for any choice
of $\alpha_{1},\alpha_{2}$, the measure $\mu$ cannot be spectrally degenerate
along $\Gamma_{1}$ when the latter is finite. Also, we deduce from [70,
Equation (9.20)] that

$\zeta_{i}(R)G_{i}(e,e|\zeta_{i}(R))=\alpha_{i}RG(e,e|R)=\alpha_{i}\theta.$

As $\alpha_{2}\rightarrow 0$ we get $\alpha_{1}\rightarrow 1$, so by [70,
Theorem 9.19] we have

$\zeta_{2}(R)\leq\zeta_{2}(R)G_{2}(e,e|\zeta_{2}(R))=\alpha_{2}\theta\leq\alpha_{2}\overline{\theta}\leq\alpha_{2}\theta_{1}/\alpha_{1}\to
0.$

hence for $\alpha_{2}$ small enough we may arrange that $\zeta_{2}(R)<R_{2}$,
so that $\mu$ is not either spectrally degenerate along $\Gamma_{2}$.

Next, without loss of generality we assume that both $\Gamma_{1}$ and
$\Gamma_{2}$ are infinite. Then, notice that both $\Gamma_{1}$ and
$\Gamma_{2}$ contain an element of order at least 3. Indeed, if without loss
of generality we assume towards contradiction that $\Gamma_{1}$ does not, then
every element of $\Gamma_{1}$ has order 2, which forces $\Gamma_{1}$ to be
abelian. As $\Gamma_{1}$ is finitely generated, we must have that
$\Gamma_{1}=(\mathbb{Z}/2\mathbb{Z})^{k}$. This implies that $\Gamma_{1}$ is
finite, which is a contradiction. Thus, both $\Gamma_{1}$ and $\Gamma_{2}$
contain an element of order at least 3. Now, by [70, Lemma 17.9], for each
$i=1,2$ there exists a probability measure $\mu_{i}$ on $\Gamma_{i}$ such that
$\Psi_{i}(\theta_{i})<1/2$. Fixing $(\alpha_{1},\alpha_{2})$ such that
$\theta_{1}/\alpha_{1}=\theta_{2}/\alpha_{2}=\overline{\theta}$, we have by
equation (4.7) that $\Psi(\overline{\theta})<0$, so that $\mu$ is spectrally
non-degenerate. ∎

Our next goal is to provide examples where spectral non-degeneracy is
automatic for relatively hyperbolic groups with virtually nilpotent parabolic
subgroups of low homogeneous dimension.

By a celebrated result of Gromov [42], finitely generated virtually nilpotent
groups are exactly discrete groups of polynomial volume growth. That is, balls
grow asymptotically like $n^{d}$. Bass [7] and Guivarc’h [43] independently
identified $d$ as the homogeneous dimension of $\Gamma$.

######  Definition 4.3.

Let $\Gamma$ be a nilpotent group. Let $\Gamma_{1}=\Gamma$ and
$\Gamma_{n}=[\Gamma_{n-1},\Gamma]$. Let $N_{\Gamma}$ be the nilpotency class
of $\Gamma$, which is the largest integer such that $\Gamma_{N}$ is not
trivial.

  1. (1)

The _rank_ of $\Gamma$ is given by

$\mathrm{rank}(\Gamma)=\sum_{k=1}^{N_{\Gamma}}\mathrm{rank}\left(\Gamma_{k}/\Gamma_{k+1}\right).$

  2. (2)

The _homogeneous dimension_ of $\Gamma$ is given by

$d=\sum_{k=1}^{N_{\Gamma}}k\cdot\mathrm{rank}\big{(}\Gamma_{k}/\Gamma_{k+1}\big{)}.$

When $\Gamma$ is virtually nilpotent, we define its homogeneous dimension as
the homogeneous dimension of a finite index nilpotent subgroup.

Note that when $\Gamma$ is nilpotent, all groups $\Gamma_{k}/\Gamma_{k+1}$ are
finitely generated abelian groups and must therefore have a well defined rank.
Moreover, when $\Gamma$ is virtually nilpotent the above definitions turn out
to be independent of the finite index nilpotent subgroup. This can be proved
by direct computations, but this also follows from the fact that $d$ is the
degree of the growth of balls, which is invariant under quasi-isometry, and
therefore invariant up to finite index.

In [33, Proposition 6.1], it is proved that if $\Gamma$ is hyperbolic relative
to virtually _abelian_ parabolic subgroups of rank at most $4$, then every
finitely supported, admissible and symmetric probability measure is spectrally
non-degenerate. We extend this result to relatively hyperbolic groups with
respect to virtually _nilpotent_ parabolic subgroups of homogenous dimension
at most $4$.

######  Proposition 4.4.

Let $\Gamma$ be hyperbolic relative to a finite collection of virtually
nilpotent parabolic subgroups $\Omega$, and let $\mu$ be a finitely supported,
admissible and symmetric probability measure on $\Gamma$. Let $H\in\Omega$ be
virtually nilpotent of homogeneous dimension at most 4. Then, $\mu$ is not
spectrally degenerate along $H$.

###### Proof.

Let $\eta\geq 0$. We consider the first return kernel $P_{R,H,\eta}$ to the
$\eta$-neighborhood $N_{\eta}(H)$ of $H$, associated with the measure $R\mu$.
We let $G_{R,H,\eta}$ be the corresponding Green function. By assumption, $H$
is virtually nilpotent. As in [33, Section 4], we may identify the
$\eta$-neighborhood of $H$ as $\mathcal{N}\times\\{1...,N_{\eta}\\}$, where
$\mathcal{N}$ is a finite index nilpotent subgroup of $H$. We then write
$\mathcal{A}$ for the abelianization of $\mathcal{N}$ which is thus a finitely
generated abelian group of rank $d_{\mathcal{A}}$ and we set $\pi$ the
projection from $\mathcal{N}$ to $\mathbb{Z}^{d_{\mathcal{A}}}$. Following
[33], for fixed $u\in\mathbb{R}^{d_{\mathcal{A}}}$, and
$j,k\in\\{1,...,N_{\eta}\\}$, we define

$F_{j,k}(u)=\sum_{x\in\mathcal{N}}P_{R,H,\eta}((e,j),(x,k))\mathrm{e}^{\pi(x)\cdot
u}.$

The matrix $F(u)$ with entries $F_{j,k}(u)$ is irreducible and has a dominant
eigenvalue that we denote by $\lambda(u)$. We also consider a left eigenvector
$\nu(u)$ and a right eigenvector $C(u)$ associated with $\lambda(u)$ and we
normalize them by declaring that $\nu(u)\cdot C(u)=1$.

By definition, spectral degeneracy along $H$ means that for $\eta=0$, the
spectral radius of $P_{R,H,\eta}$ is 1. By [33, Lemma 4.9], this implies the
same thing for every positive $\eta$. Consequently, by [33, Lemma 4.8], the
minimum of the function $\lambda(u)$ is 1. Also by [33, Proposition 4.10], the
1-Martin boundary of $P_{R,H,\eta}$ is reduced to a point, and this implies by
[33, Lemma 4.5] that the set of $u$ such that $\lambda(u)=1$ is reduced to a
point. Since the initial random walk determined by $\mu$ is symmetric, we see
that $F(u)$ is symmetric and so the minimum of $\lambda$ is necessarily
reached at $u=0$. We conclude that

(4.8) $\nu(0)F(0)C(0)=1.$

Now define the averaged transition kernel on $\mathcal{N}$ by setting

$\tilde{P}(x,y):=\sum_{j,k}\nu_{j}(0)P_{R,H,\eta}((x,j),(y,k))C_{k}(0).$

Equation (4.8) can then be reformulated as

$\sum_{x\in\mathcal{N}}\tilde{P}(e,x)=1.$

Moreover, as the initial random walk determined by $\mu$ is invariant by a
subgroup action, we get that the first return kernel $P_{R,H,\eta}$ is
$\mathcal{N}$-invariant, and hence so is $\tilde{P}$. Therefore, $\tilde{P}$
is a random walk on $\mathcal{N}$ which is determined by some symmetric
probability measure $\mu_{\mathcal{N}}$ on $\mathcal{N}$.

By [33, Lemma 4.6], if $\eta$ is chosen large enough, then $P_{R,H,\eta}$ has
exponential moments. We fix such an $\eta$. It follows that $\tilde{P}$ also
has exponential moments, since the coordinates of $\nu(0)$ and $C(0)$ are
finite. Finally, by [33, Proposition 6.3] we get that,

$\sum_{j,k}\sum_{x\in\mathcal{N}}G_{R,H,\eta}((e,j),(x,k)|1)G_{R,H,\eta}((x,k),(e,j)|1)$

is finite. We let $G_{R,H,\eta}(x,y|1)$ be the matrix whose $(j,k)$ entry is
given by $G_{R,H,\eta}((x,j)(y,k)|1)$. Denoting by $\widetilde{G}$ the Green
function associated with $\tilde{P}$, we get that

$\sum_{x\in\mathcal{N}}\widetilde{G}(e,x|1)\widetilde{G}(x,e|1)=\sum_{x\in\mathcal{N}}\nu\cdot
G_{R,H,\eta}(e,x|1)\cdot C\cdot\nu\cdot G_{R,H,\eta}(x,e|1)\cdot C.$

Recalling that $\nu$ and $C$ are normalized so that $\nu\cdot C=1$ and that
they have bounded coordinates, we get that

$\sum_{x\in\mathcal{N}}\widetilde{G}(e,x|1)\widetilde{G}(x,e|1)$

is finite. By [33, Lemma 6.2], we get that
$\frac{d}{dt}|_{t=1}\widetilde{G}(e,e|t)$ is finite as well.

To summarize, we constructed a symmetric probability measure
$\mu_{\mathcal{N}}$ whose support (which is not necessarily finite) generates
$\mathcal{N}$ as a semigroup, where $\mathcal{N}$ is a finite index nilpotent
subgroup of $H$ with exponential moments, and such that the associated Green
function has finite derivative at $1$. Let $\nu_{H}$ be an auxiliary finitely
supported, admissible, symmetric probability measure on $H$. By Pittet and
Saloff-Coste comparison theorems [59] (see also [70, Theorem 15.1]), we have

$\mu_{\mathcal{N}}(e)^{*n}\succeq\nu_{H}^{*n}(e),$

which means that there exist $C_{1},C_{2},C_{3}>0$ such that sufficiently
large $n\in\mathbb{N}$ we have,

$\mu_{\mathcal{N}}^{*n}(e)\geq C_{1}\sup\big{\\{}\nu_{H}^{*k}(e),C_{2}n\leq
k\leq C_{3}n\big{\\}}.$

Since $\nu_{H}$ has finite support, by a result of Alexopoulos [1, Corollary
1.17], there exists $C>0$ such that $\nu_{H}^{*k}(e)\sim Ck^{-d/2}$ as
$k\rightarrow\infty$. Thus, we find that

$\mu_{\mathcal{N}}^{*n}(e)\geq C^{\prime}n^{-d/2}$

for some constant $C^{\prime}$, where $d$ is the homogeneous dimension of $H$.
When $d\leq 4$, this implies that the derivative at 1 of the Green function
$t\mapsto\widetilde{G}(e,e|t)$ is infinite, arriving at a contradiction.
Hence, we see that $\mu$ is not spectrally degenerate. ∎

Let $M$ be a geometrically finite Riemannian manifold of pinched negative
curvature. Then, the fundamental group $\pi_{1}(M)$ is relatively hyperbolic
with respect to the cusp stabilizers, see [10] and [12].

