arXiv:1711.11307v2 [] 26 Jan 2018

THE MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC
GROUPS WITH VIRTUALLY ABELIAN PARABOLIC
SUBGROUPS
MATTHIEU DUSSAULE, ILYA GEKHTMAN, VICTOR GERASIMOV,
AND LEONID POTYAGAILO
Abstract. Given a probability measure on a finitely generated group, its
Martin boundary is a way to compactify the group using the Green’s function
of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively
hyperbolic groups with virtually abelian parabolic subgroups. In particular, in
the case of nonuniform lattices in the real hyperbolic space Hn , we show that
the Martin boundary coincides with the CAT (0) boundary of the truncated
space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.

1. Introduction and Statement of results
1.1. Random walks on relatively hyperbolic groups. A probability measure
µ on a countable group Γ determines a Γ-invariant Markov chain with transition
probabilities p(x, y) = µ(x−1 y), called a random walk.
Connecting asymptotic properties of this random walk to the geometry of Cayley
graphs of Γ has been a fruitful line of research. One way to do this is through relating
the Green’s function of µ to some natural metric on Γ, and the probabilistically
defined Martin boundary of µ to some geometric boundary of Γ.
The Green’s function G of (Γ, µ) is defined as
G(x, y) =

∞
X

µ∗n (x−1 y).

n=0

It describes the amount of time a random path starting at x is expected to spend
at y. We now fix a base point o in Γ. For each y ∈ Γ the function Ky : Γ → R
defined by Ky (x) = G(x, y)/G(o, y) is called a Martin kernel. The compactification
of Γ in the space of functions Γ → R of the Martin kernels Ky : y ∈ Γ is called
the Martin compactification Γµ and ∂µ Γ = Γµ \ Γ is called the Martin boundary.
These definitions also make sense for more general measures µ (see Section 2).
Giving a geometric description of the Martin boundary is often a difficult problem. Margulis showed that for centered finitely supported random walks on nilpotent groups, the Martin boundary always is trivial [17]. On the other hand, for
noncentered random walks with exponential moment on abelian groups of rank k,
Ney and Spitzer [18] showed that the Martin boundary is homeomorphic to a sphere
of dimension k − 1.
For hyperbolic groups with finitely supported measures, Ancona [1] proved that
the Green’s function is roughly multiplicative along word geodesics. He used this
to identify the Martin boundary with the Gromov boundary.
1

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M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

A finitely generated group Γ is called hyperbolic relative to a system of subgroups
Ω if Γ admits a convergence action on a compact metric space T such that every
point of T is either conical or bounded parabolic and the elements of Ω are the
stabilizers of the parabolic points. We will give more details in Section 3.
We will always assume that the action Γ y T is minimal and nonelementary,
that is T contains more than two points. We again refer to Section 3 for more
details.
For relatively hyperbolic groups with finitely supported measures, Gekhtman,
Gerasimov, Potyagailo and Yang proved a generalization of Ancona’s multiplicativity estimate, see [9, Theorem 5.2]. Its most general formulation uses the Floyd
distance, which is a rescaling of the word distance. It will be defined in Section 3.1.
The authors of [9] used this estimate to show that the Martin boundary covers
the Bowditch boundary -the compact set T of the above definition, or equivalently,
the boundary of a proper geodesic Gromov hyperbolic space X on which Γ acts
geometrically finitely. Moreover the preimage of any conical point is a singleton.
Determining the Martin boundary is thus reduced to determining the preimage of
parabolic points.
Dussaule [7] gave a geometric description of the Martin boundary of finitely
supported random walks on free products of abelian groups, identifying it with the
visual boundary of a CAT (0) space on which the group acts cocompactly.
The key technical result of [7] extends results of Ney and Spitzer [18] to more
general chains. It states that the Martin boundary of non-centered -or strictly subMarkov- chains on Zk × {1, ..., N }, N ∈ N, is a sphere of dimension k − 1. We give
a precise formulation below (see Proposition 4.6).
In this paper, we extend these two results of [9] and [7] to determine the Martin
boundary of finitely supported measures on groups relatively hyperbolic with respect to virtually abelian subgroups. These include the following well-known classes
of groups:
• geometrically finite Kleinian groups,
• limit groups,
• groups acting freely on Rn trees.
If Γ is a compactification of Γ, define the corresponding boundary as ∂Γ := Γ\ Γ.
Call a boundary ∂Γ of Γ a Z-boundary if the following holds (see Section 3.3 for
the details): an unbounded sequence (gn ) of Γ converges to a point in ∂Γ if and
only if either (gn ) converges to a conical point in the Bowditch boundary, or there
is a coset gP of a parabolic subgroup P such that one (and thus any) closest point
projections of (gn ) to gP converges to a point in the CAT (0) boundary of gP .
Our main result is the following.
Theorem 1.1. Let Γ be a hyperbolic group relative to a collection of infinite virtually abelian subgroups. Let µ be a measure on Γ whose finite support generates Γ
as a semigroup. Then, the Martin boundary is a Z-boundary.
We also prove the following.
Theorem 1.2. Let Γ be a hyperbolic group relative to a collection of infinite virtually abelian subgroups. Let µ be a measure on Γ whose finite support generates Γ
as a semigroup. Then, every point of the Martin boundary is minimal.

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We recover from Theorem 1.1, as a particular case, that the Martin boundary of a non-virtually cyclic hyperbolic group is the Gromov boundary. Since two
Z-boundaries are equivariantly homeomorphic (see Lemma 3.4 for a precise statement), Theorem 1.1 gives a complete description of the Martin boundary up to
homeomorphism.
There is a particularly simple geometric construction of a Z-boundary when Γ
is a nonuniform lattice in the real hyperbolic n-space Hn . By removing from Hn a
Γ-equivariant collection of disjoint horoballs based at parabolic points, we obtain
a CAT (0) space (for the shortest-path metric) on which Γ acts cocompactly. One
can easily check that the visual boundary of this CAT (0) space is a Z-boundary. In
particular, when n = 3, the Z-boundary is homeomorphic to the sphere S2 with a
countable and dense set of discs removed. Thus, the Z-boundary is homeomorphic
to the Sierpinski carpet.
Dahmani [5] proposed a construction of a Z-boundary for any relatively hyperbolic group with virtually abelian parabolic subgroups, using the so-called coned-off
graph of Γ. See the appendix for details. Our result implies that both these constructions are equivariantly homeomorphic to the Martin boundary.
Our proofs of Theorems 1.1 and 1.2 use both the deviation inequalities of [9]
and the generalization of Ney and Spitzer from [7]. Roughly stated, we show that
the preimage of a parabolic point on the Bowditch boundary, with stabilizer P , is
homemorphic to the Martin boundary of a neighborhood of P with a finite (though
not probability) measure induced by first return times, which by the result of [7]
is a sphere of the appropriate dimension. Actually the proof is a little bit more
technically involved and we now give some more details.
To show that the Martin boundary ∂µ Γ is a Z-boundary, we have to deal with
two types of trajectories, namely those converging to a conical point in the Bowditch
boundary and those whose projections converge to a point in the geometric boundary of a parabolic subgroup.
For the first type of trajectories, we use relative Ancona inequalities (Theorem 4.3). It was already proved in [9] that such sequences (gn ) converge in the
Martin boundary. Basically, when (gn ) converges to a conical limit point, one can
choose an increasing number of transition points on a geodesic from a base point
o to (gn ). This number of transition points tends to infinity. Relative Ancona
inequalities roughly state that the random walk follows geodesic along transition
points with large probability. Forcing the paths to go through an increasing number of bounded neighborhoods of transition points then leads to convergence of the
Martin kernels.
For the second type of trajectories, we want to study the induced random walk on
a parabolic subgroup P . However, this random walk is not finitely supported so we
cannot apply Dussaule’s result as it is stated above. Using properties of the Floyd
metric and results of Gerasimov and Potyagailo relating these with the geometry
of the Cayley graph, we prove that the induced random walk on a sufficiently large
neighborhood of P has large exponential moments. Using more precise statements
of [7], we prove that this is enough to conclude: if (gn ) converges in the geometric
boundary of P , then it converges in the Martin boundary and two different points
in the geometric boundary give rise to two different points in the Martin boundary.

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M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

1.2. Organization of the paper. The paper is divided into five main parts, besides the introduction. Section 2 is devoted to give the necessary probabilistic
background on random walks, Markov chains, and their Martin boundaries.
Section 3 is about relatively hyperbolic groups. In Sections 3.1 and 3.2 we give
different equivalent definitions of those groups and state results of Gerasimov and
Potyagailo about the interplay between the Floyd distance and the geometry of the
Cayley graph of such groups. We also define properly what is a Z-boundary and
what is the geometric compactification of a parabolic subgroup in Section 3.3.
In Section 4, we give the necessary geometric background on Martin boundaries
for the proof of our main theorem. In Section 4.1, we state the relative Ancona
inequalities obtained in [9]. These inequalities will be used throughout the proofs,
especially when we deal with trajectories converging to conical limit points. In
Section 4.2, we state the results of Dussaule about Martin boundaries of chains on
Zk × {1, ..., N }. This part is a bit technical and we extend his results to deal later
with trajectories converging in the geometric boundary of a parabolic subgroup.
Corollary 4.10 could be taken as a black box to prove Proposition 5.5 in the following
section.
In Section 5, we prove our main theorem, Theorem 1.1. We first deal with conical
limit points in Section 5.1, using results of Section 4.1 and then with parabolic
subgroups in Section 5.2, using results of Section 4.2.
In Section 6, we prove Theorem 1.2, that is, the Martin boundary is minimal.
Again, we will deal separately with trajectories converging to conical limit points
and trajectories converging in the boundary of parabolic subgroups.
In the appendix, we give a geometric construction of a Z-boundary, using a
construction of Dahmani in [5].
1.3. Acknowledgements. We would like to thank Sebastien Gouëzel for helpful
advice. We also thank Wenyuan Yang for helpful conversations.
2. Martin boundaries of random walks
Let us give here a proper definition of the Martin boundary and the minimal
Martin boundary. In this paper, we deal with random walks on groups, but during
the proofs, we will restrict the random walk to thickenings of peripheral subgroups
and we will not get actual random walks. Thus, we need to define Martin boundaries
for more general transition kernels.
Consider a countable space E and equip E with the discrete topology. Fix some
base point o in E. Consider a transition kernel p on E with finite total mass, that
is p : E × E → R+ satisfies
X
∀x ∈ E,
p(x, y) < +∞.
y∈E

It is often required that the total mass is 1 and in that case, the transition kernel
defines a Markov chain on E. In general, we will say that p defines a chain on E
and we will sometimes assume that this chain is sub-Markov, that is the total mass
is smaller than 1. If µ is a probability measure on a finitely generated group Γ,
then p(g, h) = µ(g −1 h) is a probability transition kernel and the Markov chain is
the random walk associated to µ.

