Fractal Models for Normal Subgroups of Schottky Groups

For a normal subgroup $N$ of the free group $\F_d$ with at least two generators we introduce the radial limit set $\Lr(N,\Phi)$ of $N$ with respect to a graph directed Markov system $\Phi$ associated to $\F_d$. These sets are shown to provide fractal models of radial limit sets of normal subgroups of Kleinian groups of Schottky type. Our main result states that if $\Phi$ is symmetric and linear, then we have that $\dim_{H}(\Lr(N,\Phi))=\dim_{H} \Lr(\F_d,\Phi))$ if and only if the quotient group $\F_d /N$ is amenable, where $\dim_{H}$ denotes the Hausdorff dimension. This extends a result of Brooks for normal subgroups of Kleinian groups to a large class of fractal sets. Moreover, we show that if $\F_d /N$ is non-amenable then $\dim_{H}(\Lr(N,\Phi))>\dim_{H}(\Lr(\F_d,\Phi))/2$, which extends results by Falk and Stratmann and by Roblin.


INTRODUCTION AND STATEMENT OF RESULTS
In this paper we introduce and investigate linear models for the Poincaré series and the radial limit set of normal subgroups of Kleinian groups of Schottky type. Here, a linear model means a linear graph directed Markov system (GDMS) associated to the free group F d = g 1 , . . . , g d on d ≥ 2 generators. Precise definitions are given in Section 2.2, but briefly, such a system Φ is given by the vertex set V := {g 1 , g −1 1 . . . , g d , g −1 d }, edge set E := {(v, w) ∈ V 2 : v = w −1 } and by a family of contracting similarities {φ (v,w) : (v, w) ∈ E} of the Euclidean space R d , for d ≥ 1, such that for each (v, w) ∈ E the contraction ratio of the similarity φ (v,w) is independent of w. We denote this ratio by c Φ (v). Also, we say that Φ is symmetric if c Φ (g) = c Φ g −1 for all g ∈ V . In order to state our first two main results, we must also make two further definitions. For this, we extend c Φ to a function where n ∈ N and (v 1 , . . . , v n ) ∈ V n refers to the unique representation of g as a reduced word. Also, for each subgroup H of F d , we introduce the Poincaré series of H and the exponent of convergence of H with respect to Φ which are defined for s ≥ 0 by Our first main result gives a relation between amenability and the exponent of convergence.
Our second main result gives a lower bound for the exponent of convergence δ (N, Φ).
can be removed from Brooks' Theorem. In fact, it was shown by Sharp in [Sha07, Theorem 2] that if G is a finitely generated Fuchsian groups, that is for m = 1, and if N is a normal subgroup of G, then amenability of G/N implies δ (G) = δ (N). Recently, Stadlbauer [Sta13] showed that the equivalence in (1.1) extends to the class of essentially free Kleinian groups with arbitrary exponent of convergence δ (G).
Finally, let us turn our attention to limit sets of Kleinian groups. For a Kleinian group G, the radial limit set L r (G) and the uniformly radial limit set L ur (G) (see Definition 5.1) are both subsets of the boundary S := z ∈ R m+1 : |z| = 1 of D m+1 . By a theorem of Bishop  We would like to point that there is a close analogy between the results on radial limit sets of Kleinian groups stated in (1.3) and (1.4), and our results in the context of linear GDMSs associated to free groups stated in Proposition 1.3 and Corollary 1.4.
Let us now further clarify the relation between GDMSs associated to free groups and Kleinian groups of Schottky type (see Definition 5.2). For this, recall that a Kleinian group of Schottky type G = g 1 , . . . , g d is isomorphic to a free group. In Definition 5.3 we introduce a canonical GDMS Φ G associated to G. We will then show in Proposition 5.6 that for every non-trivial normal subgroup N of G we have that L r (N) = Λ r (N, Φ G ) and L ur (N) = Λ ur (N, Φ G ).