######  Corollary 4.5.

Let $M$ be a geometrically finite Riemannian manifold of pinched negative
curvature and let $\mu$ be a finitely supported, admissible and symmetric
probability measure on $\pi_{1}(M)$. Assume that either

  1. (1)

$M$ has constant negative curvature and $\mathrm{dim}(M)\leq 5$; or,

  2. (2)

$\mathrm{dim}(M)\leq 4$.

Then $\mu$ is a spectrally non-degenerate random walk on $\pi_{1}(M)$.

###### Proof.

There are two cases. Either $M$ has constant negative curvature, or variable
negative curvature. If $M$ has constant negative curvature, the case is
treated as in [33, Theorem 1.5]. The result follows from the fact that
horospheres can be given a Euclidean structure of dimension at most
$\mathrm{dim}(M)-1$. Assuming that $\mathrm{dim}(M)\leq 5$, this implies that
cusp stabilizers are virtually abelian of rank at most $4$ (see [61, Chapter
5] for more details).

Thus, let us consider the case of variable negative curvature. Let $H$ be a
parabolic subgroup of $\pi_{1}(M)$. The assumption that $M$ is geometrically
finite implies that $H$ acts co-compactly on horospheres centered at the
parabolic point fixed determining $H$. By [14, Main Theorem, Section 1.5], $H$
contains a finite index torsion free nilpotent group $\Gamma$ of rank
$\mathrm{dim}(M)-1$.

Let $N_{\Gamma}$ be the nilpotency class of $\Gamma$ and recall that we
defined the rank of a virtually nilpotent group in Definition 4.3. Let us
prove that whenever $\mathrm{rank}(\Gamma)\leq 3$, the homogeneous dimension
of $\Gamma$ is at most $4$. First, we may assume that $\Gamma$ is not
virtually cyclic, for otherwise its homogeneous dimension is at most $1$.
Then, according to [32, Lemma 6.6], the rank of $\Gamma_{1}/\Gamma_{2}$ is at
least 2. We first deduce that $N_{\Gamma}$ is at most 2. Furthermore, if
$\mathrm{rank}(\Gamma)=2$, then we necessarily have that $\Gamma$ is virtually
$\mathbb{Z}^{2}$ and its homogeneous dimension is then also $2$. Assuming now
that $\mathrm{rank}(\Gamma)=3$, we only have two possibilities. Either
$\mathrm{rank}(\Gamma_{1}/\Gamma_{2})=3$, in which case $\Gamma$ is virtually
$\mathbb{Z}^{3}$ and has homogeneous dimension $3$, or
$\mathrm{rank}(\Gamma_{1}/\Gamma_{2})=2$ and
$\mathrm{rank}(\Gamma_{2}/\Gamma_{3})=1$, in which case the homogeneous
dimension is $4$. ∎

######  Example 4.6.

In case where $M$ has varying negative curvature, the given bound
$\mathrm{dim}\ M\leq 4$ is optimal to ensure that the homogeneous dimension of
parabolic subgroups is at most $4$. Indeed, consider the discrete Heisenberg
group $\mathrm{H}_{3}(\mathbb{Z})$, and consider
$\Gamma=\mathrm{H}_{3}(\mathbb{Z})\times\mathbb{Z}$. Then, $\Gamma$ is
nilpotent and has homogeneous dimension $5$. Furthermore, $\Gamma$ is a
lattice in the nilpotent Lie group
$\mathcal{N}=\mathrm{H}_{3}(\mathbb{R})\times\mathbb{R}$, which has
topological dimension $4$. Now, $\mathcal{N}/\Gamma$ is a nilmanifold, so by
[57, Corollary 6] we get that $\Gamma$ can be realized as a cusp stabilizer of
a pinched negatively curved manifold of finite volume and of dimension $5$.

When negative curvature is constant, the bound $\mathrm{dim}(M)\leq 5$ is also
optimal, as for any finite volume non-compact hyperbolic manifold of dimension
$6$, the cusp stabilizers are virtually $\mathbb{Z}^{5}$.

Thus, we see that spectrally non-degenerate random walks on relatively
hyperbolic are abundant. In [70], spectrally non-degenerate adapted random
walks on free products are referred to as the “typical case”. We conjecture
that if $\Gamma$ is relatively hyperbolic, and if we take $\mu_{n}$ as the
uniform measure on $n$-balls $B_{n}$ (around the identity element $e$) in
$\operatorname{Gr}(\Gamma)$, then for large enough $n$ the random walk
determined by $\mu_{n}$ will be spectrally non-degenerate.

##  5\. Asymptotics of Ratio limits

Suppose that $\Gamma$ is a non-elementary hyperbolic relative to a finite
collection of subgroups $\Omega$, and that $\mu$ is a finitely supported,
admissible, aperiodic and symmetric probability measure on $\Gamma$. To
analyze the ratio-limit boundary for random walks we will need to know the
asymptotic behavior of derivatives of Green function as $r\rightarrow R$. This
will lead us to a formula for the ratio-limit kernels which involves the
following iterated Green sums.

Following [28, 31], we say that the random walk determined by $\mu$ is
convergent if $\lim_{r\rightarrow R}G^{\prime}(e,e|r)$ exists. This occurs if
and only if $\lim_{r\rightarrow R}G^{\prime}(x,y|r)$ exists for any
$x,y\in\Gamma$. If $\mu$ is not convergent, we will say that it is divergent.
When $\Gamma$ is hyperbolic relative to virtually abelian subgroups and $\mu$
is convergent, we define its spectral degeneracy rank $d$ as the smallest rank
of a parabolic subgroup along which $\mu$ is spectrally degenerate.

For $x,y\in\Gamma$ let us denote

$I^{(s)}(x,y|r):=\sum_{x_{1},...,x_{s}\in\Gamma}G(x,x_{1}|r)G(x_{1},x_{2}|r)...G(x_{s-1},x_{s}|r)G(x_{s},y|r).$

By [31, Proposition 5.2], if there is $s\in\mathbb{N}$ which is the smallest
integer with $\underset{r\rightarrow R}{\lim}G^{(s)}(x,y|r)=\infty$ (or
equivalently $\underset{r\rightarrow R}{\lim}I^{(s)}(x,y|r)=\infty$), then
there is a constant $C>0$ independent of $x,y\in\Gamma$ so that

$I^{(s)}(x,y|r)\underset{r\rightarrow R}{\sim}C\cdot G^{(s)}(x,y|r).$

To arrive at a formula for the ratio-limit kernels involving the expressions
$I^{(s)}(x,y|r)$, we will need to relate the asymptotic behavior for higher
derivatives of Green functions with that of $P^{n}(x,y)$. This was established
by the second author and collaborators in a sequence of papers [28, 29, 31].

  1. (1)

When the random walk is spectrally non-degenerate, by [29, Corollary 7.1], we
know that $s=1$ is the smallest integer such that we have
$\underset{r\rightarrow R}{\lim}G^{(s)}(x,y|r)=\infty$ and for any
$x,y\in\Gamma$ there exists $\beta(x,y)>0$ such that

$G^{\prime}(x,y|r)\underset{r\rightarrow
R}{\sim}\frac{\beta(x,y)}{\sqrt{R-r}};\ \ \text{or,}$

  2. (2)

when $\Gamma$ is hyperbolic relative to virtually abelian subgroups and $\mu$
convergent, with spectral degeneracy rank $d$, by [31, Corollary 6.2], we know
that $s=\lceil\frac{d}{2}\rceil-1$ is the smallest integer such that we have
$\underset{r\rightarrow R}{\lim}G^{(s)}(x,y|r)=\infty$ and for any
$x,y\in\Gamma$ there exists $\beta(x,y)>0$ such that either

(5.1) $G^{(s)}(x,y|r)\underset{r\rightarrow
R}{\sim}\frac{\beta(x,y)}{\sqrt{R-r}},\ \text{if }d\ \text{is odd, or}$ (5.2)
$G^{(s)}(x,y|r)\underset{r\rightarrow
R}{\sim}\beta(x,y)\cdot\log\Big{(}\frac{1}{{R-r}}\Big{)}\ \text{if }d\
\text{is even}.$

In the case where equation (5.1) is satisfied, following the calculations in
[41, Section 9] (or equivalently those that are based on Karamata’s Tauberian
theorem [8, Corollary 1.7.3]), there is a constant $C>0$ independent of
$x,y\in\Gamma$ such that

(5.3)
$P^{n}(x,y)\underset{n\rightarrow\infty}{\sim}C\cdot\beta(x,y)R^{-n}n^{-\frac{d}{2}}.$

In the case where equation (5.2) is satisfied, following the calculation
performed after [31, Corollary 6.1]), we also get a constant $C>0$ independent
of $x,y\in\Gamma$ such that equation (5.3) holds.

Since we need to know that the constant $C>0$ is independent of
$x,y\in\Gamma$, let us see how we obtain equation (5.3) from equation (5.2).
Indeed, taking the slowly varying function “$\log$” in [8, Corollary 1.7.3] we
get

$\sum_{k=0}^{n}k^{s}R^{k}P^{k}(x,y)\underset{n\rightarrow\infty}{\sim}R^{s}\cdot\beta(x,y)\cdot\log(n)$

By [41, Corollary 9.4], for $x\neq y$ there exist $\delta>0$ and a non-
increasing sequences $(q_{n}(x,y))_{n}$ such that

$R^{n}P^{n}(x,x)=q_{n}(x,x)+O(e^{-\delta n}),\ \text{and}$
$R^{n}(P^{n}(x,x)+P^{n}(x,y))=q_{n}(x,y)+O(e^{-\delta n}).$

Thus, we deduce that

$\sum_{k=0}^{n}k^{s}q_{k}(x,x)\underset{n\rightarrow\infty}{\sim}R^{s}\beta(x,y)\log(n)$

Hence, by the same proof as in [41, Lemma 9.5] (with $\beta=1$) we get that

$n^{s}q_{n}(x,x)\underset{n\rightarrow\infty}{\sim}R^{s}\beta(x,x)n^{-1},$

so we get that

$P^{n}(x,x)\underset{n\rightarrow\infty}{\sim}\beta(x,x)R^{s-n}n^{-s-1}.$

Now, since

$G^{(s)}(x,x|r)+G^{(s)}(x,y|r)\underset{r\rightarrow
R}{\sim}(\beta(x,x)+\beta(x,y))\log\Big{(}\frac{1}{R-r}\Big{)},$

we may again apply [8, Corollary 1.7.3] to get that

$\sum_{k=0}^{n}k^{s}q_{k}(x,y)\underset{n\rightarrow\infty}{\sim}(\beta(x,x)+\beta(x,y))\log(n),$

so that again the same proof as in [41, Lemma 9.5] (with $\beta=1$) implies
that

$n^{s}q_{n}(x,y)\underset{n\rightarrow\infty}{\sim}R^{s}\cdot(\beta(x,x)+\beta(x,y))n^{-1}.$

Thus,

$P^{n}(x,x)+P^{n}(x,y)\underset{n\rightarrow\infty}{\sim}(\beta(x,x)+\beta(x,y))R^{s-n}n^{-s-1},$

so, we finally get that for any $x,y\in\Gamma$

$P^{n}(x,y)\underset{n\rightarrow\infty}{\sim}\beta(x,y)R^{s-n}n^{-\frac{d}{2}}.$

Hence, we obtain equation (5.3). Such local limit theorems are then used to
obtain the following result.

######  Corollary 5.1.

Let $\Gamma$ be a finitely generated non-elementary relatively hyperbolic
group, and assume that $\mu$ is a finitely supported, admissible, aperiodic
and symmetric probability measure on $\Gamma$. Suppose either that

  1. (1)

the random walk is spectrally non-degenerate, in which case we take $s=1$ or;

  2. (2)

that $\Gamma$ is hyperbolic relative to virtually abelian subgroups, with
$\mu$ convergent, with spectral degeneracy rank $d$, in which case we take
$s=\lceil\frac{d}{2}\rceil-1$.