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

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Define in this context the Green’s function G as
X
G(x, y) =
p(n) (x, y) ∈ [0, +∞],
n≥0

where p(n) is the nth convolution power of p, i.e.
X
p(x, x1 )p(x1 , x2 ) · · · p(xn−1 , y).
p(n) (x, y) =
x1 ,...,xn−1

Definition 2.1. Say that the chain defined by p is finitely supported if for every
x ∈ E, the set of y ∈ E such that p(x, y) > 0 is finite.
Definition 2.2. Say that the chain defined by p is irreducible if for every x, y ∈ E,
there exists n such that p(n) (x, y) > 0.
For a Markov chain, this means that one can go from any x ∈ E to any y ∈ E with
positive probability. In this setting, the Green’s function G(x, y) is closely related
to the probability that a µ-governed path starting at x ever reaches y. Indeed, the
strong Markov property shows that the latter quantity is equal to G(x,y)
G(y,y) .
Notice that in the case of a random walk on a group Γ, the Green’s function
is invariant under left multiplication, so that G(x, x) = G(o, o) for every x. Thus,
up to some constant, G(x, y) is the probability to go from x to y. Moreover, the
irreducibility of the chain is equivalent to the condition that the support of the
measure µ generates Γ as a semigroup.
In particular, in the context of Theorem 1.1, the transition kernel defined by the
probability measure µ is irreducible.
We will also use the following definition during our proofs.
Definition 2.3. Say that the chain defined by p is strongly irreducible if for every
x, y ∈ E, there exists n0 such that ∀n ≥ n0 , p(n) (x, y) > 0.
We will also assume that the chain is transient, meaning that the Green’s function
is everywhere finite. For a Markov chain, this just means that almost surely, a path
starting at x returns to x only a finite number of times.
Consider an irreducible transient chain p. For y ∈ E, define the Martin kernel
based at y as
G(x, y)
.
Ky (x) =
G(o, y)
The Martin compactification of E with respect to p (and o) is a compact space
containing E as an open and dense space, whose topology is described as follows.
A sequence (yn ) in E converges to a point in the Martin compactification if and only
if the sequence Kyn converges pointwise. Up to isomorphism, it does not depend
on the base point o and we denote it by E M . We also define the Martin boundary
as ∂M E = E M \ E. We refer to [19] for a complete construction of the Martin
compactification.
Seeing the Martin kernel K as a function of two variables x and y, the Martin
compactification is then the smallest compact space M in which E is open and dense
and such that K can be continuously extended to the space E × M . If ỹ ∈ E M ,
denote by Kỹ the extension of the Martin kernel.

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M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

In the particular case of a symmetric Markov chain, that is a Markov chain
satisfying p(x, y) = p(y, x), the Green’s distance, which was defined by Brofferio
and Blachère in [3] as
dG (x, g) = −lnP(x → y),
is actually a metric and the Martin compactification of E with respect to the Markov
chain p is the horofunction compactification of E for this metric.
Now, assume that E = Γ is a finitely generated group and that the transition
kernel p is defined by a probability measure µ. In that case, denote by Γµ the
Martin compactification and by ∂µ Γ the Martin boundary. The action by (left)
multiplication of Γ on itself extends to an action of Γ on Γµ .
One important aspect of the Martin boundary is its relation with harmonic
functions. Recall that if p is a transition kernel on a countable space E, a harmonic
function is a function φ : E → R such that pφ = φ, that is,
X
∀x ∈ E, φ(x) =
p(x, y)φ(y).
y∈E

We have the following key property (see [19, Theorem 4.1]).
Proposition 2.4. Let p be a irreducible transient transition kernel on a countable
space E. For any non-negative harmonic function φ, there exists a measure ν on
the Martin boundary ∂M E of E such that
Z
Kx̃ (x)dν(x̃).
∀x ∈ E, φ(x) =
∂M E

Let φ be a non-negative harmonic function. It is called minimal if any other nonnegative harmonic function ψ such that ψ(x) ≤ φ(x) for every x ∈ E is proportional
to φ. The minimal Martin boundary is the set
m
E = {x̃ ∈ ∂M E, Kx̃ (·) is minimal harmonic}.
∂M

It is thus a subset of the full Martin boundary ∂M E. A classical representation
theorem of Choquet shows that for any non-negative harmonic function φ, one can
m
choose the support of the measure ν lying in ∂M
E. The measure ν is then unique
(see the first section of [19]). In other words, for any such function φ, there exists
m
a unique measure µφ on ∂M
E such that
Z
Kx̃ (x)dµφ (x̃).
∀x ∈ E, φ(x) =
mE
∂M

3. Relatively hyperbolic groups
3.1. Relative hyperbolicity and the Floyd metric. Let Γ be a finitely generated group. The action of Γ on a compact Hausdorff space T is called a convergence
action if the induced action on triples of distinct points of T is properly discontinuous. Suppose Γ y T is a convergence action. The set of accumulation points ΛΓ
of any orbit Γ · x (x ∈ T ) is called the limit set of the action. As long as ΛΓ has
more than two points, it is uncountable and the unique minimal closed Γ-invariant
subset of T . The action is then said to be nonelementary. In this case, the orbit of
every point in ΛΓ is infinite. The action is minimal if ΛΓ = T .
A point ζ ∈ ΛΓ is called conical if there is a sequence (gn ) of Γ and distinct points
α, β ∈ ΛΓ such that gn ζ → α and gn η → β for all η ∈ T \ {ζ}. The point ζ ∈ ΛΓ is
called bounded parabolic if the stabilizer of ζ in Γ is infinite and acts cocompactly

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

7

on ΛΓ \ {ζ}. The stabilizers of bounded parabolic points are called (maximal)
parabolic subgroups. The convergence action Γ y T is called geometrically finite
if every point of ΛΓ ⊂ T is either conical or bounded parabolic. Yaman [23]
proved that if Γ y T is a minimal geometrically finite action, there is a proper
Gromov hyperbolic space X on which Γ acts by isometries and a Γ-equivariant
homeomorphism T → ∂X.
Suppose now Ω is a collection of subgroups of Γ. We say Γ is hyperbolic relative
to Ω if there exists some compactum T on which Γ acts minimally and geometrically finitely and the maximal parabolic subgroups are the elements of Ω. Such a
compactum is then unique up to Γ-equivariant homeomorphism [4] and is called
the Bowditch boundary of (Γ, Ω). The group Γ is said to be relatively hyperbolic
if it is hyperbolic relative to some collection of subgroups, or equivalently if it admits a geometrically finite convergence action on some compactum. The group Γ
is nonelementary relatively hyperbolic if it admits a nonelementary geometrically
finite convergence action on some infinite compactum.
A useful fact is the following. Let Γ be a group hyperbolic relative to a collection
of parabolic subgroups Ω. The set Ω is invariant under conjugacy, since the set of
parabolic limit points is invariant under translation. There is a finite number of
conjugacy classes of elements of Γ (see [21, Theorem 1B]).
P
Let f : R+ → R+ be a function satisfying two conditions:
n>0 fn < ∞; and
there exists a λ ∈ (0, 1) such that 1 > fn+1 /fn > λ for all n∈N. The function f is
called the rescaling function.
Pick a basepoint o ∈ Γ and rescale CS Γ by declaring the length of an edge σ to
be f (d(o, σ)). The induced shortpath metric on CS Γ is called the Floyd metric with
respect to the basepoint o and Floyd function f and denoted by δof (., .). Its Cauchy
completion (whose topology does not depend on the basepoint) is called the Floyd
compactification Γf and ∂f Γ = Γf \ Γ is called the Floyd boundary. Karlsson
showed that the action of a group on its Floyd boundary is always convergence
[14, Theorem 2]. On the other hand, if Γ is relatively hyperbolic and if the Floyd
function f is not decreasing exponentially too fast, Gerasimov [10, Map theorem]
proved that there is continuous Γ-equivariant surjection (Floyd map) from the Floyd
boundary to the Bowditch boundary. Furthermore, Gerasimov and Potyagailo [11,
Theorem A] proved that the preimage of any conical point by this map is a singleton
and the preimage of a parabolic fixed point p is the limit set for the action of its
stabilizer Γp on ∂f Γ. In particular if Γp is an amenable non-virtually cyclic group
then its limit set on the Floyd boundary is a point. Consequently, when Γ is
hyperbolic relative to a collection of infinite amenable subgroups which are not
virtually cyclic, the Floyd boundary is homeomorphic to the Bowditch boundary.
If α is a (finite or infinite) geodesic in CS Γ for the Cayley metric, a point p ∈ α
is said to be (ǫ, R)-deep if there is a γ ∈ Γ, P ∈ Ω such that the length R-segment
of α around p is contained in the ǫ-neighborhood of γP . Otherwise, p ∈ α is called
an (ǫ, R)-transition point of α. Gerasimov and Potyagailo prove the following key
property.
Proposition 3.1. [12, Corollary 5.10] For each ǫ > 0, R > 0 and D > 0 there is a
number δ > 0 such that if y is within word distance D of an (ǫ, R)-transition point
of a word geodesic from x to z then δyf (x, z) > δ.