This shows that our fractal models of radial limit sets of Kleinian groups of Schottky type can be thought of as a replacement of the conformal generators of the Kleinian group by similarity maps.
Our main results show that several important properties of Kleinian groups extend to these fractal models.
Let us now end this introductory section by briefly summarizing the methods used to obtain our results and how this paper is organized. Theorem 1.1 and Theorem 1.2 are based on and extend results of Woess [Woe00] and Ortner and Woess [OW07], which in turn refer back to work of Pólya [Pól21] and Kesten [Kes59b,Kes59a]. Specifically, we provide generalizations of [OW07] for weighted graphs. Our new thermodynamic formalism for group-extended Markov systems (see Section 3) characterizes amenability of discrete groups in terms of topological pressure and the spectral radius of the Perron-Frobenius operator acting on a certain L 2 -space.
The paper is organized as follows. In Section 2 we collect the necessary background on thermodynamic formalism, GDMSs and random walks on graphs. In Section 3, we prove a thermodynamic formalism for group-extended Markov systems, which is also of independent interest. Using the results of Section 3 we prove our main results in Section 4. Finally, in Section 5 we provide the background on Kleinian groups of Schottky type, which has motivated our results.
After having finished this paper, Stadlbauer ([Sta13]) proved a partial extension of Theorem 3.21 (see Remark 3.23). Moreover, in [Jae12] the author has extended Lemma 4.1 and Theorem 1.2 in order to give a short new proof of (1.2) for Kleinian groups.
Acknowledgement. Parts of this paper constitute certain parts of the author's doctoral thesis supervised by Marc Kesseböhmer at the University of Bremen. The author would like to express his deep gratitude to Marc Kesseböhmer and Bernd Stratmann for their support and many fruitful discussions. The author thanks an anonymous referee for the careful reading of the manuscript and for valuable comments on the exposition of this paper. Final thanks go to Sara Munday for helping to improve the presentation of the paper significantly.
Moreover, we equip I N with the product topology of the discrete topology on I and the Markov shift Σ ⊂ I N is equipped with the subspace topology. The latter topology on Σ is the weakest topology on Σ such that for each j ∈ N the canonical projection on the j-th coordinate p j : Σ → I is continuous.
A countable basis for this topology on Σ is given by the cylinder sets {[ω] : ω ∈ Σ * }. We will use the following metric generating the topology on Σ. For α > 0 fixed, we define the metric d α on Σ given by For a function f : Σ → R and n ∈ N 0 , we use the notation S n f : Σ → R to denote the ergodic sum of f with respect to the left-shift map σ , in other words, S n f := ∑ n−1 i=0 f • σ i . Furthermore, the following function spaces will be crucial throughout.
Definition 2.1. We say that a function f : Σ → R is bounded whenever f ∞ := sup ω∈Σ | f (ω)| is finite. We denote by C b (Σ) the real vector space of bounded continuous functions on Σ. We say that f : Σ → R is α-Hölder continuous, for some α > 0, if where for each n ∈ N we let The function f is called Hölder continuous if there exists α > 0 such that f is α-Hölder continuous.
For α > 0 we also introduce the real vector space which we assume to be equipped with the norm · α which is given by We need the following notion of pressure, which was originally introduced in [JKL10, Definition Definition 2.2. For ϕ, ψ : Σ → R with ψ ≥ 0, C ⊂ Σ * and η > 0, the ψ-induced pressure of ϕ (with respect to C ) is given by Remark. It was shown in [JKL10,Theorem 2.4] that the definition of P ψ (ϕ, C ) is in fact independent of the choice of η > 0. For this reason, we do not refer to η > 0 in the definition of the induced pressure.
Notation. If ψ and/or C is left out in the notation of induced pressure, then we tacitly assume that ψ = 1 and/or C = Σ * , that is, we let P(ϕ) := P 1 (ϕ, Σ * ).