Then, the ratio-limit kernel satisfies

$H(x,y)=\lim_{r\rightarrow R}\frac{I^{(s)}(x,y|r)}{I^{(s)}(e,y|r)}.$

###### Proof.

By combining equations (5.1) and (5.3) in the first case, and either (5.1) and
(5.3) when $d$ is odd or (5.2) and (5.3) when $d$ is even in the second case,
we get that

$H(x,y)=\frac{\beta(x,y)}{\beta(e,y)}=\lim_{r\rightarrow
R}\frac{I^{(s)}(x,y|r)}{I^{(s)}(e,y|r)}.\qed$

In [72, Question 7.1(a)], it is asked under which conditions the conclusion of
Corollary 5.1 holds with $s=1$. We give the following sufficient condition.

######  Proposition 5.2.

Let $\Gamma$ be a finitely generated non-elementary relatively hyperbolic
group, and assume that $\mu$ is a finitely supported, admissible and symmetric
probability measure on $\Gamma$. Let $0<s\in\mathbb{N}$ be so that for any
$0\leq k<s$ we have that $\lim_{r\rightarrow R}G^{(s)}(x,y|r)$ exists. Then,
for all $0\leq k<s$ the function $\mathfrak{H}_{k}$ given for $x,y\in\Gamma$
by

$\mathfrak{H}_{k}(x,y):=\lim_{r\rightarrow
R}\frac{I^{(k)}(x,y|r)}{I^{(k)}(e,y|r)}$

is not $R^{-1}$-harmonic. In particular, if there exists $s\in\mathbb{N}$
which is the smallest integer satisfying $H(x,y)=\mathfrak{H}_{s}(x,y)$ (for
some non-principle ultrafilter $\omega$), then $\lim_{r\rightarrow
R}G^{(s)}(x,y|r)=\infty$.

###### Proof.

Define inductively

$F_{1}(x,y|r):=\frac{d}{dr}(rG(x,y|r)),$

and for $k\geq 2$,

$F_{k}(x,y|r):=\frac{d}{dr}(r^{2}F_{k-1}(x,y|r)).$

By [28, Lemma 3.2] we have that $F_{k}(x,y|r)=k!r^{k-1}I^{(k)}(x,y|r)$. Thus,
we have for all $0\leq k<s$ that

$\mathfrak{H}_{k}(x,y)=\frac{F_{k}(x,y|R)}{F_{k}(x,y|R)}.$

We will prove by induction on $k$, for $0\leq k<s$, that for any $y\in\Gamma$
the function $x\mapsto\mathfrak{H}_{k}(x,y)$ is $R^{-1}$-superharmonic but not
$R^{-1}$-harmonic.

For the base case $k=0$ we get that
$\mathfrak{H}_{0}(x,y)=K_{R}(x,y)=\frac{G(x,y|R)}{G(e,y|R)}$ is the Martin
kernel, and a straightforward calculation shows that for each $y\in\Gamma$,
the function $x\mapsto G(x,y|R)$ is $R^{-1}$-superharmonic but not
$R^{-1}$-harmonic at $y$. Thus, $\mathfrak{H}_{0}(x,y)$ is
$R^{-1}$-superharmonic but not $R^{-1}$-harmonic.

Now let $0<k<s-1$ and assume that $\mathfrak{H}_{k}$ is $R^{-1}$-superharmonic
but not $R^{-1}$-harmonic. Then, since $F_{k}(x,y|r)$ only involves Green
derivatives of order at most $k$, its derivative
$\frac{d}{dr}|_{r=R}(F_{k}(x,y|r))$ at $r=R$ would involve Green derivatives
of order at most $k+1$, and hence converges as $r\rightarrow R$. Thus,
$F^{\prime}_{k}(x,y|R)$ is well-defined and is still $R^{-1}$-superharmonic in
$x$. By inductive definition we get

$F_{k+1}(x,y|R)-R\sum_{z\in\Gamma}P(x,z)F_{k+1}(z,y|R)=$
$2R\Big{[}F_{k}(x,y|R)-R\sum_{z\in\Gamma}P(x,z)F_{k}(z,y|R)\Big{]}+$
$R^{2}\Big{[}F^{\prime}_{k}(x,y|R)-R\sum_{z\in\Gamma}P(x,z)F^{\prime}_{k}(z,y|R)\Big{]}.$

Since $F_{k}(x,y|R)$ and $F^{\prime}_{k}(x,y|R)$ are $R^{-1}$-superharmonic in
$x$, the above expression is non-negative, but $F_{k}(x,y|R)$ is not
$R^{-1}$-harmonic in $x$, so we get that $F_{k+1}(x,y|R)$ is also
$R^{-1}$-superharmonic but not $R^{-1}$ harmonic in $x$. Since all expressions
involved in the definition of $x\mapsto\mathfrak{H}_{k+1}(x,y)$ are finite, we
see that for all $y\in\Gamma$, this function is not $R^{-1}$-harmonic.

Finally, if $s\in\mathbb{N}$ is the smallest integer such that
$H(x,y)=\mathfrak{H}_{s}(x,y)$, since $x\mapsto H(x,y)$ is $R^{-1}$-harmonic,
we must have that $\lim_{r\rightarrow R}G^{(s)}(x,y|r)=\infty$ by the above. ∎

As we will see, there are many examples with $s=1$ for which the conclusion of
Corollary 5.1 can fail, even for free products of abelian groups. For
instance, consider $\Gamma=\Gamma_{1}*\Gamma_{2}$ with a convergent
probability measure $\mu$ on it. That is, for $x,y\in\Gamma$ we have that
$\lim_{r\to R}G(x,y|r)$ and $\lim_{r\to R}I^{(1)}(x,y|r)$ exist for all
$x,y\in\Gamma$. Such convergent measures $\mu$ do exist, see for instance [15,
Section 7, Example D] for examples with $\Gamma_{i}=\mathbb{Z}^{d_{i}}$. We
will also construct such in Examples 5.3 and 5.4 below. Thus, by Proposition
5.2, we see that the conclusion of Corollary 5.1 would fail in these examples
for $s=1$. As we saw, in many cases there is a smallest $s\in\mathbb{N}$ such
that $I^{(s)}(x,y|r)$ tends to infinity as $r\to R$. Our next goal is to
construct examples of random walks such that there is no such $s$, i.e.
$\lim_{r\rightarrow R}G^{(s)}(x,y|r)$ exists for _any_ $s\in\mathbb{N}$. Using
Proposition 5.2, we will be able to deduce that the conclusion of Corollary
5.1 can fail for _any_ $s\in\mathbb{N}$. We will need the following result of
Varapoulos.

Let $\Gamma$ be a group of exponential volume growth and let $\mu$ be a
finitely supported, admissible and symmetric probability measure on $\Gamma$.
Then, by [65, Theorem 1] there are constants $c,C>0$ such that

$\mu^{*n}(e)\leq C\mathrm{e}^{-cn^{1/3}}.$

More generally (see [69, Theorem 6.7]), if $n$-balls around $e$ in
$\operatorname{Gr}(\Gamma)$ satisfy

(5.4) $|B_{n}|\geq\mathrm{e}^{cn^{\alpha}}$

for some $0<\alpha\leq 1$, then

(5.5) $\mu^{*n}(e)\leq
C\mathrm{exp}\left(-cn^{\frac{\alpha}{\alpha+2}}\right).$

######  Example 5.3.

Consider a finitely generated amenable group $\Gamma_{1}$ and set
$\Gamma_{2}=\mathbb{Z}$. Consider a finitely supported, admissible and
symmetric probability measures $\mu_{i}$, and let $\mu$ be the adapted
probability measure on $\Gamma=\Gamma_{1}*\Gamma_{2}$ given by

$\mu=\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2}.$

Note that since both $\Gamma_{1}$ and $\Gamma_{2}$ are amenable, we have that
the spectral radii of $\mu_{1}$ and $\mu_{2}$ are $R_{1}=R_{2}=1$.

Denote by $T^{+}_{e}$ the first return time to $e$ according to the random
walk determined by $\mu$. Let

$U(r)=\sum_{n\geq 1}\mathbb{P}_{e}[T^{+}_{e}=n]r^{n}$

be the generating function of first return time to $e$. Then, according to
[70, Lemma 1.13 (a)] we have that $G(e,e|r)(1-U(r))=1$. Also, by [70, Equation
(9.14)], the function $\Psi$ defined by equation (4.6) satisfies

$\Psi(t)=\frac{1}{1+rU^{\prime}(r)-U(r)}$

for $t:=rG(e,e|r)\leq\theta=RG(e,e|R)$. Using $G(e,e|r)(1-U(r))=1$, we get

$\Psi(t)=\frac{G(e,e|r)^{2}}{rG^{\prime}(e,e|r)+G(e,e|r)}.$

Since $\Gamma$ is non-amenable, we have that $G(e,e|R)$ is finite, so we get
that that $\Psi(\theta)=0$ if and only if $\lim_{r\rightarrow
R}G^{\prime}(e,e|r)=\infty$.

Our first goal is to prove that for a suitable choice of
$(\alpha_{1},\alpha_{2})$, the random walk is convergent, or equivalently that
$\Psi(\theta)\neq 0$. Rewriting $\Psi$ as

$\Psi(t)=\frac{1}{1+\sum_{n\geq 0}(n-1)\mathbb{P}_{e}[T^{+}_{e}=n]r^{n}},$

it follows that $\Psi$ is decreasing. Thus, it is enough to prove that for a
suitable choice of $(\alpha_{1},\alpha_{2})$ we have
$\Psi(\overline{\theta})>0$, where $\overline{\theta}$ is defined in equation
(4.4).

Note that in our example, since the random walk on $\Gamma_{2}=\mathbb{Z}$
determined by $\mu_{2}$ must be recurrent, we have $G_{2}(e,e|1)=\infty$. In
particular, we have $\overline{\theta}=\theta_{1}/\alpha_{1}$. Therefore, by
equation (4.7), where $\Psi_{1},\Psi_{2}$ are defined similarly for
$\mu_{1},\mu_{2}$ respectively, we have,

$\Psi(\overline{\theta})=\Psi_{1}(\theta_{1})+\Psi_{2}(\alpha_{2}\theta_{1}/\alpha_{1})-1=\Psi_{1}(\theta_{1})+\Psi_{2}\left(\frac{(1-\alpha_{1})\theta_{1}}{\alpha_{1}}\right)-1.$

Suppose now that $\Gamma_{1}$ satisfies (5.4). By the upper bound in equation
(5.5), for every $s\geq 0$, we have

(5.6) $G_{1}^{(s)}(e,e|1)<\infty.$

Consequently, for $s=1$, we get that $\Psi_{1}(\theta_{1})>0$. Now, as
$\alpha_{1}$ tends to 1, $\frac{(1-\alpha_{1})\theta_{1}}{\alpha_{1}}$ tends
to 0 and since $\Psi_{2}(0)=1$, we see that $\Psi(\overline{\theta})$ tends to
$\Psi_{1}(\theta_{1})$. In particular, for small enough $\alpha_{1}$,
$\Psi(\overline{\theta})>0$ and so the random walk determined by $\mu$ is
convergent for that choice of $\alpha_{1}$.