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3.2. Other viewpoints on relative hyperbolicity. We defined relative hyperbolicity in terms of convergence actions. There are two useful equivalent definitions.
The first one is stated in terms of actions on hyperbolic spaces.
Definition 3.2. [4] A group Γ is hyperbolic relative to a collection of subgroups Ω if
it acts by isometries and properly discontinuously on a path-metric proper hyperbolic
space X such that the action on ∂X is geometrically finite and the maximal parabolic
subgroups are exactly the elements of Ω.
As stated above, Yaman proved that this notion of relative hyperbolicity coincides with the one we gave in terms of convergence actions.
Relative hyperbolic groups were first introduced by Gromov in [13] as generalizations of finite covolume Kleinian groups with parabolic elements. The space X
plays the role of the hyperbolic space Hn , although it is not CAT (0) in general.
Actually, a large part of the intuition for relative hyperbolic groups comes from
this action on a hyperbolic space X and the analogy with Kleinian groups acting
on Hn . During the proofs, it will be more convenient to use the Floyd metric, but
one could recover our results using the hyperbolicity of X.
The second definition of relative hyperbolicity is based on the geometry of the
Cayley graph. It was introduced by Farb in [8]. We will not use this point of view
during the proofs and so we do not include this definition here. However, we will
use Farb’s point of view in the construction of a Z-boundary in the appendix. We
will give more details there.
3.3. Geometric compactifications. We now give a precise definition of a Zboundary. We first define the geometric boundary of an infinite, virtually abelian,
finitely generated group. Let P be such a subgroup, so that there exists a subgroup
of P isomorphic to Zk , for some k ≥ 1, with finite index in P . Then, any section
P/Zk → P provides an identification between P and Zk ×{1, ..., N } for some N ≥ 1.
Let (gn ) be a sequence in P and identify gn with (zn , jn ) ∈ Zk × {1, ..., N }. Say
that the sequence (gn ) converges to a point in the boundary of P if zn tends to
infinity and kzznn k converges to some point in the sphere Sk−1 . This defines what
we call the geometric boundary ∂P of P . This construction coincides with the
CAT (0) boundary of P : the visual boundary of a CAT (0) space Rk on which P
acts properly and cocompactly.
More generally, if F is a finite set, we define the geometric boundary of the
product P × F as follows. First identify P with Zk × {1, ..., N } as before. This
provides an identification between P × F and Zk × {1, ..., N ′ } for some other integer
N ′ ≥ 1. As above, a sequence (gn ) in P × F is said to converge in the geometric
boundary if its projection onto Zk under this identification converges in the CAT (0)
boundary of Zk . This slight generalization will be useful in the following. Indeed,
for technical reasons, when studying sequences converging in ∂P , we will not restrict
the random walk to parabolic subgroups but to bounded neighborhoods of them.
Suppose now that Γ is hyperbolic relative to a collection Ω of infinite subgroups,
each of which is virtually abelian. For A ⊂ Γ and g ∈ Γ, let
projA (g) = {h ∈ A : d(h, g) = d(A, g)}
be the set of closest point projections of g to A.
For another subset F ⊂ Γ let
projA (F ) = ∪g∈F projA (g).

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Let πA : Γ → A be a function with πA (g) ∈ projA (g).
If P is a coset of a parabolic subgroup of Γ, the diameter of projA (g) is finite and
bounded independently of g ∈ Γ [4, Lemma 5.1]. In particular, if (gn ) is a sequence
in Γ, the convergence of πP (gn ) to the geometric boundary of P does not depend
on the choice of πP .
We will use boundaries throughout the paper. We fix the following terminology.
A compactification Γ of Γ is a metrizable compact space, containing Γ as an open
and dense space, endowed with a group action Γ y Γ that extends the action by
left multiplication on Γ. Then, ∂Γ := Γ \ Γ is called a boundary of Γ.
Definition 3.3. Let Γ be a group hyperbolic relative to a collection Ω of virtually
abelian subgroups. Fix a finite full set of representatives of parabolic subgroups
Ω0 ⊂ Ω for the action Γ y Ω by conjugacy.
A Z-boundary of (Γ, Ω) is a boundary ∂Γ such that the following holds.
• The identity on Γ extends to a continuous equivariant surjective map
Γ ∪ ∂Γ → Γ ∪ ∂B Γ,
where ∂B Γ is the Bowditch boundary.
• A sequence (gn ) in Γ converges to a point in ∂Γ if and only if (gn ) tends
to infinity and either (gn ) converges to a conical point in the Bowditch
boundary or there exist g ∈ Γ and a parabolic subgroup P ∈ Ω0 such that
g −1 πgP (gn ) converges in the geometric boundary of P .
In other words, a sequence converges in the Z-boundary if it converges to a conical
limit point or if its projection converges in the geometric boundary of a coset of a
parabolic subgroup.
The term Z-boundary was coined by Bestvina [2] in a more general context. In
the context of relatively hyperbolic groups with virtually abelian parabolic subgroups, Dahmani [5] gave an equivalent explicit construction. In the appendix we
show that Dahmani’s construction is equivalent to Definition 3.3, justifying our use
of the term.
Lemma 3.4. Let Γ be a group hyperbolic relative to a collection Ω of infinite
virtually abelian subgroups. Up to equivariant homeomorphism, the Z-boundary
does not depend on the set of coset representatives.
Proof. Let Ω1 and Ω2 be two sets of representatives of conjugacy classes of elements
of Ω. Let ∂1 Γ and ∂2 Γ be two Z-boundaries constructed with Ω1 and Ω2 respectively. If γ ∈ ∂1 Γ, there exists a sequence (gn ) in Γ that converges to γ. If γ is
mapped to a conical limit point α in the Bowditch boundary, then (gn ) converges
in α in the Bowditch compactification. Thus, it converges to a uniquely defined
point γ̃ in ∂2 Γ.
Assume now that γ is mapped to a parabolic limit point α in the Bowditch
boundary. Then, (gn ) cannot converges to a conical limit point, so there exists
a coset gP of a parabolic subgroup P ∈ Ω1 , such that the projection (πgP gn )
converges in the geometric boundary of gP . There is a unique h ∈ Γ such that
Q = hP h−1 ∈ Ω2 . The points πgP gn lie a bounded distance away from πgh−1 Q gn ,
so that (πgh−1 Q gn ) converges in the geometric boundary of gh−1 Q. Thus, there
exists a uniquely defined point γ̃ ∈ ∂2 Γ such that (gn ) converges to γ̃.

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M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

This defines a map γ ∈ ∂1 Γ 7→ γ̃ ∈ ∂2 Γ. Similarly, one defines a map ∂2 Γ → ∂1 Γ
and by construction, the composition of these maps give the identity on ∂1 Γ and
∂2 Γ respectively.
These maps are continuous. Indeed, let (γn ) converges to γ in ∂1 Γ. By compactness, we only have to prove that the sequence (γ̃n ) has a unique limit point, which
is the image of γ. Let γ̃ be a limit point, so that there is a subsequence (γ̃σ(n) ) that
converges to γ̃. By density, there exist sequences (gn,m ) that converge to γn when
m tends to infinity, with gn,m in Γ. By construction, (gn,m ) also converges to γ˜n in
∂2 Γ. Both Γ ∪ ∂1 Γ and Γ ∪ ∂2 Γ are metrizable compact spaces. Denote by d1 and d2
corresponding distances. Then, one can choose a sequence hn := gσ(n),φ(σ(n)) , with
φ : N → N increasing, such that d1 (hn , γσ(n) ) ≤ n1 and d2 (hn , γ̃σ(n) ) ≤ n1 . Then,
(hn ) converges to γ in ∂1 Γ and converges to γ̃ in ∂2 Γ. By construction, this proves
that γ̃ is uniquely determined and is the image of γ. Thus, the map ∂1 Γ → ∂2 Γ
is continuous and similarly, the map ∂2 Γ → ∂1 Γ is continuous, which proves these
maps are homeomorphisms. By construction, they also are Γ-equivariant.

According to this lemma, verifying that the Martin boundary satisfies the conditions of Definition 3.3 completely determines it up to Γ-equivariant homeomorphism.
In [5], Dahmani gave a geometric construction of a Z-boundary for any Γ which
is hyperbolic relative to virtually abelian subgroups [5, Theorem 3.1]. We refer
to the appendix for more details. When Γ is a geometrically finite subgroup of
isometries of real hyperbolic space Hn , Bestvina [2] noted that ∂Γ coincides with
the CAT (0) boundary of the space obtained by removing from Hn a Γ-equivariant
family of open horoballs based at parabolic points of ∂Hn .
4. Topology of Martin boundaries
4.1. Ancona’s inequality for relatively hyperbolic groups. Suppose Γ is a
finitely generated group. Let µ be a probability measure whose finite support
generates Γ as a semigroup and let G be the associated Green’s function.
c
Denote by G(x, z; BR
(y)) the Green’s function from x to z conditioned by not
visiting the ball of center y and radius R, that is
X
c
G(x, z; BR
(y)) =
Px (Xk = z|∀l ∈ {1, ..., k − 1}, Xl ∈
/ BR (y)).
k≥0

When Γ is hyperbolic, Ancona [1] proved the following.
Theorem 4.1. [1] For every ǫ > 0 there is a R > 0 such that whenever x, y, z ∈ Γ
lie along a word geodesic, in that order, we have
c
G(x, z; BR
(y)) ≤ ǫG(x, z).

In other words, if y is on a word geodesic connecting x and z, a random path
between x and z passes in a bounded neighborhood of y with high probability.
Ancona used this to identify the Martin boundary of hyperbolic groups with their
Gromov boundary. To do the same for relatively hyperbolic groups, we will need
the following result of Gekhtman, Gerasimov, Potyagailo, and Yang. Let f be a
Floyd function.

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

11

Theorem 4.2. [9, Theorem 5.2] For each ǫ > 0 and δ > 0 there is a R > 0 such
f
that for all x, y, w ∈ Γ with δw
(x, y) > δ one has
c
G(x, y; BR
(w)) ≤ ǫG(x, y).