The following fact is taken from [JKL10, Remark 2.11, Remark 2.7]. Fact 2.3. Let Σ be a Markov shift over a finite alphabet. If ϕ, ψ : Σ → R are two functions such that ψ ≥ c > 0, for some c > 0, and if C ⊂ Σ * then P ψ (ϕ, C ) is equal to the unique real number s ∈ R for which P (ϕ − sψ, C ) = 0. Moreover, we have that The next definition goes back to the work of Ruelle and Bowen ([Rue69,Bow75]).
Definition 2.4. Let ϕ : Σ → R be continuous. We say that a Borel probability measure µ is a Gibbs measure for ϕ if there exists a constant C > 0 such that The Perron-Frobenius operator, which we are going to define now, provides a useful tool for guaranteeing the existence of Gibbs measures and for deriving some of the stochastic properties of these measures (see [Rue69,Bow75]).
Definition 2.5. Let Σ be a Markov shift over a finite alphabet and let ϕ : Σ → R be continuous. The Perron-Frobenius operator associated to ϕ is the operator L ϕ : The following theorem summarizes some of the main results of the thermodynamic formalism for a Markov shift Σ with a finite alphabet I (see for instance [Wal82] and [MU03, Section 2]). Here, Σ is called irreducible if for all i, j ∈ I there exists ω ∈ Σ * ∪ {∅} such that iω j ∈ Σ * . Moreover, for k ∈ N 0 , the σ -algebra generated by [ω] : ω ∈ Σ k is denoted by C (k), and we say that f : Theorem 2.6. Let Σ be an irreducible Markov shift over a finite alphabet and let ϕ : Σ → R be α-Hölder continuous, for some α > 0. Then there exists a unique Borel probability measure µ supported on Σ such that L ϕ ( f ) dµ = e P(ϕ) f dµ, for all f ∈ C b (Σ). Furthermore, µ is a Gibbs measure for ϕ and there exists a unique α-Hölder continuous function h : Σ → R + such that h dµ = 1 and L ϕ (h) = e P(ϕ) h. The measure h dµ is the unique σ -invariant Gibbs measure for ϕ and will be denoted by µ ϕ . If ϕ : Σ → R is C (k)-measurable, for some k ∈ N 0 , then h is C (max {k − 1, 1})-measurable.

Graph Directed Markov Systems.
In this section we will first recall the definition of a graph directed Markov system (GDMS), which was introduced by Mauldin and Urbański [MU03]. Subsequently, we will introduce the notion of a linear GDMS associated to a free group and certain radial limit sets.
sists of a finite vertex set V , a family of nonempty compact metric spaces (X v ) v∈V , a countable edge set E, the maps i,t : E → V defining the initial and terminal vertex of an edge, a family of injective contractions φ e : X t(e) → X i(e) with Lipschitz constants bounded by some 0 < s < 1, and an edge in- For a GDMS Φ there exists a canonical coding map π Φ : Σ Φ → ⊕ v∈V X v , which is defined by where ⊕ v∈V X v denotes the disjoint union of the sets X v , φ ω| n := φ ω 1 • · · · • φ ω n and Σ Φ denotes the Markov shift with alphabet E and incidence matrix A. We set and refer to J (Φ) as the limit set of Φ.
The following notion was introduced in [MU03, Section 4].
A is called conformal if the following conditions are satisfied.
(a) For v ∈ V , the phase space X v is a compact connected subset of a Euclidean space R D , · , for some D ≥ 1, such that X v is equal to the closure of its interior, that is , for every e ∈ E with t (e) = v. (d) (Cone property) There exist l > 0 and 0 < γ < π/2 such that for each x ∈ X ⊂ R D there exists an open cone Con(x, γ, l) ⊂ Int(X) with vertex x, central angle of measure γ and altitude l. (e) There are two constants L ≥ 1 and α > 0 such that for each e ∈ E and x, y ∈ X t(e) we have The associated geometric potential ζ Φ : A Markov shift Σ with a finite or countable alphabet I is called finitely irreducible if there exists a finite set Λ ⊂ Σ * such that for all i, j ∈ I there exists a word ω ∈ Λ ∪ {∅} such that iω j ∈ Σ * (see [MU03,Section 2]). Note that if I is finite, then Σ is finitely irreducible if and only if Σ is irreducible.