Next, we show that $\lim_{r\to R}I^{(s)}(e,e|r)$ is finite for every $s$. For
$i=1,2$, we consider the iterated Green sums for the subgroups $\Gamma_{i}$,

$I_{\Gamma_{i}}^{(s)}(e,e|r)=\sum_{x_{1},...,x_{s}\in\Gamma_{i}}G(e,x_{1}|r)...G(x_{s},e|r).$

By [28, Lemma 5.7], since $\lim_{r\to R}I^{(1)}(e,e|r)$ is finite,
$I^{(s)}(e,e|r)$ is bounded by quantities only involving $I^{(k)}(e,e|r)$ for
$k<s$ and $I_{\Gamma_{i}}^{(k)}(e,e|r)$ for $k\leq s$. Thus, by induction, to
show that $\lim_{r\to R}I^{(s)}(e,e|r)$ is finite for all $s$, it suffices to
show that $\lim_{r\to R}I_{\Gamma_{i}}^{(s)}(e,e|r)$ is finite for all $s$.

First, let us deal with $\Gamma_{1}$. Using the relation between the
derivatives of the Green function and the iterated sums given by [31,
Proposition 5.2] on the one hand and the relation between the Green functions
$G$ and $G_{i}$ given by equation (4.3) on the other hand, there are constants
$C_{1},C_{2}>0$ such that

$\displaystyle I_{\Gamma_{1}}^{(s)}(e,e|r)$
$\displaystyle=\left(\frac{G(e,e|r)}{G_{1}(e,e|\zeta_{i}(r))}\right)^{s+1}\sum_{x_{1},...,x_{s}\in\Gamma_{1}}G_{1}(e,x_{1}|\zeta_{1}(r))...G_{1}(x_{s},e|\zeta_{1}(r))$
$\displaystyle\leq
C_{1}\sum_{x_{1},...,x_{s}\in\Gamma_{1}}G_{1}(e,x_{1}|\zeta_{1}(R))...G_{1}(x_{s},e|\zeta_{1}(R))$
$\displaystyle\leq C_{1}C_{2}G_{1}^{(s)}(e,e|\zeta_{1}(R)).$

Thus, we need only prove that $G_{1}^{(s)}(e,e|\zeta_{1}(R))$ is finite for
all $s$. This follows from equation (5.6).

Second, since we have $\Gamma_{2}=\mathbb{Z}$, we deduce from Proposition 4.4
that the random walk cannot be spectrally degenerate along $\Gamma_{2}$. By
using equation (4.1), a similar calculation as above shows that we need only
show that $G_{R,\Gamma_{2}}^{(s)}(e,e|1)$ is finite for every $s\geq 0$. But
now, spectral non-degeneracy along $\Gamma_{2}$ implies that $1$ is smaller
than the radius of convergence of the Green function
$G_{R,\Gamma_{2}}(e,e|r)$, so we deduce again that $\lim_{r\to
R}I_{\Gamma_{2}}^{(s)}(e,e|r)$ is finite.

Thus, we have that $\lim_{r\rightarrow R}I^{(s)}(e,e|r)$ exists for all $s\geq
0$. A standard calculation then shows that $\lim_{r\to R}I^{(s)}(x,y|r)$
exists for all $x,y\in\Gamma$ and for all $s\geq 0$.

We also get the following examples of convergent random walks on free products
of $\mathbb{Z}^{5}$ and $\mathbb{Z}$.

######  Example 5.4.

Let $\Gamma=\Gamma_{1}*\Gamma_{2}$ where $\Gamma_{1}=\mathbb{Z}^{5}$ and
$\Gamma_{2}=\mathbb{Z}$. Suppose $\mu_{i}$ are finitely supported, admissible
and symmetric probability measures on $\Gamma_{i}$ for $i=1,2$, and let
$\mu:=\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2}$ be an adapted probability measure
on $\Gamma$. Then, we will show that for small enough $\alpha_{1}$, the random
walk $\mu$ is spectrally non-degenerate, and for large enough $\alpha_{1}$,
the random walk $\mu$ is spectrally degenerate and convergent along
$\mathbb{Z}^{5}$.

Indeed, we only need to prove that $\Psi(\bar{\theta})$ is negative for small
enough $\alpha_{1}$ and positive for large enough $\alpha_{1}$. Following the
same line of reasoning as in Example 5.3, we have

$\Psi(\bar{\theta})=\Psi_{1}(\theta_{1})+\Psi_{2}\left(\frac{1-\alpha_{1}}{\alpha_{1}}\theta_{1}\right)-1.$

Since $G_{1}(e,e|1)$ is finite, $\Psi_{1}(\theta_{1})>0$. Moreover, recall
that $\Psi_{1}$ is strictly decreasing and satisfies $\Psi_{1}(0)=1$, hence
$\Psi_{1}(\theta_{1})<1$. On the other hand,
$\lim_{t\to\theta_{2}}\Psi_{2}(t)=0$, see for instance [70, Example 9.18 (3)],
while $\Psi_{2}(0)=1$. We deduce that $\Psi(\bar{\theta})$ converges to
$\Psi_{1}(\theta_{1})>0$ as $\alpha_{1}$ tends to 1 and converges to
$\Psi_{1}(\theta_{1})-1<0$ as $\alpha_{1}$ tends to 0.

Proposition 5.2 together with the examples above show the limitation in the
strategy initiated in [72] to be applied to more general groups. This is
because this strategy relies on finding the asymptotic behaviour of
derivatives of the Green function at its singularities, which in turns uses
the fact that $\lim_{r\to R}I^{(s)}(e,e|r)=\infty$.

##  6\. Essential minimality of ratio-limit boundary.

In this section we will show that when $\Gamma$ be a non-elementary relatively
hyperbolic group, and $\mu$ is a finitely supported, admissible, aperiodic,
symmetric and spectrally non-degenerate probability measure on $\Gamma$, then
$\overline{\partial^{m}_{M,R}\Gamma}$ is the unique smallest closed
$\Gamma$-invariant subspace of $\partial_{\rho}\Gamma$. In order to establish
this we adapt a proof strategy of Woess from [72], which was used for
hyperbolic groups. Considering the radically different behavior of relatively
hyperbolic groups exhibited in Sections 4 and 5, our proofs require new ideas.
We set

$\Phi_{r}(x,y):=\frac{I^{(1)}(e,e|r)}{G(x,y|r)},$

so that by Corollary 5.1 we have

(6.1) $\frac{\Phi_{R}(x,y)}{\Phi_{R}(e,y)}:=\lim_{r\rightarrow
R}\frac{\Phi_{r}(x,y)}{\Phi_{r}(e,y)}=\frac{H(x,y)}{K_{R}(x,y)}.$

######  Proposition 6.1.

Let $\Gamma$ be a non-elementary relatively hyperbolic group, and assume that
$\mu$ is a finitely supported, admissible, aperiodic and symmetric probability
measure on $\Gamma$. Suppose further that the random walk is spectrally non-
degenerate, and that $\xi\in\partial_{M,R}\Gamma$ is a point such that
$\pi(\xi)$ is conical point in $\partial_{B}(\Gamma;\Omega)$. Then, for every
$x\in\Gamma$, as $y\rightarrow\xi$ in $\partial_{M,R}\Gamma$ we have

$\frac{\Phi_{R}(x,y)}{\Phi_{R}(e,y)}\underset{n\rightarrow\infty}{\longrightarrow}1$

###### Proof.

First, as suggested in Figure 1, we claim that if $x$ is fixed, then there
exists a constant $C(x)$ such that if two relative geodesics $[e,y]$ and
$[e,z]$ fellow-travel for a time $k$, then the relative geodesics $[x,z]$ and
$[e,y]$ fellow-travel for a time $k-C(x)$.

To prove the claim, we let $w$ be the projection of $x$ on $[e,y]$. Then, we
have by [28, Lemma 4.16] (see also [29, Lemma 2.5]) that any relative geodesic
from $x$ to $y$ passes within a bounded distance in
$\operatorname{Gr}(\Gamma)$ of $w$. Since $d(w,x)$ is bounded by $|x|$, the
claim follows, since by [28, Lemma 4.14] there exists $C(x)$ such that the two
geodesics $[e,y]$ and $[x,y]$ $C(x)$-fellow travel, with $C(x)$ depending only
on $d(w,x)\leq|x|$.

Following the computations in [29, Subsection 6.1] we decompose the sum over
$z\in\Gamma$ as follows. We write $[e,y]$ as $(e,y_{1},...,y_{n})$, where $n$
is the relative distance between $e$ and $y$ and we denote by $\Gamma_{k}$,
$k\leq n$, the set of $z$ whose projection in
$\operatorname{Gr}(\Gamma;\Omega)$ on $[e,y]$ is at $y_{k}$. If there is more
than one such projection, we choose the projection the one which is closest to
$y$. Then,

$\Phi_{r}(e,y)=\sum_{k=0}^{n}\sum_{z\in\Gamma_{k}}\frac{G(e,z|r)G(z,y|r)}{G(e,y|r)}.$

Now if $z\in\Gamma_{k}$, then $[e,z]$ and $[e,y]$ fellow travel for a time
$k-1$ by [29, Lemma 2.5]. Thus, $[x,z]$ and $[e,y]$ fellow travel for a time
$k-1-C(x)$ by what we previously showed. We then have

$\big{|}\Phi_{r}(x,y)-\Phi_{r}(e,y)\big{|}\leq\sum_{k=0}^{n}\sum_{z\in\Gamma_{k}}\frac{G(e,z|r)G(z,y|r)}{G(e,y|r)}\left|\frac{G(x,z|r)G(e,y|r)}{G(x,y|r)G(e,z|r)}-1\right|.$

$e$$y$$x$$z$ Figure 1.

Assume now that $y$ converges to a point $\xi\in\partial_{M,R}\Gamma$ whose
image $\pi(\xi)$ in Bowditch boundary is conical. Since $\pi$ is continuous,
we know that $y$ converges to $\pi(\xi)$ in the relative metric. Hence, by
strong relative Ancona inequality given in Proposition 2.9, there exists $K$
depending only on $C(x)$ (i.e. on $x$) and $0<\alpha<1$ such that for
$z\in\Gamma_{k}$,

$\left|\frac{G(x,z|r)G(e,y|r)}{G(x,y|r)G(e,z|r)}-1\right|\leq K\alpha^{k}.$

We thus find

(6.2) $\big{|}\Phi_{r}(x,y)-\Phi_{r}(e,y)\big{|}\leq
K\sum_{k=0}^{n}\alpha^{k}\sum_{z\in\Gamma_{k}}\frac{G(e,z|r)G(z,y|r)}{G(e,y|r)}.$

Let $H_{k}$ be the union of all parabolic subgroups containing
$y_{k-1}^{-1}y_{k}$. Then it is proved at the beginning of [29, Subsection
6.1] that

(6.3) $\sum_{z\in\Gamma_{k}}\frac{G(e,z|r)G(z,y|r)}{G(e,y|r)}\asymp
I^{(1)}(e,e|r)\sum_{w\in
H_{k}}\frac{G(y_{k-1},y_{k-1}w|r)G(y_{k-1}w,y_{k}|r)}{G(y_{k-1},y_{k}|r)}.$

Only the upper bound for the sum on the left is written explicitly at the
beginning of [29, Subsection 6.1], but equation (6.3) is a direct consequence
of [29, Equation (41), Proposition 6.3]. Still, let us give some details on
the proof of the lower bound for the sake of completeness. We write $y_{k-1}w$
for the projection of $z$ on $y_{k-1}H_{k}$.