The analogy with Ancona’s estimate along geodesics comes from the following
fact. Recall that if α is a word geodesic in Γ, a point p ∈ α is called an (ǫ, R)transition point if there does not exist a coset P of a parabolic subgroup such that
the length R-segment of α surrounding p is contained in the ǫ-neighborhood of P .
Theorem 4.3. [9] Let α be a geodesic in CS Γ. Then for any ǫ > 0, r > 0, D > 0,
and η > 0 there is an R > 0 such that if x, y, z ∈ Γ, α is a geodesic between x and
z, and y is within D of an (η, r)-transition point of α then
c
G(x, z; BR
(y)) ≤ ǫG(x, z).

Indeed, in this situation, the Floyd distance δyf (x, z) is bounded from below by
a universal constant, according to Proposition 3.1.
Although Theorem 4.2 holds for arbitrary finitely generated groups, its results are
vacuous when the Floyd boundary is trivial, and besides virtually cyclic groups, the
only known examples with nontrivial Floyd boundary are nonelementary relatively
hyperbolic groups.
4.2. Martin boundaries of thickened abelian groups. To understand the behavior of Kgn (g), when gn converges in the geometric boundary of a parabolic
subgroup, we will introduce the transition kernel of the first return to the corresponding subgroup P . We will then get a sub-Markov chain on P and we will show
that we can identify this first-return-chain with a Zk -invariant sub-Markov chain
on Zk × {1, ..., N } (see Lemma 5.9). We will then use results for such chains.
In [7], the author shows that under some technical assumptions, the Martin
boundary of such a chain on Zk × {1, ..., N } coincides with the geometric boundary.
In this setting, the geometric boundary is defined as follows. A sequence (zn , jn ) in
Zk ×{1, ..., N } converges to a point in the geometric boundary if zn tends to infinity
and kzznn k converges in the unit sphere Sk−1 . We now introduce the assumptions of
[7] and we will later show that they are satisfied in our setting.
Consider a Zk -invariant chain p on the product space Zk × {1, ..., N }. For every
function defined on Zk × {1, ..., N }, the {1, ..., N } coordinate will be considered as
an index. For example, the transition kernel will be written as pj1 ,j2 (z1 , z2 ), its
(n)
nth power of convolution as pj1 ,j2 (z1 , z2 ), the Green’s function as Gj1 ,j2 (z1 , z2 ) and
the Martin kernel as Kj1 ,j2 (z1 , z2 ). We can thus see these functions as the entries
of N × N matrices. Assume that the chain p is strongly irreducible, that is, for
every j1 , j2 ∈ {1, ..., N } and for every z1 , z2 ∈ Zk , there exists n0 such that for every
(n)
n ≥ n0 , pj1 ,j2 (z1 , z2 ) > 0. As we will see later (see Lemma 5.1), strong irreducibility
is not too much to ask and we will be able to reduce our study of irreducible chains
to strongly irreducible ones.
In [18], Ney and Spitzer show that the Martin boundary of a strongly irreducible,
finitely supported, noncentered random walk on Zk coincides with the CAT (0)
boundary. Their proof is based on the study of minimal harmonic functions which
are of the form z ∈ Zk 7→ eu·z for some u ∈ Rk satisfying the condition
X
p(0, z)eu·z = 1.
(1)
z∈Zk

12

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

In our setting, for u ∈ Rk , we define the N × N matrix F (u) whose entries are given
by
X
pj1 ,j2 (0, z)eu·z .
Fj1 ,j2 (u) =
z∈Zk

The entries of this matrix may be infinite. We restrict our attention to the set
where they are finite and denote this set by F0 :
F0 = {u ∈ Rk , ∀j1 , j2 ∈ {1, ..., N }, Fj1 ,j2 (u) < +∞}.
We also denote by F the interior of F0 .
Lemma 4.4. For every u ∈ F0 , the matrix F (u) has non-negative entries. Furthermore, this matrix is strongly irreducible, meaning that there exists n ≥ 0 such
that F (u)n has positive entries.
Proof. Direct calculation shows that the entries of F (u)n are given by
X (n)
pj1 ,j2 (0, z)eu·z .
Fj1 ,j2 (u)n =
z∈Zk

Strong irreducibility of F (u) is deduced from strong irreducibility of p.



Since F (u) is strongly irreducible, it follows from the Perron-Frobenius theorem
(see [20, Theorem 1.1]) that F (u) has a dominant positive eigenvalue, that is an
eigenvalue λ(u) which is positive and such that for every other eigenvalue λ ∈ C,
|λ| < λ(u). Moreover, any eigenvector associated to λ(u) has non-zero coordinates
and we can assume that every coordinate is positive. The analog of Equation (1)
will be
(2)

λ(u) = 1.

Denote by D the set where λ(u) is at most 1: D = {u ∈ F , λ(u) ≤ 1}. The two
technical assumptions of [7] on the chain p are the following.
Assumption 1. The set D is compact.
Assumption 2. The minimum of the function λ is strictly smaller than 1.
Since λ(u) is a dominant eigenvalue, it is analytic in u (see Proposition 8.20 in
[22]). For u ∈ F , denote by ∇λ(u) the gradient of λ with respect to u. We have
the following (see Lemma 3.13 in [7]).
Lemma 4.5. Under Assumptions 1 and 2, the set {u ∈ Rk , λ(u) = 1} is homeomorphic to Sk−1 and an explicit homeomorphism is given by
∇λ(u)
u ∈ {u ∈ Rk , λ(u) = 1} 7→
.
k∇λ(u)k
This homeomorphism provides a homeomorphism ϕ between the geometric boundary of Zk × {1, ..., N } and Sk−1 constructed as follows. Let (zn , jn ) be a sequence in
Zk ×{1, ..., N } converging to a point z̃ in the geometric boundary ∂(Zd ×{1, ..., N }).
Then zn tends to infinity and kzznn k converges to a point θ in the unit sphere Sk−1 .
∇λ(u)
There exists a unique u ∈ {u ∈ Rk , λ(u) = 1} such that θ = k∇λ(u)k
. Then, define
ϕ(z̃) = u.
The Martin boundary is defined up to the choice of a base point. Fix such a
base point (z0 , j0 ) ∈ Zk × {1, ..., N }. Now, we can state that the Martin boundary
coincides with the geometric boundary (see Proposition 3.17 in [7]).

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

13

Proposition 4.6. Let p be a strongly irreducible transition kernel on Zk ×{1, ..., N }
which is Zk -invariant and satisfies Assumptions 1 and 2. If zn ∈ Zk converges to
z̃ ∈ ∂Zk , let u = ϕ(z̃). Then, for every z ∈ Zk and for every j1 , j2 ∈ {1, ..., N },
there exists Cj1 > 0 which only depends on j1 such that Kj1 ,j2 (z, zn ) converges to
Cj1 eu·(z−z0 ) .
Consider now a chain p on Zk × N. If N ≥ 1, define the induced chain pN as the
chain of the first return to Zk × {1, ..., N }, that is, if (z, j), (z ′ , j ′ ) ∈ Zk × {1, ..., N },
pN ((z, j), (z ′ , j ′ )) = p((z, j), (z ′ , j ′ ))
X
X
p((z, j), (z1 , j1 ))p((z1 , j1 ), (z2 , j2 ))...p((zk , jk )(z, j ′ )).
+
k≥1 (z1 ,j1 ),...,(zk ,jk )
j1,...,jk >N

Denote by G the Green’s function associated to p and by GN the Green’s function
associated to the induced chain pN . Then, we have the following lemma.
Lemma 4.7. The restriction to Zk × {1, ..., N } of the Green’s function G coincides
with the Green’s function GN .
Proof. Let (z, j), (z ′ , j ′ ) ∈ Zk × {1, ..., N }. By definition,
X (n)
GN ((z, j), (z ′ , j ′ )) =
pN ((z, j), (z ′ , j ′ ))
=

X

X

n≥0

pN ((z, j), (z1 , j1 ))pN ((z1 , j1 ), (z2 , j2 )) · · · pN ((zn , jn ), (z ′ , j ′ ))

n≥0 (z1 ,j1 ),...,(zn ,jn )
ji ≤N
(0)

+ pN ((z, j), (z ′ , j ′ )).
We use the notation γi = (zi , ji ). Since pN is the transition kernel of the first
return to Zk × {1, ..., N },
X
X
(m−1)
(2)
(1)
(1)
, γi+1 ),
pN (γi , γi+1 ) =
p(γi , γi )p(γi , γi ) · · · p(γi
m≥0 γ (1) ,...,γ (m−1)
i

i

(l)

ji >N
(l)

(l)

where ji is the projection of γi ∈ Zk × N on the N factor.
Reorganizing the first sum, one gets
X X
p(γ, γ1 )p(γ1 , γ2 ) · · · p(γn−1 , γ ′ ) = G(γ, γ ′ ).
GN ((z, j), (z ′ , j ′ )) =
n≥0 γ1 ,...,γn−1
γi ∈Zk ×N

In other words, every trajectory from (z, j) to (z ′ , j ′ ) for the initial chain p
defines a trajectory from (z, j) to (z ′ , j ′ ) for pN , excluding every point of the path
that is not in Z × {1, ..., N } and every trajectory for pN is obtained in such a way.
Summing over all trajectories, the two Green’s functions coincide.

Let M ≥ 0. If p is a chain on Zk ×{1, ..., N }, say that p has exponential moments
up to M if for every j, j ′ ∈ {1, ..., N },
X
pj,j ′ (0, z)eMkzk < +∞.
z∈Zk

We also have the following proposition.

14

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

Proposition 4.8. Let p be a Zk -invariant, finitely supported, strongly irreducible
transition kernel on Zk × N. Then, there exist N0 ≥ 0 and M ≥ 0 such that
whenever N ≥ N0 and the chain pN has exponential moments up to M , pN satisfies
Assumption 1.
Proof. We will prove that there exist u0 ∈ Rk and M ≥ 0, independent of large
enough N , such that if the chain pN has exponential moments up to M , then
(3)

{u ∈ Rk , λ(u) ≤ 2} ⊂ {u ∈ Rk , kuk ≤ u0 } ⊂ F .