The following result from [RU08, Theorem 3.7] shows that in the sense of Hausdorff dimension, the limit set of a conformal GDMS with a finitely irreducible incidence matrix can be exhausted by its finitely generated subsystems. The last equality in Theorem 2.9 follows from [JKL10, Corollary 2.10] since the associated geometric potential of the conformal GDMS Φ is bounded away from zero by − log (s), where s denotes the uniform bound of the Lipschitz constants of the contractions of Φ (see Definition 2.7).
Theorem 2.9 (Generalized Bowen's formula). Let Φ be a conformal GDMS such that Σ Φ is finitely irreducible. We then have that Let us now give the definition of a GDMS Φ associated to the free group F d of rank d ≥ 2 and introduce the radial limit set of a normal subgroup N of F d with respect to Φ. additionally Φ is a conformal GDMS such that, for each (v, w) ∈ E, the map φ (v,w) is a similarity for which the contraction ratio is independent of w, then Φ is called a linear GDMS associated to F d .
For a subgroup H of F d and a GDMS Φ associated to F d , the radial and the uniformly radial limit set of H with respect to Φ are respectively given by such that for infinitely many n ∈ N, v 1 · · · · · v n ∈ Hγ} Remark. It is clear that if Φ is a GDMS generated by a family of similarity maps, then Φ automatically satisfies (c) and (e) in Definition 2.8 of a conformal GDMS.
2.3. Random Walks on Graphs and Amenability. In this section we collect some useful definitions and results concerning random walks on graphs. We will mainly follow [Woe00].
Definition 2.11. A graph X = (V, E) consists of a countable vertex set V and an edge set all v, w ∈ V with v = w, there exists k ∈ N and a path of length k from v to w. For a connected graph X = (V, E) and v, w ∈ V we let d X (v, w) denote the minimal length of all paths from v to w, We now recall an important property of groups, which was introduced by von Neumann [Neu29] under the German name messbar. Later, groups with this property were renamed amenable groups by Day [Day49] and also referred to as groups with full Banach mean value by Følner [Føl55].
Definition 2.12. A discrete group G is said to be amenable if there exists a finitely additive probability measure ν on the set of all subsets of G which is invariant under left multiplication by elements of G, that is, ν (A) = ν (g (A)) for all g ∈ G and A ⊂ G.
We will also require the concept of an amenable graph, which extends the concept of amenability for groups (see Proposition 2.17 below).

Definition 2.13. A graph X = (V, E) with bounded geometry is called amenable if and only if there
For the study of graphs in terms of amenability, the following definition is useful.

Definition 2.14. A rough isometry (or quasi-isometry) between two metric spaces
is a map ϕ : Y → Y ′ which has the following properties. There exist constants A, B > 0 such that for all y 1 , y 2 ∈ Y we have . For connected graphs X = (V, E) and X = (V ′ , E ′ ) with graph metrics d X and d X ′ we say that the graphs X and X ′ are roughly isometric if the metric spaces (V, d X ) and (V ′ , d X ′ ) are roughly isometric.

The next theorem states that amenability of graphs is invariant under rough isometries ([Woe00,
Theorem 4.7]).
Theorem 2.15. Let X and X ′ be graphs with bounded geometry such that X and X ′ are roughly isometric. We then have that X is amenable if and only if X ′ is amenable.
The Cayley graph of a group provides the connection between groups and graphs.
Definition 2.16. We say that a set S ⊂ G is a symmetric set of generators of the group G if S = G and if g −1 ∈ S, for all g ∈ S. For a group G and a symmetric set of generators S, the Cayley graph of G with respect to S is the graph with vertex set G and edge set E : We denote this graph by X (G, S).