Then as suggested in Figure 2, we deduce from [28, Lemma 4.16] (see also [29,
Lemma 2.5]) that a relative geodesic $[e,z]$ passes within a bounded distance
of $y_{k-1}$, then of $y_{k-1}w$. Similarly, a relative geodesic $[z,y]$
passes within a bounded distance of $y_{k-1}w$, then of $y_{k}$. Finally, by
definition, $[e,y]$ passes first at $y_{k-1}$ then at $y_{k}$. Therefore, by
weak relative Ancona inequalities,

$G(e,z|r)\asymp G(e,y_{k-1}|r)G(y_{k-1},y_{k-1}w|r)G(y_{k-1}w,z|r),$
$G(z,y|r)\asymp G(z,y_{k-1}w|r)G(y_{k-1}w,y_{k}|r)G(y_{k},y|r)$

and

$G(e,y|r)\asymp G(e,y_{k-1}|r)G(y_{k-1},y_{k}|r)G(y_{k},y|r).$

Now, summing over $z\in\Gamma_{k}$ boils down to summing over $w\in H_{k}$ and
over $z$ that projects on $y_{k-1}H_{k}$ at $y_{k-1}w$. Moreover, for fixed
$w$, the residual sum

$\Sigma=\sum_{z}G(y_{k-1}w,z|r)G(z,y_{k-1}w|r)$

is roughly asymptotic to $I^{(1)}(e,e|r)$. Indeed, by distinguishing elements
$z$ according to their projection on $y_{k-1}H_{k}$ and using weak relative
Ancona inequalities, we see that the ratio of $I^{(1)}(e,e|r)$ and $\Sigma$ is
roughly given by $\sum_{w\in H_{k}}G(e,w|r)G(w,e|r)$, and the latter is
uniformly finite (see the similar proof in [28, (20)]). All these inequalities
together imply the desired lower bound in equation (6.3).

Next, it will be convenient to use the following notations from [29]. Define
the function $\Upsilon_{r}$ on $\Gamma$ as follows. For a point $y\in\Gamma$,
let $[e,y]$ be some relative geodesic chosen according to an automaton (see
[29]), and let $y_{1}$ be the first point after $e$ in $[e,y]$. Then, we
define

$\Upsilon_{r}(y)=\sum_{w\in H_{1}}\frac{G(e,w|r)G(w,y_{1}|r)}{G(e,y_{1}|r)}.$

$y_{k-1}$$e$$y_{k}$$y$$z$$y_{k-1}w$ Figure 2.

Denote by $T$ the left shift on relative geodesics starting at $e$. That is,
if $\gamma=(e,\gamma_{1},...,\gamma_{n})$ is a relative geodesic, then
$T\gamma=(e,\gamma_{1}^{-1}\gamma_{2},...,\gamma_{1}^{-1}\gamma_{n})$. Then,
combining (6.2) and (6.3), we have

(6.4) $\big{|}\Phi_{r}(x,y)-\Phi_{r}(e,y)\big{|}\lesssim
I^{(1)}(e,e|r)K\sum_{k=0}^{n}\alpha^{k}\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}$

whereas by [29, Equation (41), Proposition 6.3] we have

(6.5) $\Phi_{r}(e,y)\asymp
I^{(1)}(e,e|r)\sum_{k=0}^{n}\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}$

Fix $\epsilon>0$. Then, there exists $k_{0}$ such that for $k\geq k_{0}$ we
have $\frac{\alpha^{k}}{1-\alpha}\leq\epsilon$. Hence, by equations (6.4) and
(6.5),

$\big{|}\Phi_{r}(x,y)-\Phi_{r}(e,y)\big{|}\lesssim
I^{(1)}(e,e|r)\sum_{k=0}^{k_{0}}\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}+\epsilon\Phi_{r}(e,y)$

Using again (6.5), we rewrite this as

(6.6)
$\left|\frac{\Phi_{r}(x,y)}{\Phi_{r}(e,y)}-1\right|\lesssim\frac{\sum_{k=0}^{k_{0}}\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}}{\sum_{k=0}^{n}\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}}+\epsilon.$

Finally, $k_{0}$ is fixed and if $n=|y|$ is big enough, for $k\leq k_{0}$ we
have that $\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}$ is bounded by a constant that
only depends on $\xi$. On the other hand, for every $k$,

$\sum_{w\in
H_{k}}\frac{G(y_{k-1},y_{k-1}w|r)G(y_{k-1}w,y_{k}|r)}{G(y_{k-1},y_{k}|r)}\gtrsim
1,$

hence as $n$ tends to infinity,
$\sum_{k=0}^{n}\Upsilon_{r}\big{(}T^{k}[e,y]\big{)}$ tends to infinity. Thus,
if we take $n=|y|$ large enough, we can arrange for the first summand in the
right-hand-side of equation (6.6) to be at most $\epsilon$, and get

$\left|\frac{\Phi_{r}(x,y)}{\Phi_{r}(e,y)}-1\right|\lesssim 2\epsilon.$

Since weak and strong relative Ancona inequalities are uniform in $r\leq R$,
this last bound is also uniform in $r$, and concludes the proof. ∎

This allows us to show, when the random walk is spectrally non-degenerate,
that $\overline{\partial_{M,R}^{m}\Gamma}$ embeds $\Gamma$-equivariantly
inside $\partial_{\rho}\Gamma$.

######  Corollary 6.2.

Let $\Gamma$ be a non-elementary hyperbolic group relative to a finite
collection of subgroups $\Omega$, and assume that $\mu$ is a finitely
supported, admissible, aperiodic and symmetric probability measure on
$\Gamma$. Suppose further that the random walk is spectrally non-degenerate.
Then, there is a bi-Lipschitz $\Gamma$-equivariant map
$\iota:\overline{\partial_{M,R}^{m}\Gamma}\rightarrow\partial_{\rho}\Gamma$
which is a homeomorphism on its image. Moreover, if we have $C_{x}=D_{x}$ for
every $x\in\Gamma$ in equations (2.1) and (2.4), then $\iota$ is isometric.

###### Proof.

Let $\pi:\partial_{M,R}\Gamma\rightarrow\partial_{B}(\Gamma;\Omega)$ be the
canonical $\Gamma$-factor map (see discussion at the end of Section 2).
Suppose that $(y_{n})$ is a sequence converging to $\xi$ in the $R$-Martin
metric $d_{M,R}$, with $\pi(\xi)$ conical. Since $\pi$ extends to a
$\Gamma$-factor from
$\Delta_{M,R}\rightarrow\Gamma\cup\partial_{B}(\Gamma;\Omega)$, we get that
$(y_{n})$ converges to $\pi(\xi)$ in the relative metric as well. By
Proposition 6.1 and equation (6.1) we see that $y_{n}\rightarrow\iota(\xi)$
according to the metric $d_{\rho}$, for some point $\iota(\xi)$ in
$\partial_{\rho}\Gamma$, so that $H(x,\iota(\xi))=K_{R}(x,\xi)$. By equations
(2.1) and (2.4), since the metrics $d_{M,R}$ and $d_{\rho}$ are defined using
$R$-Martin kernels and ratio-limit kernels respectively, we would get that
$\iota$ is bi-Lipschitz map on the set of $\xi$ for which $\pi(\xi)$ is
conical. If we take $C_{x}=D_{x}$ for every $x\in\Gamma$ in equations (2.1)
and (2.4) we actually get that $\iota$ is an isometry. By Corollary 3.2 we
know that the pre-images under $\pi$ of conical points are dense in
$\overline{\partial_{M,R}^{m}\Gamma}$, so we may extend $\iota$ to a bi-
Lipschitz map from $\overline{\partial_{M,R}^{m}\Gamma}$ to
$\partial_{\rho}\Gamma$, and if $\iota$ is isometric on conical points, it
would extend to an isometry from $\overline{\partial_{M,R}^{m}\Gamma}$ to
$\partial_{\rho}\Gamma$.

The above relation $H(x,\iota(\xi))=K_{R}(x,\xi)$ together with
$K_{R}(x,g\xi)K_{R}(g^{-1},\xi)=K_{R}(g^{-1}x,\xi)$ for the Martin kernel and
$H(x,g\xi)H(g^{-1},\xi)=H(g^{-1}x,\xi)$ for the ratio limit kernel show that
$\iota$ is $\Gamma$-equivariant on the set of $\xi$ such that $\pi(\xi)$ is
conical. Since $\Gamma$ acts on both boundaries via homeomorphisms, if
$\xi_{n}$ converges to $\xi$ and if $g\in\Gamma$, then $g\xi_{n}$ converges to
$g\xi$. Now, the set of $\xi$ such that $\pi(\xi)$ is conical is dense in
$\overline{\partial_{M,R}^{m}\Gamma}$. Letting $\xi_{n}$ with $\pi(\xi_{n})$
conical tend to $\xi\in\overline{\partial_{M,R}^{m}\Gamma}$, we see that

$\iota(g\xi)=\lim_{n}\iota(g\xi_{n})=\lim_{n}g\iota(\xi_{n})=g\iota(\xi).$

Therefore, the extension of $\iota$ to $\overline{\partial_{M,R}^{m}\Gamma}$
is still $\Gamma$-equivariant.

In any case, since $\iota$ is bi-Lipschitz and
$\overline{\partial_{M,R}^{m}\Gamma}$ is compact, we deduce that $\iota$ is a
homeomorphism on its image. ∎

######  Corollary 6.3.

Let $\Gamma$ be a non-elementary hyperbolic group relative to a finite
collection of subgroups $\Omega$, and assume that $\mu$ is aperiodic,
symmetric probability measure with finite support generating $\Gamma$. Suppose
further that the random walk is spectrally non-degenerate. Then
$\theta|_{\partial_{\rho}\Gamma}:\partial_{\rho}\Gamma\rightarrow\partial_{\rho}^{r}\Gamma$
is a $\Gamma$-equivariant homeomorphism.

###### Proof.

Since $R_{\mu}$ is normal, it acts trivially on $\Gamma/R_{\mu}$, and
therefore also on $\Delta_{\rho}^{r}\Gamma$. By [72, Lemma 6.2(ii)] we know
that
$\theta|_{\partial_{\rho}\Gamma}:\partial_{\rho}\Gamma\rightarrow\Delta_{\rho}^{r}\Gamma$
is injective, so that $R_{\mu}$ acts trivially also on
$\partial_{\rho}\Gamma$. By Corollary 6.2 we get that $R_{\mu}$ acts trivially
on $\overline{\partial^{m}_{M,R}\Gamma}$. Since the pre-image of every point
in $\partial_{B}(\Gamma;\Omega)$ contains a minimal point (see end of Section
2) we see that the restriction of $\pi$ to
$\overline{\partial^{m}_{M,R}\Gamma}$ is still onto
$\partial_{B}(\Gamma;\Omega)$, and we get that $R_{\mu}$ acts trivially on
$\partial_{B}(\Gamma;\Omega)$. Now, since the action of $\Gamma$ on the space
of distinct triples is properly discontinuous, it follows that the kernel of
$\Gamma\curvearrowright\partial_{B}(\Gamma;\Omega)$ is finite, so that
$R_{\mu}$ is also finite.

Now let $\xi\in\partial_{\rho}\Gamma$ be a boundary element, and $(g_{n})$ a
sequence in $\Gamma$ such that $g_{n}\rightarrow\xi$ in $\Delta_{\rho}\Gamma$.
As $R_{\mu}$ is finite, for any finite subset $F\subset\Gamma/R_{\mu}$ we must
have that $g_{n}R_{\mu}$ is eventually not in $F$. Hence, we must have that
$g_{n}R_{\mu}=\theta(g_{n})\rightarrow\theta(\xi)$ is not in $F$, so that
$\theta(\xi)\in\partial_{\rho}^{r}\Gamma$. Since $\theta$ is surjective,
$\theta|_{\partial_{\rho}\Gamma}$ must be onto $\partial_{\rho}^{r}\Gamma$,
and is therefore a homeomorphism. ∎

Combining Corollary 6.2 with results from Section 3, we obtain one of the main
results of the paper, showing that $\partial_{\rho}\Gamma$ is essentially
minimal. That is, $\overline{\partial^{m}_{M,R}\Gamma}$ is the unique smallest
closed $\Gamma$-invariant subspace of $\partial_{\rho}\Gamma$.