Denote by (e1 , ..., ek ) the canonical basis in Rk and fix j0 ∈ {1, ..., N }. Since
the chain p is strongly irreducible, there exists ni such that for every n ≥ ni ,
p(n) ((0, 1), (ei , 1)) > 0, so that there is a path of length n from (0, 1) to (ei , 1). This
path stays in Zk × {1, ..., Ni } for some Ni , since the chain p is finitely supported.
(n)
Thus, for every N ≥ Ni , the restricted chain pNi satisfies pNi ((0, 1), (ei , 1)) > 0, if
n ≥ ni . Thus, there exist a > 0, N0 and n0 ≥ 0 such that for every i ∈ {1, ..., k},
(n)
pN (0, ei ) ≥ a for N ≥ N0 , n ≥ n0 and a does not depend on N0 .
Now, let u ∈ Rd and fix L ≥ 0. There exists u0 such that if kuk ≥ u0 , then
at least one of the eu·ei will be larger than La , so that Fj0 ,j0 (u)n0 ≥ L. Since a
does not depend on N0 , u0 does not depend on N0 . Moreover, if p has exponential
moments up to u0 +1, then F (u) has finite entries for u0 ≤ kuk ≤ u0 +1 and so does
F (u)n0 . Let v(u) be an eingenvector associated to λ(u). Then, it is an eigenvector
of F (u)n0 associated to λ(u)n0 . Since F (u) is strongly irreducible, v(u) has nonzero coordinates and we can even choose v(u) with strictly positive coordinates.
Denote by v(u)(j0 ) its j0 th coordinate. Then, v(u)(j0 )λ(u)n0 ≥ Fj0 ,j0 (u)n0 v(u)(j0 )
so that λ(u)n0 ≥ Fj0 ,j0 (u)n0 ≥ L.
Consequently, λ(u)n0 , hence λ(u), can be made arbitrarily large, when enlarging
kuk. Moreover, if p has sufficiently large exponential moments, then λ(u) is well
defined for arbitrarily large kuk. Equation (3) follows from these two facts. This
proves that the sub-level λ(u) ≤ 1, is bounded, thus compact and contained in the
open set λ(u) < 2, which is included in F . Thus, Assumption 1 is satisfied.

Remark 4.1. It will be important in further applications that the M in the statement of this proposition is independent of N . Indeed, in the next section we will
be in a similar situation and we will prove that for every K, for large enough N ,
the chain pN has exponential moments up to K. We will then choose N1 so that if
N ≥ N1 , the chain pN has exponential up to M , so that for N ≥ N0 , N1 , the chain
pN satisfies Assumption 1.
We will also use the following.
Lemma 4.9. Let p be a Zk -invariant, strongly irreducible chain on Zk × {1, ..., N }.
If p is (strictly) sub-Markov, then it satisfies Assumption 2.
Proof. The fact that the chain is strictly sub-Markov means that the matrix F (0)
defined above is strictly sub-stochastic. In particular, its dominant eigenvalue λ(0)
satisfies λ(0) < 1 and the minimum of λ is strictly less than 1, so Assumption 2 is
satisfied.

Combining the explicit formula given in Proposition 4.6 together with Proposition 4.8 and Lemma 4.9, we obtain the following corollary which describes convergence in the Martin boundary for a chain on Zk × N. It will be important in the

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

15

proof of Theorem 1.1 in the next section. Indeed, we will identify the random walk
on Γ with a Zk -invariant Markov chain on Zk × N. To identify the Martin boundary
of the induced chain pN , we will verify the conditions of this corollary.
Corollary 4.10. Let p be a Zk -invariant, finitely supported, strongly irreducible
transition kernel on Zk × N such that:
a)For large enough N , the induced chain pN on Zk × {1, ..., N } is strictly subMarkov.
b)For all M there exists an N0 > 0 such that for N > N0 , the chain pN has
exponential moments up to M .
Then, a sequence (zn , jn ) in Zk × N, with sup(jn ) < +∞, converges to a point
in the Martin boundary of p if and only if kzn k tends to infinity and kzznn k converges
to a point of Sk−1 .
Thus, the fact that (zn , jn ) converges to a point in the Martin boundary does
not depend on the sequence (jn ). In particular, (zn , jn ) converges in the Martin
boundary if and only if its projection (zn , 0) on Zk × {0} converges in the Martin
boundary and the limits are the same.

5. Convergence of Martin kernels: proof of Theorem 1.1
Let Γ be hyperbolic relative to a collection Ω of infinite virtually abelian subgroups. Let µ be a measure on Γ whose finite support generates Γ as a semigroup.
In this section we prove that the Martin boundary is a Z-boundary, proving Theorem 1.1. Recall that ∂P denotes the geometric boundary of a maximal parabolic
subgroup P defined in Section 3.3. We fix a finite set Ω0 of representatives of conjugacy classes of Ω. We will deal separately with sequences converging to conical
limit points and sequences converging in g∂P for some coset gP of a parabolic
subgroup P . For the second case, we will apply results of Section 4.2. It will be
more convenient to deal with strongly irreducible chains. Thus, we first show that
we can reduce to such chains.
In a very general context, consider a chain p on a countable space E. Define the
modified chain p̃ on E by
p̃(x1 , x2 ) =

1
1
∆(x1 , x2 ) + p(x1 , x2 ),
2
2

where ∆(x1 , x2 ) = 0 if x1 6= x2 and 1 otherwise. Denote by p̃(n) the nth convolution
power of p̃. Also denote by G̃ the associated Green’s function:
X
G̃(x1 , x2 ) =
p̃(n) (x1 , x2 ).
n≥0

We have the following (see [7, Lemma 3.20]).
Lemma 5.1. With these notations,
kernels are the same.

1
2 G̃(x1 , x2 )

= G(x1 , x2 ) and thus the Martin

In our context, this means we can assume that µ(e) > 0. Now, since the random
walk is irreducible, if it satisfies µ(e) > 0, it is strongly irreducible.

16

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

5.1. Convergence to conical limit points. We first study conical limit points.
We prove the following.
Proposition 5.2. Consider a sequence (gn ) of Γ that converges to a conical point α
in the Bowditch boundary. Then, (gn ) converges to a point in the Martin boundary.
This is a consequence of the following results of [9], which were recalled in the
introduction.
Proposition 5.3. [9, Theorem 6.3] The identity map Γ → Γ extends to a continuous equivariant surjection F from the Martin compactification to the Bowditch
compactification.
Proposition 5.4. [9, Corollary 6.13] The preimage F −1 (α) of a conical limit point
α consists of a single point.
In the case of a free product, the analog of conical limit points are infinite words.
In [7], the author also proves Proposition 5.2 in this context but the proof is simpler
and slightly differs from the proofs in [9]. In both cases, the key point is that the
random walk tracks relative geodesics when it converges to conical limit points.
This idea is encoded in relative Ancona inequalities in [9] (see Theorem 4.3) and in
the use of transitional sets in [7].
5.2. Convergence in parabolic subgroups. In this section, we prove the convergence of the Martin kernels Kgn (·) when (gn ) converges in the boundary of a
coset of a parabolic subgroup. We begin by fixing notations. Recall that we fixed
a finite set Ω0 of conjugacy classes of parabolic subgroups such that every maximal
parabolic subgroup is conjugated to one in Ω0 . Fix a parabolic subgroup P ∈ Ω0
and assume that (gn ) is such that the closest projection πP gn of gn in P converges
to a point in ∂P . We first recall the definition of the geometric boundary of P from
Section 3.3.
Parabolic subgroups are assumed to be finitely generated virtually abelian groups.
Denote by k the rank of P , that is, assume that P contains a subgroup isomorphic
to Zk with finite index. Any section P/Zk → P provides an identification between
P and Zk × {1, ..., N }.
Identify then gn with (zn , jn ) ∈ Zk × {1, ..., N }. By definition, the sequence
(gn ) converges to a point in the boundary ∂P of P if zn tends to infinity and kzznn k
converges to some point in Sk−1 . Denote the corresponding point in Sk−1 by θ and
say that (gn ) converges to θ.
Proposition 5.5. If (gn ) is a sequence such that (πP gn ) converges to a point of
∂P , then (gn ) and (πP gn ) both converge in the Martin boundary to the same point.
Since a sequence (gn ) converges in g∂P if and only if (g −1 gn ) converges in ∂P
and since (gn ) converges in the Martin boundary if and only if (g −1 gn ) converges
in the Martin boundary, the same result will hold for any coset gP of P .
To prove this proposition, we will adapt the arguments given in [7] to our situation. We will follow the same strategy as in Section 5 of [7]. Namely, we will first
prove the proposition assuming (gn ) stays in a fixed neighborhood of P . Finally,
we will use the relative Ancona inequalities to reduce to the case where (gn ) does
stay in such a neighborhood.

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

17

Proposition 5.6. Suppose (gn ) is a sequence with supn d(gn , P ) < ∞. Then (gn )
converges to a point in the Martin boundary ∂µ Γ if and only if (πP gn ) converges to
a point of ∂P and in that case, the limit of (gn ) and (πP gn ) in the Martin boundary
are the same.
Let η ≥ 0. We denote the η-neighborhood of P by Nη (P ). To prove this
proposition, we introduce the chain p corresponding to the first return to Nη (P ),
defined as in Section 2, that is, for γ1 , γ2 ∈ Nη (P ) denote by p(γ1 , γ2 ) the probability
that the µ-random walk starting at γ1 returns to Nη (P ), and first does so at γ2 . In
other words p(γ1 , γ2 ) = G(γ1 , γ2 ; Nηc (P )). We will see that the probability that the
µ-random walk starting at γ1 never goes back to Nη (P ) is positive (see Lemma 5.9).
Thus, p is not a probability transition kernel and defines a sub-Markov chain on
Nη (P ). Nevertheless, one can still define the Green’s function associated to p as
X
Gp (γ1 , γ2 ) =
p(n) (γ1 , γ2 ), γ1 , γ2 ∈ Nη (P ),
n≥0

where p(n) is the nth power of convolution of p.
Lemma 5.7. The Green’s function Gp coincides with the restriction to Nη (P ) of
the Green’s function Gµ associated to the initial random walk.
Proof. This is given by Lemma 4.7. Recall that the proof is based on the following.
Every trajectory from γ to γ ′ for the random walk defines a trajectory from γ to γ ′
for p, excluding every point of the path that is not in Nη (P ) and every trajectory
for p is obtained in such a way. Summing over all trajectories, the two Green’s
functions coincide.