Next proposition shows that amenability of groups and graphs is compatible ([Woe00, Proposition 12.4]).

Proposition 2.17. A finitely generated group G is amenable if and only if one (and hence every) Cayley graph X (G, S) of G with respect to a finite symmetric set of generators S ⊂ G is amenable.
Let us now relate amenability of graphs to spectral properties of transition operators.
Definition 2.18. For a finite or countably infinite discrete vertex set V , we say that the matrix P = The following definitions introduce the concept of a transition matrix to be adapted to a graph (see [Woe00, (1.20, 1.21)]).
Definition 2.19. For a connected graph X = (V, E) and a transition matrix P = (p (v, w)) ∈ R V ×V on V , we say that P is uniformly irreducible with respect to X if there exist K ∈ N and ε > 0 such that for all v, w ∈ V satisfying v ∼ w there exists k ∈ N with k ≤ K such that p (k) (v, w) ≥ ε. We say that P has bounded range with respect to X if there exists R > 0 such that p (v, w) = 0 whenever and that the norm of this operator is less or equal to one. For the spectral radius ρ (P) of the operator P on ℓ 2 (V, ν) we cite the following result from [OW07]. This result has a rather long history going back to [Kes59b,Kes59a] Theorem 2.20 (Ortner, Woess). Let X = (V, E) be a graph with bounded geometry and let P denote a transition matrix on V such that P is uniformly irreducible with respect to X and has bounded range with respect to X. If there exists a P-invariant Borel measure ν on V and a constant C ≥ 1 such that C −1 ≤ ν (w) ≤ C, for all w ∈ V , then we have that ρ (P) = 1 if and only if X is amenable.

THERMODYNAMIC FORMALISM FOR GROUP-EXTENDED MARKOV SYSTEMS
Throughout this section our setting is as follows.
(2) G is a countable discrete group G with Haar measure (counting measure) λ .
(3) Ψ : I * → G is a semigroup homomorphism such that the following property holds. For all (4) ϕ : Σ → R denotes a Hölder continuous potential with σ -invariant Gibbs measure µ ϕ , denotes the Perron-Frobenius operator associated to ϕ, and h : Σ → R denotes the unique Hölder continuous eigenfunction of L ϕ with corresponding eigenvalue e P(ϕ) whose existence is guaranteed by Theorem 2.6.
In this section we will address the following problem.
It turns out that in order to investigate Problem 3.1 it is helpful to consider group-extended Markov systems (defined below), which were studied in ([AD00, AD02]) for certain abelian groups.
is called a group-extended Markov system. We let π 1 : Σ × G → Σ and π 2 : Σ × G → G denote the projections to the first and to the second factor of Σ × G.
Remark. Throughout, we assume that Σ×G is equipped with the product topology. Note that by item (3) of our standing assumptions we have that the group-extended Markov system (Σ × G, σ ⋊ Ψ) is topologically transitive if and only if Ψ (Σ * ) = G.
3.1. Perron-Frobenius Theory. In this section, we investigate the relationship between the pressure P ϕ, Ψ −1 {id} ∩ Σ * and the spectral radius of a Perron-Frobenius operator associated to (Σ × G, σ ⋊ Ψ), which will be introduced in Definition 3.4 below. Combining this with results concerning transition operators of random walks on graphs, which will be given in Section 3.2, we are able to give a complete answer to Problem 3.1 for potentials ϕ depending only on a finite number of coordinates (see Theorem 3.21).
Let us begin by stating the following lemma. The proof is straightforward and is thus left to the reader.
Next, we define the Koopman operator ( [Koo31,LM94]) and the Perron-Frobenius operator associated to the group-extended Markov system (Σ × G, σ ⋊ Ψ). Note that the previous lemma ensures that these operators are well-defined. We denote by L 2 Σ × G, µ ϕ × λ the Hilbert space of realvalued functions on Σ × G which are square-integrable with respect to µ ϕ × λ .
and the Perron-Frobenius operator L ϕ•π 1 : and U * : The proof of the next lemma is straightforward and therefore omitted.