######  Theorem 6.4.

Let $\Gamma$ be a non-elementary hyperbolic group relative to a finite
collection of subgroups $\Omega$, and assume that $\mu$ is aperiodic,
symmetric probability measure with finite support generating $\Gamma$. Suppose
further that the random walk is spectrally non-degenerate. Then
$\Gamma\curvearrowright\partial_{\rho}\Gamma$ is essentially minimal, and its
unique smallest closed $\Gamma$-invariant subspace is
$\Gamma\curvearrowright\overline{\partial^{m}_{M,R}\Gamma}$.

###### Proof.

Let
$\iota:\overline{\partial_{M,R}^{m}\Gamma}\rightarrow\partial_{\rho}\Gamma$ be
the map constructed in Corollary 6.2, and
$\eta\in\partial_{\rho}\Gamma\setminus\iota(\overline{\partial_{M,R}^{m}\Gamma})$.
By the discussion on $r$-harmonic functions in Section 2 we know that
$\varphi:\operatorname{Prob}(\partial_{M,R}\Gamma)\rightarrow\mathcal{H}^{+}(P,R^{-1})$
given by $\varphi(\nu)=u_{\nu}$ is a surjective continuous map.

Let us denote by $\nu^{\eta}$ the unique representing measure for $x\mapsto
H(x,\eta)$ satisfying $\nu^{\eta}(\partial_{M,R}^{m}\Gamma)=1$. For every
$g\in\Gamma$ we have that,

$\displaystyle H(g^{-1},\eta)H(x,g\eta)$
$\displaystyle=H(g^{-1}x,\eta)=\int_{\partial_{M,R}\Gamma}K_{R}(g^{-1}x,\xi)d\nu^{\eta}(\xi)$
$\displaystyle=\int_{\partial_{M,R}\Gamma}K_{R}(x,g\xi)K_{R}(g^{-1},\xi)d\nu^{\eta}(\xi)$
$\displaystyle=\int_{\partial_{M,R}\Gamma}K_{R}(x,\xi)K_{R}(g^{-1},g^{-1}\xi)d\nu^{\eta}(g^{-1}\xi).$

Hence, by uniqueness of representing measures with full measures on
$\partial_{M,R}^{m}\Gamma$, and as $\partial_{M,R}^{m}\Gamma$ is a
$\Gamma$-invariant set, we get the Radon-Nikodym derivative

$\frac{dg_{*}\nu^{\eta}}{d\nu^{g\eta}}(\xi)=\frac{K_{R}(g^{-1},g^{-1}\xi)}{H(g^{-1},\eta)}.$

By Proposition 3.3 we know that
$\Gamma\curvearrowright\overline{\partial_{M,R}^{m}\Gamma}$ is minimal and
strongly proximal, and contains all minimal points. Since the support of
$\nu^{\eta}$ is contained in $\overline{\partial_{M,R}^{m}\Gamma}$, by strong
proximality of $\Gamma\curvearrowright\overline{\partial_{M,R}^{m}\Gamma}$
there exists a net $(g_{\alpha})$ in $\Gamma$ so that
$(g_{\alpha})_{*}\nu^{\eta}$ converges to some dirac measure $\delta_{\xi}$
for a minimal point $\xi\in\partial_{M,R}^{m}\Gamma$. Thus, since
$\nu^{g\eta}$ are all probability measures, it also follows that
$\nu^{g_{\alpha}\eta}$ converges to $\delta_{\xi}$. Since
$\varphi:\operatorname{Prob}(\partial_{M,R}\Gamma)\rightarrow\mathcal{H}^{+}(P,R^{-1})$
is continuous, we get for all $x\in\Gamma$ that
$H(x,g_{\alpha}\eta)\rightarrow K_{R}(x,\xi)$. But now, if without loss of
generality (up to taking a subnet) $g_{\alpha}\eta$ converges to some cluster
point $\eta^{\prime}\in\partial_{\rho}\Gamma$, then
$H(x,\eta^{\prime})=K_{R}(x,\xi)$ for all $x\in\Gamma$, so that
$\iota(\eta^{\prime})=\xi$ is a minimal point. Since
$\Gamma\curvearrowright\iota(\overline{\partial_{M,R}^{m}\Gamma})$ is a
minimal action, we see that $\overline{\partial_{M,R}^{m}\Gamma}$ must be the
unique smallest closed $\Gamma$-invariant subspace of $\partial_{\rho}\Gamma$.
∎

##  7\. Distinct ratio-limit boundaries on $\mathbb{Z}^{5}*\mathbb{Z}$

In this section, we provide an example of a group endowed with two probability
measures such that the resulting ratio limit boundaries are not equivariantly
homeomorphic. More specifically, we take $\Gamma=\Gamma_{1}*\Gamma_{2}$ where
$\Gamma_{1}=\mathbb{Z}^{5}$ and $\Gamma_{2}=\mathbb{Z}$, and
$\mu=\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2}$ will be an finitely supported,
admissible, adapted and symmetric probability measures. By Example 5.4 we may
fix $\alpha_{1}$ large enough so that the random walk is convergent. We
introduce the function $\Phi_{r}^{(2)}$ defined by

(7.1)
$\Phi_{r}^{(2)}(x,y)=\frac{I^{(2)}(x,y|r)}{G(x,y|r)}=\frac{1}{G(x,y|r)}\sum_{z_{1},z_{2}\in\Gamma}G(x,z_{1}|r)G(z_{1},z_{2}|r)G(z_{2},y|r).$

By Corollary 5.1 (2) we have that,

$\frac{\Phi_{R}^{(2)}(x,y)}{\Phi_{R}^{(2)}(e,y)}:=\lim_{r\to
R}\frac{\Phi_{r}^{(2)}(x,y)}{\Phi_{r}^{(2)}(e,y)}=\frac{H(x,y)}{K(x,y|R)}.$

Our goal is to prove the following analogous of Proposition 6.1 in this
special case. The proof follows the same line as the proof of Proposition 6.1.
The main difference is that we need to deal with $\Phi_{r}^{(2)}$ instead of
$\Phi_{r}$. However, in our specific context, most geometric arguments are
replaced with combinatorial ones that use the free product structure. We will
use the more convenient terminology of free products rather than the
terminology of relatively hyperbolic groups, which is better suited for our
proof.

######  Proposition 7.1.

Let $\Gamma=\Gamma_{1}*\Gamma_{2}$ where $\Gamma_{1}=\mathbb{Z}^{5}$ and
$\Gamma_{2}=\mathbb{Z}$. Let $\mu_{1},\mu_{2}$ be finitely supported,
admissible and symmetric probability measures on $\Gamma_{1},\Gamma_{2}$
respectively, such that the adapted probability measure
$\mu=\alpha_{1}\mu_{1}+\alpha_{2}\mu_{2}$ is convergent and spectrally
degenerate along $\Gamma_{1}$. Let $\xi\in\partial_{M,R}\Gamma$ be a point
such that $\pi(\xi)$ is conical in $\partial_{B}(\Gamma;\Omega)$. Then for
every $x\in\Gamma$, as $y\rightarrow\xi$ in $\partial_{M,R}\Gamma$ we have

$\frac{H(x,y)}{K_{R}(x,y)}=\frac{\Phi_{R}^{(2)}(x,y)}{\Phi_{R}^{(2)}(e,y)}\underset{y\rightarrow\xi}{\longrightarrow}1.$

###### Proof.

Before we begin our proof, we describe our overall strategy. Our first goal is
to prove that

$\Phi_{r}^{(2)}(e,y)-\Phi_{r}^{(2)}(x,y)\sim\tilde{\Phi}_{r}^{(2)}(e,y)-\tilde{\Phi}_{r}^{(2)}(x,y),$

where $\tilde{\Phi}_{r}^{(2)}$ is a twisted Birkhoff sum that replaces the
expression $\sum\Upsilon_{r}(T^{k}[e,y])$ in the proof of Proposition 6.1.
This twisted Birkhoff sum involves the second derivatives of the induced Green
functions on free factors. Unlike $\sum\Upsilon_{r}(T^{k}[e,y])$, it tends to
infinity as $r$ goes to $R$, even for fixed $y$ and we need to understand its
asymptotic behavior in $r$. To do so, we use classical results about the
expansion of the Green function at its spectral radius in $\mathbb{Z}^{d}$ and
show that $\tilde{\Phi}_{r}^{(2)}(e,y)$ behaves like
$(1-\zeta_{1}(r))^{-1/2}$. Finally, as $y$ tends to a conical limit point, we
prove that for fixed $x$, the terms in
$\tilde{\Phi}_{r}^{(2)}(e,y)-\tilde{\Phi}_{r}^{(2)}(x,y)$ simplify each other
so
$(1-\zeta_{1}(r))^{-1/2}\big{(}\tilde{\Phi}_{r}^{(2)}(e,y)-\tilde{\Phi}_{r}^{(2)}(x,y)\big{)}$
remains bounded, while $(1-\zeta_{1}(r))^{-1/2}\tilde{\Phi}_{r}^{(2)}(e,y)$
goes to infinity. We conclude that $\lim_{r\to
R}\frac{\tilde{\Phi}_{r}^{(2)}(e,y)-\tilde{\Phi}_{r}^{(2)}(x,y)}{\tilde{\Phi}_{r}^{(2)}(e,y)}$
converges to 0 as $y$ converges to a conical limit point, so that
$H(x,y)/K_{R}(x,y)$ converges to 1.

We first compute $\Phi^{(2)}_{r}(e,y)$ as defined in (7.1). We write $y$ in
its normal form $y=s_{1}...s_{n}$, where $s_{i}$ and $s_{i+1}$ for
$i=1,...,n-1$ belong to different parabolic subgroups (either $\mathbb{Z}^{5}$
or $\mathbb{Z}$), and we write $y_{j}=s_{1}...s_{j}$.

$e$$y_{j-1}$$y_{j}$$y_{j-1}w_{1}$$z_{1}$$z_{2}$$y_{k-1}w_{2}$$y_{k-1}$$y_{k}$$y$
Figure 3.

Fix two points $z_{1},z_{2}\in\Gamma$, and let $y_{j-1}$ be the largest common
prefix of $y$ and $z_{1}$. Similarly, let $y_{k-1}$ be the largest common
prefix of $y$ and $z_{2}$. We also write $w_{1}$ and $w_{2}$ for the next
letter (either after $y_{j-1}$ or $y_{k-1}$) in the normal form of $z_{1}$ and
$z_{2}$. In other words, $z_{1}$ and $z_{2}$ can be written as

$z_{1}=y_{j-1}w_{1}z^{\prime}_{1}$ $z_{2}=y_{k-1}w_{2}z^{\prime}_{2},$

with $w_{i}\in\mathbb{Z}^{5}$ or $w_{i}\in\mathbb{Z}$ and no prefix of
$z_{1}^{\prime},z_{2}^{\prime}$ in the same subgroup that $w_{1}$ and $w_{2}$
belong to.