We also have the following property.
Lemma 5.8. The chain p is strongly irreducible.
Proof. The proof is based on the same idea as the proof of Lemma 5.7. First,
the initial random walk is irreducible. Now, every trajectory for p comes from a
trajectory for the random walk on the whole group, after excluding points that do
not stay in the neighborhood of P . Thus, there is a positive proportion (for p) of
paths from any point γ ∈ Nη (P ) to any other point γ ′ ∈ Nη (P ). This proves that
p is irreducible. Now, recall that we assumed that µ(e) > 0 (see Lemma 5.1), so
that p(γ, γ) > 0 and thus p is strongly irreducible.

In light of Lemma 5.7, to prove Proposition 5.6, it suffices to show that a sequence
satisfying its conditions converges to a point in the Martin boundary of Nη (P ) with
the induced process p (for η large enough).
We first notice that, as a set, Γ can be identified P -equivariantly with P × N.
Indeed, P acts by left multiplication on Γ and the quotient is countable. We
order elements in the quotient according to their distance to P . It follows that
the η-neighborhood Nη (P ) can be P -equivariantly identified with P × {1, ..., N }.
Moreover, if η ′ ≤ η, the set P × {1, ..., N ′ } identified with Nη′ (P ) is a subset of
P × {1, ..., N } identified with Nη (P ).
Now, identifying P with Zk × F , where F is finite, the group Γ can be identified
with Zk × N. Thus, the µ-random walk can be considered as a Zk -invariant Markov
chain q on Zk × N and the restriction of the random walk to Nη (P ) coincides with
the restriction of the chain q to Zk × {1, ..., N }.

18

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

Let (γn ) be a sequence in Nη (P ) and identify γn with (zn , jn ) ∈ Zk × {1, ..., N }.
Notice that the projection of (γn ) to P converges in the geometric boundary ∂P of
P if and only if ((zn , jn )) converges in the geometric boundary of Zk × {1, ..., N },
since in both cases, the sequence converges in the geometric boundary if and only
if (zn ) tends to infinity and kzznn k converges to a point in the sphere.
To prove Proposition 5.6, it suffices to show that the Markov chain q on Zk × N
and its induced chain p on Zk × {1, .., N } satisfy the conditions of Corollary 4.10.
Thus, we just need to show that for large enough η, the induced chain on Nη (P )
has sufficiently large exponential moments and is strictly sub-Markov.
Lemma 5.9. The induced chain p is strictly sub-Markov.
Proof. It suffices to show that there exists γ ∈ Nη (P ) such that
X
p(γ, γ ′ ) < 1.
γ ′ ∈Nη (P )

This follows from the fact that the µ-random walk starting at γ with d(γ, P ) = η
has a positive probability of never returning to Nη (P ). This, in turn, follows from
the fact that the random walk almost surely converges to a conical point (see for
example [9, Theorem 9.8, Theorem 9.14]).

For M ≥ 0, recall that p is said to have exponential moments up to M if for
every j, j ′ ∈ {1, ..., N },
X
pj,j ′ (0, z)eMkzk < +∞.
z∈Zk

Proposition 5.10. Let M ≥ 0. For large enough η, p has exponential moments
up to M .
The proof of Proposition 5.10 will be divided into several steps. We will use the
following two results which are related to hyperbolic properties of the geometry
of the Cayley graph CS Γ. Every maximal parabolic subgroup P of a relatively
hyperbolic group is quasiconvex (see for example [11, Corollary 3.9]) so we have the
following.
Lemma 5.11. [4, Lemma 5.1] There exists a constant c0 such that the following
holds. If g1 , g2 ∈ Γ and if gi′ ∈ projP gi for i = 1, 2 then,
d(g1′ , g2′ ) ≤ d(g1 , g2 ) + c0 .
Lemma 5.12. There exists an a0 > 0 such that the function ρ : R+ → R+ defined
by
ρ(η) = inf{d(g1 , g2 ) : d(πP (g1 ), πP (g2 )) ≥ a0 , d(gi , P ) > η}
tends to infinity as η → ∞.
Proof. By Proposition 8.5 of [12], there are constants a0 , D > 0 (independent of
the parabolic subgroup P ) such that if α is a geodesic with d(α, P ) ≥ D, then
diam(projP (α)) < a0 .
Now, consider g1 , g2 ∈ Γ with d(g1 , P ) > η and d(g1 , g2 ) ≤ η − D. Let α be a
geodesic connecting g1 and g2 . By the triangle inequality we have d(α, P ) ≥ D and
so
diam(projP (α)) < a0 .

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

19

In particular d(πP (g1 ), πP (g1 )) < a0 . Thus, the conditions d(g1 , P ) > η and
d(πP (g1 ), πP (g1 )) ≥ a0 imply that d(g1 , g2 ) > η − D, completing the proof, since η
tends to infinity.

The following classical lemma is due to Kesten (see [15]). It holds as soon as Γ
is nonamenable.
Lemma 5.13 (Kesten). Denote by µ∗n the nth power of convolution of the measure
µ. There exists α > 0 such that for every g ∈ Γ,
µ∗n (g) ≤ e−αn .
We can now prove Proposition 5.10. Let z ∈ Zk and j, j ′ ∈ {1, ..., N }. If the first
return to Nη (P ) starting at (0, j) is at (z, j ′ ), there is a path Z0 , ..., Zn+1 such that
Z0 = (0, j), Zn+1 = (z, j ′ ) and Zl ∈
/ Nη (P ) for 1 ≤ l ≤ n. Since d(Zl , Zl+1 ) ≤ r(µ),
where r(µ) only depends on µ, if η ≥ 3r(µ), then Z0 , Zn+1 ∈
/ N2η/3 (P ) as soon as
n ≥ 1, which will hold if kzk is large enough. Moreover, any geodesic from Zl to
Zl+1 stays outside of Nη/3 (P ), for 0 ≤ l ≤ n.
Define a path φ from Z0 to Zn+1 by gluing together geodesics from Zl to Zl+1 .
Then, the length of φ is smaller or equal to nr(µ). The parabolic subgroup P
together with the word distance is quasi-isometric to its subgroup Zk together with
the euclidean distance. In particular, the word distance between 0 and z is larger
than Λkzk, where Λ only depends on the quasi-isometry parameters.
Using Lemma 5.11, we can choose consecutive points y1 , ..., yl on φ which project
on P on points ỹ1 , ..., ỹl such that the distance between ỹi and ỹi+1 is between a0
and 2a0 , for 1 ≤ i ≤ l − 1 and such that l ≥ Λkzk
2a0 . (where a0 is the constant in
Lemma 5.12). By gluing together a path from 0 to ỹ1 , paths from ỹi to ỹi+1 and
a path fom ỹl to z, we get a path from 0 to z inside P whose length is thus larger
than Λkzk. We deduce from Lemma 5.12 that d(yj , yj+1 ) ≥ ρ(η/3), so that the
length of φ is larger or equal to 2aΛ0 ρ(η/3)kzk, where ρ(η/3) tends to infinity when
η tends to infinity.
Fix R0 ≥ 0. Then, for large enough η, n ≥ R0 kzk. Summing over all paths from
γ = (0, j) to γz = (z, j ′ ) that stay outside Nη (P ), we have
X
µ∗n (γ −1 γz ).
pj,j ′ (0, z) ≤
n≥R0 kzk

Lemma 5.13 shows that
pj,j ′ (0, z) ≤

X

n≥R0 kzk

e−αn ≤ e−αR0 kzk

X

e−αn .

n≥0

To prove Proposition 5.10, it suffices to choose R0 so that R0 α > M .



We can now complete the proof of Proposition 5.6, using Corollary 4.10. Indeed,
Lemma 5.8 shows that the induced chain on (arbitrary) bounded neighborhoods
of P is strongly irreducible, while Lemma 5.9 shows that it is strictly sub-Markov
(Condition a) of Corollary 4.10) and Proposition 5.10 shows that it has sufficiently
high exponential moment (Condition b) of Corollary 4.10).
Thus, Corollary 4.10 implies that the Martin compactification of the induced
chain on bounded neighborhoods of P coincides with the geometric compactification
of P , and together with Lemma 5.7 this implies Proposition 5.6.


20

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

To prove Proposition 5.5, we now show that we can reduce to the case of a
sequence that stays in a uniform neighborhood of the parabolic subgroup P . The
proof is based on the following strategy. Assume that the sequence (gn ) leaves every
bounded neighborhood of P (but its projections to P still converge to a point θ ∈
∂P ). Proposition 5.6 applied to (πP gn ) guarantees that (πP gn ) converges to a point
α in the Martin boundary. We want to prove that the same is true for gn . In other
words, we want to prove that Kgn converges pointwise to a function Kα . Relative
Ancona inequalities show that to go from a basepoint o or from an arbitraty point g
to gn , the random walk visits πP (gn ) with high probability. Thus, G(g, gn ) is close
to G(g, πP (gn ))G(πP (gn ), gn ) and G(o, gn ) is close to G(o, πP (gn ))G(πP (gn ), gn ),
so that Kgn (g) is close to KπP (gn ) (g). Convergence for Kgn (g) then follows from
convergence for KπP (gn ) (g).
We now give a formal proof. Let zn = πP gn be a projection point of gn to
P . By Lemma 8.2 of [9], there is a (uniform, depending only on Γ) δ > 0 with
lim inf n→∞ δzfn (x, gn ) > δ for all x ∈ Γ. Let ǫ > 0. Consider any x ∈ Γ. By
Theorem 4.2, there is a η > 0 such that for large enough n,
G(x, gn ; Bηc (zn )) ≤ ǫG(x, gn )

(4)
and

G(o, gn ; Bηc (zn )) ≤ ǫG(o, gn ).