(1) U is an isometry, so we have that U = ρ (U) = 1, where ρ denotes the spectral radius of U.
Remark. The representation of L ϕ•π 1 in Lemma 3.5 (2) extends Definition 2.5 of the Perron-Frobenius operator for Markov shifts with a finite alphabet.
The next lemma gives relationships between P ϕ, Ψ −1 {id} ∩ Σ * and L ϕ•π 1 . Before stating the lemma, let us fix some notation. We write ½ A for the characteristic function of a set A and we use {π 2 = g} to denote the set π −1 2 {g}, for each g ∈ G. Further, let B (Σ × G) denote the Borel σ -algebra on Σ × G.
Lemma 3.6. For all sets A, B ∈ B (Σ × G) and for each n ∈ N we have that Moreover, for all g, g ′ ∈ G we have that Proof. For the first assertion, observe that by the definition of L ϕ•π 1 we have that Since the continuous function h : Σ → R + is bounded away from zero and infinity on the compact set Σ, the first assertion follows.
The second assertion follows from the first, if we set A := {π 2 = g} and B := {π 2 = g ′ } and use the Gibbs property (2.1) of µ ϕ .
As an immediate consequence of the previous lemma, we obtain the following upper bound for P ϕ, Ψ −1 g −1 g ′ ∩ Σ * in terms of the spectral radius of L ϕ•π 1 .
Proof. By the Cauchy-Schwarz inequality and Gelfand's formula ([Rud73, Theorem 10.13]) for the spectral radius, we have that Combining the above inequality with the second assertion of Lemma 3.6 completes the proof.

Recall that for a closed linear subspace
The following lemma will be crucial in order to obtain equality in the inequality stated in Corollary 3.7. The lemma extends a result of Gerl (see [Ger88] and also [Woe00, Lemma 10.1]).

be a self-adjoint bounded linear operator on V , which is
positive and which satisfies ker (T ) ∩V + = {0}. We then have that

Proof.
Since T is self-adjoint, it follows that T = ρ (T ). As in the proof of Corollary 3.7, one immediately verifies that Let us first give an outline for the proof of the opposite inequality. We will first prove that for all , is non-decreasing. This will then imply that the following limits exist and are equal:

From this we obtain for every
Subsequently, we make use of the fact that Combining this observation with the estimate in (3.3), we conclude that Let us now turn to the details. We first verify that for every f ∈ V + with f = 0, the sequence (a n ) n∈N 0 of positive real numbers, given for n ∈ N 0 by a n : Using that T is self-adjoint and applying the Cauchy-Schwarz inequality, we have for n ∈ N 0 that Since (T n f , T n f ) = 0 for all n ∈ N 0 by our hypothesis, we can multiply both sides of (3.4) by which proves that (a n ) n∈N 0 is non-decreasing. Hence, we have that lim n→∞ a n ∈ R + ∪ {∞} exists. Observing that log (T n f , T n f ) is equal to the telescoping sum log ( f , f ) + ∑ n−1 j=0 log a j and using that lim n→∞ log (a n ) is equal to its Cesàro mean, we deduce that log a j = lim n→∞ log a n , which proves (3.1). Since (T n f , T n f ) 1/n ≤ T 2 max f 2 2 , 1 , for all n ∈ N, we have that the limits in (3.1) are both finite.