Any path from $e$ to $z_{1}$ necessarily has to pass through $y_{j-1}$ and
$y_{j-1}w_{1}$. By [68, Equation (3.3)] we may decomposing such a path to get

$\frac{G(e,z_{1}|r)}{G(e,e|r)}=\frac{G(e,y_{j-1}|r)}{G(e,e|r)}\frac{G(y_{j-1},y_{j-1}w_{1}|r)}{G(e,e|r)}\frac{G(y_{j-1}w_{1},z_{1}|r)}{G(e,e|r)}.$

We rewrite this as

(7.2)
$G(e,e|r)^{2}G(e,z_{1}|r)=G(e,y_{j-1}|r)G(e,w_{1}|r)G(e,z^{\prime}_{1}|r).$

Similarly, any path from $z_{2}$ to $y$ has to pass through $y_{k-1}w_{2}$ and
$y_{k}$, hence

$\frac{G(z_{2},y|r)}{G(e,e|r)}=\frac{G(z_{2},y_{k-1}w_{2}|r)}{G(e,e|r)}\frac{G(y_{k-1}w_{2},y_{k}|r)}{G(e,e|r)}\frac{G(y_{k}w_{1},y|r)}{G(e,e|r)},$

We rewrite this as

(7.3)
$G(e,e|r)^{2}G(z_{2},y|r)=G(z_{2}^{\prime},e|r)G(w_{2},s_{k}|r)G(y_{k},y|r).$

We have two cases to consider, either $j=k$ or $j\neq k$. Our goal is
ultimately to prove that the dominant term in $\Phi^{(2)}(e,y)$ is the sub-sum
over $z_{1}$ and $z_{2}$ such that $j=k$. We will prove that the sub-sum over
$z_{1}$ and $z_{2}$ such that $j\neq k$ converges to a finite limit as $r$
tends to $R$, while the sub-sum over $z_{1}$ and $z_{2}$ such that $j=k$
diverges as $r$ tends to $R$.

We first assume that $j<k$ and consider Figure 3. Then, using [68, Equation
(3.3)] again, we similarly get that for any path from $z_{1}$ to $z_{2}$ has
to pass through $y_{j-1}w_{1}$, $y_{j}$, $y_{k-1}$ and $y_{k-1}w_{2}$ so we
have

(7.4)
$G(e,e|r)^{4}G(z_{1},z_{2}|r)=G(z_{1}^{\prime},e|r)G(w_{1},s_{j}|r)G(y_{j},y_{k-1}|r)G(e,w_{2}|r)G(e,z_{2}^{\prime}|r).$

In the same manner, any path from $e$ to $y$ has to pass through $y_{j-1}$,
$y_{j}$, $y_{k-1}$ and $y_{k}$, so

(7.5)
$G(e,e|r)^{4}G(e,y|r)=G(e,y_{j-1}|r)G(e,s_{j}|r)G(y_{j},y_{k-1}|r)G(e,s_{k}|r)G(y_{k},y|r).$

By multiplying (7.2), (7.3), (7.4) and dividing by (7.5) we get

(7.6)
$\begin{split}G(e,e|r)^{4}\frac{G(e,z_{1}|r)G(z_{1},z_{2}|r)G(z_{2},y|r)}{G(e,y|r)}=&\
\frac{G(e,w_{1}|r)G(w_{1},s_{j}|r)}{G(e,s_{j}|r)}\frac{G(e,w_{2}|r)G(w_{2},s_{k}|r)}{G(e,s_{k}|r)}\cdot\\\
&\hskip
2.84544ptG(e,z^{\prime}_{1}|r)G(z^{\prime}_{1},e|r)G(e,z^{\prime}_{2}|r)G(z^{\prime}_{2},e|r).\end{split}$

We now sum (7.6) over all $z_{1}$ and $z_{2}$ such that $j<k$. This is
equivalent to summing over $k$ and then over $j<k$, $w_{1}$ in the same free
factor as $s_{j}$, $z^{\prime}_{1}$ that start with a letter in the other free
factor than $s_{j}$, $w_{2}$ in the same free factor as $s_{k}$, and
$z^{\prime}_{2}$ that start with a letter in the other free factor than
$s_{k}$. For $\ell=1,...,n$ we set $\Gamma_{i_{\ell}}$ the free factor to
which such that $s_{\ell}$ belongs to, and we write
$\Gamma_{i_{\ell}}^{\perp}$ for the set of elements that start with a letter
not in $\Gamma_{i_{\ell}}$. By changing summation as above we obtain that the
following expression is equal to the one obtained by summing equation (7.6)
over $z_{1},z_{2}$, $j<k$

$\displaystyle\sum_{k\leq n}\sum_{j<k}$
$\displaystyle\sum_{w_{1}\in\Gamma_{i_{j}}}\frac{G(e,w_{1}|r)G(w_{1},s_{j}|r)}{G(e,s_{j}|r)}\sum_{w_{2}\in\Gamma_{i_{k}}}\frac{G(e,w_{2}|r)G(w_{2},s_{k}|r)}{G(e,s_{k}|r)}\cdot$
$\displaystyle\hskip
14.22636pt\sum_{z^{\prime}_{1}\in\Gamma_{i_{j}}^{\perp}}G(e,z_{1}^{\prime}|r)G(z^{\prime}_{1},e|r)\sum_{z^{\prime}_{2}\in\Gamma_{i_{k}}^{\perp}}G(e,z_{2}^{\prime}|r)G(z^{\prime}_{2},e|r).$

$e$$y_{k-1}$$y_{k}$$y_{k-1}w_{2}$$z_{2}$$z_{1}$$y_{j-1}w_{1}$$y_{j-1}$$y_{j}$$y$
Figure 4.

Since the random walk is convergent we know that $I^{(1)}(e,e|R)$ is finite.
Consequently, the two last inner sums in equation (7.6) over $z_{1}^{\prime}$
and $z_{2}^{\prime}$ converge to a finite limit as $r$ tends to $R$. Next, by
[33, Proposition 6.3], the sums over $w_{1}$ and $w_{2}$ also converge to a
finite limit as $r$ tends to $R$. We conclude that the sum over all $z_{1}$
and $z_{2}$ such that $j<k$ converges to a finite limit as $r$ tends to $R$.

Second, if we assume that $j>k$ a similar analysis (see Figure 4) yields

(7.7)
$\begin{split}G(e,e|r)^{8}\frac{G(e,z_{1}|r)G(z_{1},z_{2}|r)G(z_{2},y|r)}{G(e,y|r)}=&\
G(e,w_{1}|r)G(w_{1},e|r)G(s_{k},w_{2}|r)G(w_{2},s_{k}|r)\\\ &\hskip
2.84544ptG(e,z^{\prime}_{1}|r)G(z^{\prime}_{1},e|r)G(e,z^{\prime}_{2}|r)G(z^{\prime}_{2},e|r).\\\
&\hskip 2.84544ptG(y_{k},y_{j-1}|r)G(y_{j-1},y_{k}|r)\end{split}$

Summing the left hand side of equation (7.7) over $z_{1}$ and $z_{2}$ such
that $j>k$ is equivalent to summing the right hand side over $j>k$, $w_{1}$,
$w_{2}$, $z^{\prime}_{1}$ and $z_{2}^{\prime}$ as before, and we again find
that this quantity converges to a finite limit as $r$ tends to $R$.

Thus, we are left with analyzing the case where $j=k$.

$y_{k-1}$$e$$y_{k}$$y$$z_{2}$$z_{1}$$y_{k-1}w_{1}$$y_{k-1}w_{2}$ Figure 5.

This time, similarly to before, following paths in Figure 5 we obtain

(7.8)
$\begin{split}G(e,e|r)^{4}\frac{G(e,z_{1}|r)G(z_{1},z_{2}|r)G(z_{2},y|r)}{G(e,y|r)}=&\
\frac{G(e,w_{1}|r)G(w_{1},w_{2}|r)G(w_{2},s_{k}|r)}{G(e,s_{k}|r)}\cdot\\\
&\hskip
2.84544ptG(e,z^{\prime}_{1}|r)G(z^{\prime}_{1},e|r)G(e,z^{\prime}_{2}|r)G(z^{\prime}_{2},e|r).\end{split}$

As before, summing over $z_{1}$ and $z_{2}$ boils down to summing over $k$,
$w_{1},w_{2}\in\Gamma_{i_{k}}$ and $z_{1},z_{2}\in\Gamma_{i_{k}}^{\perp}$.
Hence, we get that the expression obtained by summing equation (7.8) is equal
to

$\displaystyle\sum_{k\leq n}$
$\displaystyle\sum_{w_{1},w_{2}\in\Gamma_{i_{k}}}\frac{G(e,w_{1}|r)G(w_{1},w_{2}|r)G(w_{2},s_{k}|r)}{G(e,s_{k}|r)}$
$\displaystyle\hskip
14.22636pt\sum_{z^{\prime}_{1},z^{\prime}_{2}\in\Gamma_{i_{k}}^{\perp}}G(e,z_{1}^{\prime}|r)G(z^{\prime}_{1},e|r)G(e,z_{2}^{\prime}|r)G(z^{\prime}_{2},e|r).$

We use again that the random walk is convergent to deduce that the inner sums
over $z_{1}^{\prime}$ and $z_{2}^{\prime}$ converge to a finite limit as $r$
tends to $R$.

Thus, we need only focus on the sum over $w_{1}$ and $w_{2}$ given by

$\displaystyle\sum_{k\leq n}$
$\displaystyle\sum_{w_{1},w_{2}\in\Gamma_{i_{k}}}\frac{G(e,w_{1}|r)G(w_{1},w_{2}|r)G(w_{2},s_{k}|r)}{G(e,s_{k}|r)}.$

By (4.3), we have

$\sum_{w_{1},w_{2}\in\Gamma_{i_{k}}}\frac{G(e,w_{1}|r)G(w_{1},w_{2}|r)G(w_{2},s_{k}|r)}{G(e,s_{k}|r)}=\frac{G(e,e|r)^{2}}{G_{i_{k}}(e,e|\zeta_{i_{k}}(r))^{2}}\frac{I_{i_{k}}^{(2)}(e,s_{k}|\zeta_{i_{k}}(r))}{G_{i_{k}}(e,s_{k}|\zeta_{i_{k}}(r))}.$

We set

$\alpha_{k}(r):=\frac{G(e,e|r)^{2}}{G_{i_{k}}(e,e|\zeta_{i_{k}}(r))^{2}}\frac{1}{G(e,e|r)^{4}}\sum_{z^{\prime}_{1},z^{\prime}_{2}\in\Gamma_{i_{k}}^{\perp}}G(e,z_{1}^{\prime}|r)G(z^{\prime}_{1},e|r)G(e,z_{2}^{\prime}|r)G(z^{\prime}_{2},e|r)$

and

$\tilde{\Phi}_{r}^{(2)}(e,y):=\sum_{k=1}^{n}\alpha_{k}(r)\frac{I_{i_{k}}^{(2)}(e,s_{k}|\zeta_{i_{k}}(r))}{G_{i_{k}}(e,s_{k}|\zeta_{i_{k}}(r))},$

Thus, by summing $z_{1}$ and $z_{2}$ in the left hand side of equation (7.8)
in the case where $j=k$ together with the convergence of appropriate
summations of the expression in equations (7.6) for $j<k$ and (7.7) for $j>k$
we find that

$\Phi_{r}^{(2)}(e,y)=\tilde{\Phi}_{r}^{(2)}(e,y)+O(1),r\to R,$

and the coefficients $\alpha_{k}(r)$ converge to some finite positive limit
$\alpha_{k}(R)$ as $r$ tends to $R$. Now, the random walk is spectrally
degenerate along $\Gamma_{1}=\mathbb{Z}^{5}$, hence
$I^{(2)}_{1}(e,g|\zeta_{1}(R))$ is infinite for $g\in\Gamma_{1}$. Therefore,
as long as the normal form of $y$ has one letter in $\Gamma_{1}$ (which
necessarily happens as soon as it has more than one letter), we get that
$\tilde{\Phi}_{r}^{(2)}(e,y)$ goes to infinity as $r$ tends to $R$. Hence, we
see that for fixed $y\in\Gamma$, as $r$ tends to $R$ we have that

(7.9) $\Phi_{r}^{(2)}(e,y)\sim\tilde{\Phi}_{r}^{(2)}(e,y),$

which may not be uniform in $y$.