(5)

Assume η is also large enough to satisfy Proposition 5.10. Decomposing a path
from o to gn according to its last visit to Bη (zn ), we can write
X
G(o, un )G(un , gn ; Bηc (zn )) + G(o, gn ; Bηc (zn ))
(6)
G(o, gn ) =
un ∈Bη (zn )

and similarly,
(7)

G(x, gn ) =

X

G(x, un )G(un , gn ; Bηc (zn )) + G(x, gn ; Bηc (zn ))

un ∈Bη (zn )

By Proposition 5.6 we know that for any un ∈ B(zn , η), G(x, un )/G(o, un ) converges to some Kα (x) where α ∈ ∂µ Γ is independent of (un ) and so, for large enough
n, we have
(8)

(1 − ǫ)Kα (x) ≤

G(x, un )
≤ (1 + ǫ)Kα (x).
G(o, un )

Combining (4), (7) and (8), we obtain for all large n that
X
(1 + ǫ)Kα (x)G(o, un )G(un , gn ; Bηc (zn )) + ǫG(x, gn ),
G(x, gn ) ≤
un ∈Bη (zn )

so that
(1 − ǫ)G(x, gn ) ≤ (1 + ǫ)Kα (x)

X

G(o, un )G(un , gn ; Bηc (zn ))

un ∈Bη (zn )

and then, using (6), (1 − ǫ)G(x, gn ) ≤ (1 + ǫ)Kα (x)G(o, gn ). Similarly, using (5),
(6), (7) and (8), we get a lower bound, so that for large enough n,
1−ǫ
G(x, gn )
1+ǫ
Kα (x) ≤
≤
Kα (x).
1+ǫ
G(o, gn )
1−ǫ

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

21

Since ǫ > 0 is arbitrary we get that (gn ) converges to α in the Martin boundary,
completing the proof of Proposition 5.5.

We have shown that if (gn ) is a sequence in Γ such that either:
(1) (gn ) converges to a conical point of the Bowditch boundary, or
(2) For some maximal parabolic subgroup P , the projections (πP gn ) converge
to a point of ∂P ,
then (gn ) converges in the Martin compactification. As explained above, the same
holds if, for some coset gP of a parabolic group P , (πgP gn ) converges in g∂P .
To complete the proof that the Martin boundary is a Z-boundary we need to
show the converse: namely that if (gn ) converges to a point in the Martin boundary,
then it satisfies either (1) or (2).
Suppose (gn ) converges to a point in the Martin boundary. By Proposition 5.3,
(gn ) converges to a point α in the Bowditch boundary. If α is conical, then (1)
holds. Suppose now that α is parabolic, with stabilizer P . If (2) is not satisfied,
there are subsequences (hn ) and (h′n ) of (gn ) with (πP hn ) and (πP h′n ) converging to
different points of ∂P . By Proposition 5.6, (πP hn ) and (πP h′n ) converge to different
points α and α′ in the Martin boundary. Furthermore, by Proposition 5.5, (hn )
converges to the same point in the Martin boundary as (πP hn ) and (h′n ) converges
to the same point in the Martin boundary as (πP h′n ). Thus (hn ) and (h′n ) converge
to different points of the Martin boundary, contradicting our assumption on (gn ).
This proves that the Martin boundary is a Z-bondary, ending the proof of Theorem 1.1. Indeed, as stated above, the authors of [9] already proved that the identity
of Γ extends to an equivariant surjective map from the Martin compactification to
the Bowditch compactification.

Using Theorem 1.1 and [9, Theorem 1.3], we recover the following corollary,
which is interesting independently of the random walks context. However, it could
also be deduced from previous results of Gerasimov and Gerasimov-Potyagailo.
Corollary 5.14. Let Γ be a finitely generated group, hyperbolic relative to a collection of infinite virtually abelian subgroups. Then, there exists an equivariant and
continuous surjective map from any Z-boundary to the Floyd boundary of Γ.
Since the Martin boundary does not depend on different peripheral structures,
but only on the random walk, our argument also implies the following.
Corollary 5.15. If Γ is hyperbolic relative to two different collections of infinite
virtually abelian subgroups, then two corresponding Z-boundaries, constructed for
each relatively hyperbolic structure, are equivariantly homeomorphic,
We conclude the discussion with few questions related to the above argument.
Since Z-boundaries can be defined for more general relatively hyperbolic groups, it
seems reasonable to ask the following question.
1. Is Corollary 5.14 true for more general relatively hyperbolic groups? We
conjecture that there are possible counter-examples.
The following question, motivated by the proof of Proposition 5.5, also seems to
be interesting.
2. For which classes of groups P does the following hold ? Assume that Γ is
hyperbolic relative to P and that (xn ) is a sequence of elements of Γ leaving every

22

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

compact. Then, if yn is the projection of xn onto P (or actually its projection onto
any horosphere based at P , see [12] for the precise definition of a horosphere in this
n ) G(x,yn )
context), one has that G(x,x
G(o,xn ) G(o,yn ) converges to 1.
6. Minimality
In this section we prove Theorem 1.2 from the introduction, namely the minimality of the Martin boundary. We will use the following result of Dussaule [7,
Proposition 6.3].
Proposition 6.1. Let p be a strongly irreducible transition kernel on Zk ×{1, ..., N }
which is Zk -invariant and satisfies Assumptions 1 and 2 of section 4.2. Let θ0 6= θ1
be two points in the Martin boundary ∂Zk . There exists a neighborhood U of θ1
in ∂Zd and a sequence (xn , kn ) of Zd × {1, ..., N } such that for every θ in U,
Kθ ((xn , kn )) tends to infinity, uniformly in θ and Kθ0 ((xn , kn )) converges to 0.
Applied to our situation, this gives the following.
Corollary 6.2. There exists an η0 > 0 such that for η > η0 the following holds.
For any distinct α0 , α1 ∈ ∂P there exists a neighborhood U of α1 in ∂P and and
a sequence (gn ) of Nη P such that Kα (gn ) tends to infinity, uniformly over α ∈ U
and Kα0 (gn ) converges to 0.
We now prove the following.
Theorem 6.3. Let Γ be hyperbolic relative to a collection of virtually abelian subgroups. Let µ be a probability measure on Γ whose finite support generates Γ as a
semigroup. Then every point of the Martin boundary ∂µ Γ corresponds to a minimal
harmonic function.
Proof. By Theorem 1.1, ∂µ Γ is a Z-boundary. This means that there is a Γequivariant surjective map F : ∂µ Γ → ∂B Γ such that if a ∈ ∂B Γ is conical, F −1 (a)
is a single point and if a ∈ ∂B Γ is parabolic, F −1 (a) = ∂P where P is the stabilizer
of a and ∂P denotes its CAT (0) boundary. Notice that F : ∂µ Γ → ∂B Γ is the same
map as the the map ψ : ∂M Γ → ∂B Γ constructed in [9] (see the discussion after
Theorem 1.3 there).
Let α0 ∈ ∂µ Γ. Then Kα0 is a positive harmonic function. By the Choquet
representation theorem, there exists a finite Borel measure ν0 = ν α0 on the Martin
boundary, with support contained in the minimal Martin boundary ∂µm Γ such that
for all g ∈ Γ
Z
Kα0 (g) =

Kα (g)dν0 (α).

mΓ
∂µ

To prove minimality of α0 it suffices to show that the support of ν0 consists of
the single point α0 . In Corollary 7.9 of [9] the authors prove the following result,
which is based on relative Ancona inequalities (see Theorem 4.2).
Lemma 6.4. The support of Kα0 is contained in F −1 (F (α0 )).
If F (α0 ) is conical, then F −1 (F (α0 )) is a single point. Lemma 6.4 then implies
that the support of ν0 is a single point so that α0 is minimal.
On the other hand, if α0 is a parabolic point of the Bowditch boundary with
stabilizer P , Theorem 1.1 implies that F −1 (F (α0 )) = ∂P . Thus we know ν0 is
supported on ∂P ∩ ∂µm Γ.

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

23

Now, suppose α1 is a point of ∂P distinct from α0 . By Corollary 6.2, there
exists a neighborhood U of α1 and a sequence (gn ) such that Kα (gn ) tends to
infinity uniformly over α ∈ U and Kα0 (gn ) converges to 0. Thus, for large enough
n and for all α ∈ U , we have Kα (gn ) ≥ 1. Then by definition,
Z
Z
Kα (gn )dν0 (α) ≥ ν0 (U).
Kα (gn )dν0 (α) ≥
Kα0 (gn ) =
α∈∂P

α∈U

As Kα0 (gn ) → 0 as n → ∞ it follows that ν0 (U) = 0 so that the support of ν0
does not contain α. We conclude that the support of ν0 consists only of α0 , so Kα0
must be minimal.