It remains to prove that (3.3) holds for every f ∈ D with f = 0. By definition of D, there exists a finite set G 0 ⊂ G such that f = ∑ g∈G 0 f ½ {π 2 =g} . Since T is positive and self-adjoint, we conclude that Finally, raising both sides of the previous inequality to the power 1/n and let n tend to infinity gives lim n→∞ (T n f , T n f ) 1/n ≤ max Regarding the requirements of the previous proposition, we prove the following for L ϕ•π 1 . Lemma 3.9. Let V be a closed L ϕ•π 1 -invariant linear subspace of L 2 Σ × G, µ ϕ × λ and suppose * is a positive operator and if there exists g ∈ V with g > 0, then Proof. Clearly, by definition of L ϕ•π 1 , we have that L ϕ•π 1 V is positive. Now let f ∈ ker L ϕ•π 1 V ∩ V + . Since µ ϕ is a fixed point of L * ϕ , one deduces by the monotone convergence theorem and by the definition of L ϕ•π 1 that f d µ ϕ × λ = L ϕ•π 1 ( f ) d µ ϕ × λ . Hence, f ∈ ker L ϕ•π 1 V ∩ V + implies f d µ ϕ × λ = 0 and so, f = 0.
We now turn our attention to the adjoint operator and using that L ϕ•π 1 is positive, we obtain that − 2 2 and so, L ϕ•π 1 V * is positive. Now let f ∈ ker L ϕ•π 1 V * ∩ V + be given and assume that there exists g ∈ V with g > 0. We then have that Since g > 0, we have L ϕ•π 1 (g) > 0, which implies that f = 0. The proof is complete.
It turns out that the Perron-Frobenius operator is not self-adjoint in general. In fact, as we will see in the following remark, this operator is self-adjoint only in very special cases. Therefore, we will introduce the notion of an asymptotically self-adjoint operator in Definition 3.10 below.
Remark. We observe that the requirement that L ϕ•π 1 V is self-adjoint, for some closed linear sub- has at most two elements. To prove this, let i j ∈ Σ 2 be given. By the Gibbs property (2.1) of µ ϕ we have that µ ϕ [i j] > 0. Setting C := max h min h e −P(ϕ) we deduce from Lemma 3.6 that Using that L ϕ•π 1 V is self-adjoint and again by Lemma 3.6, we conclude that Combining the previous two estimates, we conclude that The following definition introduces a concept which is slightly weaker than self-adjointness.
Definition 3.10. Let V be a closed linear subspace of L 2 Σ × G, µ ϕ × λ and let T : V → V be a positive bounded linear operator. We say that T is asymptotically self-adjoint if there exist sequences (c m ) m∈N ∈ (R + ) N and (N m ) m∈N ∈ N N 0 with the property that lim m→∞ (c m ) 1/m = 1, lim m→∞ m −1 N m = 0, such that for all non-negative functions f , g ∈ V and for all n ∈ N we have Remark. Note that T is asymptotically self-adjoint if and only if T * is asymptotically self-adjoint. We also remark that it clearly suffices to verify (3.5) on a norm-dense subset of non-negative func- The next proposition shows that if L ϕ•π 1 V is asymptotically self-adjoint, for some closed linear subspace V of L 2 Σ × G, µ ϕ × λ , then we can relate the supremum of P ϕ, Ψ −1 {g} ∩ Σ * , for g ∈ G, to the spectral radius of L ϕ•π 1 V . The proof, which makes use of Lemma 3.8 and Lemma 3.9, is inspired by [OW07, Proposition 1.6].
In the following definition, we introduce certain important closed linear subspaces of the space denotes the product σ -algebra of C ( j) and the Borel σ -algebra B (G) on G.
Note that V j is a Hilbert space for each j ∈ N 0 . The next lemma gives an invariance property for V j with respect to L ϕ•π 1 for C (k)-measurable potentials ϕ.
Proof. If f is C ( j)-measurable, j ∈ N 0 , then it follows from Lemma 3.5 (2) that L ϕ•π 1 ( f ) is given by Note that the right-hand side of the previous equation depends only on g ∈ G and on the elements ω 1 , . . . , ω max{k−1, j−1,1} ∈ I. Consequently, for j ∈ N with j ≥ k − 1, we have that V j is L ϕ•π 1invariant.
The remaining assertion follows immediately from the definition of U.
We need the following notion of symmetry.