We now consider $\Phi_{r}^{(2)}(x,y)$. By writing $x^{-1}y$ in its normal form
$x^{-1}y=t_{1}...t_{m}$ with $t_{i}\in\mathbb{Z}^{5}$ or $t_{i}\in\mathbb{Z}$,
we may reduce to the case where $x=e$. Hence, similarly we get that

$\tilde{\Phi}_{r}^{(2)}(x,y)=\sum_{k=1}^{m}\beta_{k}(r)\frac{I_{i_{k}}^{(2)}(e,t_{k}|\zeta_{i_{k}}(r))}{G_{i_{k}}(e,t_{k}|\zeta_{i_{k}}(r))}$

with coefficients $\beta_{k}$ such that as $r\to R$, $\beta_{k}(r)$ converges
to a finite limit and

$\Phi_{r}^{(2)}(x,y)\sim\tilde{\Phi}_{r}^{(2)}(x,y).$

We now let $y_{l-1}$ be the largest common prefix of $x$ and $y$ and again
write $x^{-1}y=t_{1}...t_{m}$ in its normal form.

$y_{l-1}$$e$$y_{l}$$y$$x$ Figure 6.

In particular, $s_{k}=t_{k+m-n}$ and $\alpha_{k}=\beta_{k+m-n}$ for $k\geq
l+1$. Consequently, the terms in $\tilde{\Phi}_{r}^{(2)}(e,y)$ and
$\tilde{\Phi}_{r}^{(2)}(x,y)$ eventually simplify each others and so as $r\to
R$ we get

$\Phi_{r}^{(2)}(x,y)-\Phi_{r}^{(2)}(e,y)=\tilde{\Phi}_{r}^{(2)}(x,y_{l})-\tilde{\Phi}_{r}^{(2)}(e,y_{l})+O(1).$

Indeed, this is exactly because setting $k_{0}=m-n$, we have
$s_{k}=t_{k+k_{0}}$ and $\alpha_{k}=\beta_{k+k_{0}}$ for $k\geq l+1$, so that

$\alpha_{k}\frac{I_{i_{k}}^{(2)}(e,s_{k}|\zeta_{i_{k}}(r))}{G_{i_{k}}(e,s_{k}|\zeta_{i_{k}}(r))}=\beta_{k+k_{0}}\frac{I_{i_{k+k_{0}}}^{(2)}(e,t_{k+k_{0}}|\zeta_{i_{k+k_{0}}}(r))}{G_{i_{k+k_{0}}}(e,t_{k+k_{0}}|\zeta_{i_{k+k_{0}}}(r))}.$

Consequently, the terms after $k=l+1$ cancel each other when taking the
difference.

Since we have,

$\frac{H(x,y)}{K_{R}(x,y)}-1=\lim_{r\to
R}\frac{\Phi_{r}^{(2)}(x,y)-\Phi_{r}^{(2)}(e,y)}{\Phi_{r}^{(2)}(x,y)}=\lim_{r\to
R}\frac{\tilde{\Phi}_{r}^{(2)}(x,y_{l})-\tilde{\Phi}_{r}^{(2)}(e,y_{l})}{\tilde{\Phi}_{r}^{(2)}(e,y)},$

in order to conclude the proof we need only show that

(7.10) $\lim_{y\to\xi}\lim_{r\to
R}\frac{\tilde{\Phi}_{r}^{(2)}(x,y_{l})-\tilde{\Phi}_{r}^{(2)}(e,y_{l})}{\tilde{\Phi}_{r}^{(2)}(e,y)}=0.$

It follows from [70, Theorem 13.10] that for every
$g\in\Gamma_{1}=\mathbb{Z}^{5}$,

(7.11)
$\bigg{|}C_{0}n^{5/2}\mu_{1}^{(n)}(g)-\mathrm{e}^{-\frac{\Sigma(g)}{2n}}\bigg{|}\underset{n\to\infty}{\longrightarrow}0,$

where $C_{0}>0$ is a constant and $\Sigma$ is a quadratic form associated with
the random walk on $\mathbb{Z}^{5}$, and the convergence is uniform in
$g\in\mathbb{Z}^{5}$. Consequently, since $1=R_{1}>\zeta_{1}(r)^{n}$, we get
that

$\bigg{|}C_{0}n^{2}\mu_{1}^{(n)}(g)\zeta_{1}(r)^{n}-\frac{1}{\sqrt{n}}\zeta_{1}(r)^{n}\mathrm{e}^{-\frac{\Sigma(g)}{2n}}\bigg{|}\underset{n\to\infty}{\longrightarrow}0.$

Thus, by Karamata’s Tauberian theorem [8, Corollary 1.7.3] we get that as
$r\to R$

$G_{1}^{(2)}(e,g|\zeta_{1}(r))\sim C_{g}\cdot\sum_{n\geq
1}\frac{1}{\sqrt{n}}\zeta_{1}(r)^{n}\mathrm{e}^{-\frac{\Sigma(g)}{2n}}\sim
C^{\prime}_{g}(1-\zeta_{1}(r))^{-1/2},$

where $C_{g}$ and $C_{g}^{\prime}$ depend only on $g$. Since $G_{1}^{(2)}$ is
roughly the same as $I_{1}^{(2)}$, we deduce that

(7.12) $I_{1}^{(2)}(e,e|\zeta_{1}(r))\asymp(1-\zeta_{1}(r))^{-1/2},$

where the implicit constant is independent of $r$. More generally, as $r\to R$
we have

(7.13) $I_{1}^{(2)}(e,g|\zeta_{1}(r))\asymp(1-\zeta_{1}(r))^{-1/2},$

where the implicit constant only depends on $g$.

Now, for any $g,h\in\Gamma_{1}=\mathbb{Z}^{5}$, we have

$G_{1}(h,g|\zeta_{1}(r))\geq
C_{1}G_{1}(h,e|\zeta_{1}(r))G_{1}(e,g|\zeta_{1}(r))$

where the constant $C_{1}$ is independent of $g,h$ and $r$. We deduce that

$\displaystyle I_{1}^{(2)}(e,g|\zeta_{1}(r))$ $\displaystyle\
=\sum_{w_{1},w_{2}\in\Gamma_{1}}G(e,w_{1}|\zeta_{1}(r))G(w_{1},w_{2}|\zeta_{2}(r))G(w_{2},g|\zeta_{2}(r))$
$\displaystyle\ \geq C_{1}\cdot
I_{1}^{(2)}(e,e|\zeta_{1}(r))G(e,g|\zeta_{1}(r)).$

Consequently, letting $k$ be such that $i_{k}=1$, so that
$s_{k}\in\Gamma_{1}=\mathbb{Z}^{5}$, we have

$I_{1}^{(2)}(e,e|\zeta_{1}(r))\lesssim\frac{I_{1}^{(2)}(e,s_{k}|\zeta_{1}(r))}{G_{1}(e,s_{k}|\zeta_{1}(r))}$

where the implicit constant is independent of $s_{k}$ and $r$. Thus, by
equation (7.12),

$(1-\zeta_{1}(r))^{-1/2}\lesssim\frac{I_{1}^{(2)}(e,s_{k}|\zeta_{1}(r))}{G_{1}(e,s_{k}|\zeta_{1}(r))}.$

Also, by (7.13),

$\frac{I_{1}^{(2)}(e,s_{k}|\zeta_{1}(r))}{G_{1}(e,s_{k}|\zeta_{1}(r))}\lesssim(1-\zeta_{1}(r))^{-1/2}C_{k},$

where $C_{k}$ only depends on $s_{k}$. We conclude that

(7.14)
$1\lesssim(1-\zeta_{1}(r))^{1/2}\frac{I_{i_{k}}^{(2)}(e,s_{k}|\zeta_{i_{k}}(r))}{G_{i_{k}}(e,s_{k}|\zeta_{i_{k}}(r))}\lesssim
C_{k}.$

On the other hand, the random walk is not spectrally degenerate along
$\Gamma_{2}=\mathbb{Z}$. Thus, if $i_{k}=2$, then

$\frac{I_{2}^{(2)}(e,s_{k}|\zeta_{2}(R))}{G_{2}(e,s_{k}|\zeta_{2}(R))}$

is finite, hence

(7.15)
$(1-\zeta_{1}(r))^{1/2}\frac{I_{i_{k}}^{(2)}(e,s_{k}|\zeta_{i_{k}}(r))}{G_{i_{k}}(e,s_{k}|\zeta_{i_{k}}(r))}\underset{r\to
R}{\longrightarrow}0$

Finally, let $\xi$ be a conical limit point. In our context of free products,
this means that $\xi$ is an infinite word, and $y$ converges to $\xi$ if and
only if for every $k$, the prefix of length $k$ of $y$ in its normal form
eventually agrees with the prefix of length $k$ of $\xi$. We deduce that for
fixed $x$, the largest common prefix $y_{l-1}$ of $x$ and $y$, as well as its
successor $y_{l}$ only depend on $\xi$ and not on $y$, as soon as $|y|$ is
large enough. Combining (7.14) and (7.15), we get

$(1-\zeta_{1}(r))^{1/2}\big{|}\tilde{\Phi}_{r}^{(2)}(x,y_{l})-\tilde{\Phi}_{r}^{(2)}(e,y_{l})\big{|}\asymp
C(x,\xi),$

where $C(x,\xi)$ is a finite constant only depending on $x$ and $\xi$.

Also, as $y$ converges to $\xi$, we get that $i_{k}=1$ occurs infinitely many
times, that is $s_{k}\in\Gamma_{1}$ for arbitrarily many $k$, hence by (7.12)

$\liminf_{r\to
R}(1-\zeta_{1}(r))^{1/2}\tilde{\Phi}_{r}^{(2)}(e,y)\underset{|y|\to\infty}{\longrightarrow}\infty.$

Therefore,

$\lim_{r\to
R}\frac{\tilde{\Phi}_{r}^{(2)}(x,y_{l})-\tilde{\Phi}_{r}^{(2)}(e,y_{l})}{\tilde{\Phi}_{r}^{(2)}(e,y)}=\lim_{r\to
R}\frac{(1-\zeta(r))^{1/2}\big{(}\tilde{\Phi}_{r}^{(2)}(x,y_{l})-\tilde{\Phi}_{r}^{(2)}(e,y_{l})\big{)}}{(1-\zeta(r))^{1/2}\tilde{\Phi}_{r}^{(2)}(e,y)}$

converges to 0 as $|y|$ tends to infinity, which proves (7.10) and thus
concludes the proof. ∎

######  Example 7.2.

Consider Example 5.4, where we constructed two probability measures $\mu$ and
$\mu^{\prime}$ on $\Gamma=\mathbb{Z}^{5}*\mathbb{Z}$ such that $\mu$ is
convergent and spectrally degenerate along $\mathbb{Z}^{5}$ and $\mu^{\prime}$
is spectrally non-degenerate. Then the $R$-Martin boundaries
$\partial_{M,R}(\Gamma,\mu)$ and $\partial_{M,R}(\Gamma,\mu^{\prime})$
associated with $\mu$ and $\mu^{\prime}$ are not homeomorphic.

Note first that since the parabolic subgroups $\mathbb{Z}^{5}$ and
$\mathbb{Z}$ are virtually abelian, by [33, Theorem 1.4] the $R$-Martin
boundaries are minimal, so that
$\overline{\partial_{M,R}^{m}\Gamma}=\partial_{M,R}\Gamma$ for both $\mu$ and
$\mu^{\prime}$.

Now, since $\mathbb{Z}$ is hyperbolic, we have that $\Gamma$ is also
hyperbolic relative (only) to $\mathbb{Z}^{5}$. In fact, $\Gamma$ is the HNN-