Appendix A. Construction of a Z-boundary
We now use results of Dahmani in [5] to show that if Γ is hyperbolic relative to
a collection of virtually abelian subgroups, there is a geometric construction of a
Z-boundary. His construction uses Farb’s definition of relatively hyperbolic groups
and we first give a brief outline of it.
A.1. Farb’s definition of relative hyperbolicity. Let Γ be a finitely generated
group and Ω a family of finitely generated subgroups of Γ. If CS Γ is a Cayley
graph of Γ containing the Cayley graphs of every subgroup in Ω, define the conedoff Cayley graph by adding one vertex v(gP ) for every coset of the form gP , with
g ∈ Γ and P ∈ Ω and adding one edge e(g, γ) between v(gP ) and gγ, for P in Ω and
γ in P . Following Farb in [8], the group Γ is said to be weakly hyperbolic relative
to the collection Ω if the coned-off Cayley graph defined above is hyperbolic (in the
sense of Gromov). This definition is invariant under quasi-isometry and thus does
not depend on the choice of the Cayley graph of Γ. In the following, for simplicity,
we will assume that there is only one conjugacy class in Ω, and in that case, if
P ∈ Ω, then Γ is said to be hyperbolic relative to P . We fix a Cayley graph CS Γ
containing the Cayley graph of P . The coned-off Cayley graph is quasi-isometric to
the quotient space obtained by identifying every two elements lying in the same left
coset of P . Denote by Γ̂ this quotient space which is thus hyperbolic and denote
by ∂ Γ̂ its Gromov boundary. The following is a reformulation by Dahmani in [5] of
the bounded coset penetration property of Farb in [8].
Definition A.1. Let α : [a, b] → CS Γ be a path parametrized by arc length and let
α̂ be its image in Γ̂. Re-parameter α̂ to remove all loops of length 1 (that is remove
subpaths that lie in a same coset), so that it is still parametrized by arc length.
Say that α is a relative geodesic if this new parametrization of α̂ is a geodesic.
Say that it is a (λ, c)-relative-quasi-geodesic if the new parametrization of α̂ is a
(λ, c)-relative-quasi-geodesic, that is,
1
|t − s| − c ≤ d(α̂(t), α̂(s)) ≤ λ|t − s| + c.
λ
Definition A.2. The pair (Γ, P ) satisfies the bounded coset penetration property
(BCP for short) if for all λ, c, there exists a constant r such that for every pair
(α1 , α2 ) of (λ, c)-relative-quasi-geodesic without loop, starting at the same point
and ending at the same point in CS Γ, the following holds
(1) if α1 travels more than r in a coset, then α2 enters this coset,
(2) if α1 and α2 enter the same coset, the two entering points and the two
exiting points are r-close to each other.

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M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

Say that Γ is hyperbolic relative to P if it is weakly hyperbolic relative to P and
the pair (Γ, P ) satisfies the BCP property. Similarly, one can define hyperbolicity
relative to a family of subgroups Ω. This definition of relative hyperbolicity is
equivalent to those we gave in Section 3 (see for example Appendix A in [6], where
Dahmani proves it is equivalent to the definition of Bowditch in [4]).
A.2. Dahmani’s geometric boundary. In [5], the author introduces a compactification for general relatively hyperbolic groups. This construction is a Z-boundary
in our situation. For simplicity, we again assume that there is only one conjugacy class in Ω, that is Γ is hyperbolic relative to a parabolic group P , but the
construction holds for more than one conjugacy class (see Section 6.2 in [5]).
We will need the following definition.
Definition A.3. Consider a path α in the Cayley graph CS Γ and denote by g0 the
starting point of α. The path α is called left-reduced if it immediately leaves the
coset g0 P and never returns to it.
Definition A.4. Assume that P ∪ ∂P is a compactification of P . Say that finite
sets fade at infinity if for all finite subset F of P and for all open cover U of P ∪∂P ,
all translates of F but finitely many are contained in an element of U.
Also recall that ∂ Γ̂ is the Gromov boundary of the quotient space Γ̂ obtained by
identifying every coset of P to one point. Now, choose a system of representatives
g of Γ/P and define the boundary ∂Γ of Γ as
Γ/P


G
∂Γ := 
γ̃∂P  ⊔ ∂ Γ̂.
g
γ̃∈Γ/P

Dahmani [5, Theorem 3.1] proved the following.
Theorem A.5. Assume that finite sets fade at infinity in P ∪ ∂P . Then, the
discrete topology on Γ extends to a topology on Γ ∪ ∂Γ which makes it a metrizable
compact space. With this topology, Γ is dense in Γ ∪ ∂Γ. Moreover, we have the
following characterization of convergence to the boundary. Let (γn ) be a sequence
in Γ and ξ ∈ ∂Γ.
• If ξ ∈ ∂ Γ̂, then (γn ) converges to ξ if and only if its image (γ̂n ) in Γ̂
converges to ξ (in the Gromov sense).
g , then (γn ) converges to ξ if and only if there
• If ξ ∈ γ̃∂P , with γ̃ ∈ Γ/P
is a sequence (xn ) of points in P and a sequence of left-reduced relativequasi-geodesics (with bounded parameters) from γ̃xn to γn , such that xn
converges to a point in ∂P .
We can rephrase convergence in γ̃∂P as follows. Assume that γ ∈ Γ, x ∈ P ,
and there is a left-reduced relative-quasi-geodesic (with bounded parameters) from
γ̃x to γ. According to the BCP property, if α is another relative-quasi-geodesic
with bounded parameters from γ̃P to γ, then the exiting point of α is a uniformly
bounded distance away from γ̃x. In particular, if πP γ is a closest point projection
of γ on P , then πP γ is a bounded distance away from γ̃x. Thus, a sequence (γn )
converges to a point ξ ∈ γ̃∂P if and only if the sequence of closest point projections
of γn on γ̃P converges to ξ ∈ γ̃∂P .

MARTIN BOUNDARY OF RELATIVELY HYPERBOLIC GROUPS

25

Now, recall that in our context, the parabolic subgroup P is virtually abelian.
We have already defined a compactification of P . Identifying P with Zk ×{1, ..., N },
as in Section 3.3, and identifying γ ∈ P with (z, j) ∈ Zk × {1, ..., N }, a sequence
(zn , jn ) converges in ∂P if zn tends to infinity and kzznn k converges to a point θ in
Sk−1 . Formally, one can choose sets of the form Un,m (θ) × {1, ..., N } to form a
countable system of neighborhoods of a point θ in the boundary, where
z
Un,m (θ) = Vn (θ) ∪ {z ∈ Zk , kzk ≥ m,
∈ Vn (θ)},
kzk
with Vn (θ) a neighborhood of θ in Sk−1 .
Lemma A.6. With this topology, finite sets fade at infinity in P ∪ ∂P .
Proof. First, we notice that Zk ∪ ∂Zk satisfies a fellow-traveler property. Precisely,
if (zn ) and (zn′ ) are two sequences in Zk such that kzn − zn′ k is bounded by a
constant and zn converges to a point θ in ∂Zk , that is kzn k tends to infinity and
zn
k−1
, then zn′ also converges to θ in ∂Zk . By induction,
kzn k converges to θ ∈ S
(1)

(j)

the same holds with a finite number of sequences, that is, if (zn ), ..., (zn ) are
(j )
(j )
sequences in Zk such that kzn 1 − zn 2 k is bounded for every j1 , j2 and such that
one of them converges in ∂Zk , then they all converge to the same point.
Assume by contradiction that F is a finite subset of P and U is an open cover of
P ∪ ∂P such that there are infinitely many translates of F that are not contained
in one of the open sets in U. Denotes these translates by γn · F . Let f ∈ F . The
sequence (γn · f ) tends to infinity. Up to taking a sub-sequence, it converges in
∂P . By the fellow-traveler property described above, for every f ′ ∈ F , (γn · f ′ )
converges to the same point. Eventually, they all lie in a neighborhood of this limit
point, which is a contradiction.

We use this compactification of P to construct ∂Γ as above. The left multiplication action on Γ extends to an action on Γ ∪ ∂Γ in the obvious way. In [5], the
author claims (without a proof) that the compactification Γ ∪ ∂Γ projects on the
Bowditch compactification Γ ∪ ∂B Γ. Indeed, we have the following.
Lemma A.7. With these notations, the identity map on Γ extends to an equivariant, continuous and surjective map
Γ ∪ ∂Γ → Γ ∪ ∂B Γ.
Moreover, a point ξ ∈ Γ̂ is mapped to a conical point in ∂B Γ and a point ξ ∈ γ̃∂P ,
g is mapped to a parabolic point in ∂B Γ.
for some γ̃ ∈ Γ/P

Proof. We first construct a map ∂Γ → ∂B Γ. Consider the space Y which consists
on the space X with disjoint horoballs removed at every parabolic point and endow
Y with the length metric coming from the induced distance of X on Y . The map
Γ → Y given by γ 7→ γ · o (where o is a fixed base point) is a quasi-isometry.
If a sequence (γn ) converges in ∂Γ, then its image in X tends to infinity. If it
converges to ∂ Γ̂, then the Gromov product of γn and γm based at some point in Γ̂
tends to infinity, when n and m tend to infinity. Thus, geodesics in Γ̂ from a fixed
base point to γn and to γm travel together for arbitrary large time. This provides
geodesics in the Cayley graph CS Γ that travel together for arbitrary large time
and spend arbitrary large time outside cosets of peripheral subgroups. The images
in X are quasi-geodesics that travel together for arbitrary large time and spend

26

M. DUSSAULE, I. GEKHTMAN, V. GERASIMOV, AND L. POTYAGAILO

arbitrary large time outside horoballs corresponding to parabolic points. Thus, the
image of (γn ) converges to a conical point. Now, if (γn ) converges to γ̃∂P , then its
projection on γ̃P converges to the same point and stays in the coset γ̃P of P . Thus,
the projection (γn′ ) of the image of (γn ) in X on the corresponding horosphere H
converges to the corresponding parabolic point. Denote this parabolic point by
α. A geodesic from a fixed base point o to γn follows approximately a geodesic
from o to γn′ , then from γn′ to γn (see [16, Proposition 3.4]), so that the Gromov
product of γn and α based at o converges to infinity. Thus, (γn ) also converges to
α. Thus, if γ ∈ ∂Γ, there is a uniquely defined γ̃ ∈ ∂B Γ and we can consider the
map γ ∈ ∂Γ 7→ γ̃ ∈ ∂B Γ.
By construction, if (gn ) ∈ Γ converges to γ, then (gn · o) converges to γ̃ in the
Bowditch boundary. By density of Γ in the Bowditch compactification, this map is
surjective and by compactness of the Bowditch compactification and density of Γ
in Γ ∪ ∂Γ, it is continuous (see the similar proof of Lemma 3.4). By construction,
it is equivariant and continuously extends the identity map Γ → Γ.

Theorem A.5 together with Lemmas A.6 and A.7 prove that ∂Γ is a Z-boundary.
As stated above, Dahmani’s construction still holds when there is more than one
conjugacy class of parabolic subgroups (see [5, Section 6.2]).
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