Definition 3.14. We say that ϕ : Σ → R is asymptotically symmetric with respect to Ψ if there exist sequences (c m ) m∈N ∈ (R + ) N and (N m ) m∈N ∈ N N 0 with the property that lim m (c m ) 1/m = 1, lim m m −1 N m = 0 and such that for each g ∈ G and for all n ∈ N we have Remark 3.15. If ϕ is asymptotically symmetric with respect to Ψ, then it is straightforward to verify that, for each ψ : Σ → R + Hölder continuous and c ∈ R, we have that also ϕ + log ψ − log ψ • σ + c is asymptotically symmetric with respect to Ψ. Using the Gibbs property (2.1) of µ ϕ , an equivalent way to state that ϕ is asymptotically symmetric with respect to Ψ is the following: there exist sequences (c ′ m ) m∈N ∈ (R + ) N and (N ′ m ) m∈N ∈ N N 0 with the property that lim m (c ′ m ) 1/m = 1, lim m m −1 N ′ m = 0 and such that for each g ∈ G and for all n ∈ N we have ∑ ω∈Σ n :Ψ(ω)=g Next lemma gives a necessary and sufficient condition for L ϕ•π 1 V j to be asymptotically self-adjoint.
Lemma 3.16. Let ϕ : Σ → R be C (k)-measurable, for some k ∈ N 0 . For each j ∈ N with j ≥ k − 1, we then have that ϕ is asymptotically symmetric with respect to Ψ if and only if L ϕ•π 1 V j is asymptotically self-adjoint.
Proof. We first observe that by Lemma 3.6 and by the Gibbs property (2.1) of µ ϕ , there exists K > 0 such that for all n ∈ N and for all g, g ′ ∈ G we have (3.10) unless nominator and denominator in (3.10) are both equal to zero. From (3.10) we obtain that ϕ is asymptotically symmetric with respect to Ψ if and only if there exist sequences (c m ) ∈ (R + ) N and (N m ) ∈ N N 0 , as in Definition 3.14, such that for all n ∈ N and g, g ′ ∈ G we have Since V 0 ⊂ V j for each j ∈ N 0 , we obtain that if L ϕ•π 1 V j is asymptotically self-adjoint, then ϕ is asymptotically symmetric with respect to Ψ.
For the opposite implication, let j ∈ N, j ≥ k − 1 and suppose that ϕ is asymptotically symmetric with respect to Ψ. By Lemma 3.13, we have that V j is L ϕ•π 1 -invariant. Next, we note that since ϕ is asymptotically symmetric with respect to Ψ, we have that, for each ω ∈ Σ j , there exists κ(ω) ∈ Σ * such that Ψ(ω)Ψ(κ(ω)) = id. Combining this with item (3) of our standing assumptions, we conclude that for all ω, ω ′ ∈ Σ j there exists a finite-to-one map which maps τ ∈ Σ * to an element ωγ 1 κ(ω)γ 2 τγ 3 ω ′ ∈ Σ * , where Ψ(γ i ) = id and γ i depends only on the preceding and successive symbol, for each i ∈ {1, 2, 3}. Hence, in view of Lemma 3.6 and the Gibbs property (2.1) of µ ϕ , and by using that Σ j is finite, we conclude that there exist N ∈ N and C > 0 (depending on j) such that for all n ∈ N, g, g ′ ∈ G and for all ω, ω ′ ∈ Σ j we have By first using (3.11) and then (3.12), we deduce that for all n ∈ N, g, g ′ ∈ G and for all ω, ω ′ ∈ Σ j , The following corollary is a consequence of Proposition 3.11 and clarifies the relation between sup g∈G P ϕ, Ψ −1 {g} ∩ Σ * and the spectral radius of L ϕ•π 1 V j provided that ϕ is asymptotically symmetric with respect to Ψ. Corollary 3.17. Let ϕ : Σ → R be C (k)-measurable, for some k ∈ N 0 , and suppose that ϕ is asymptotically symmetric with respect to Ψ. For each j ∈ N with j ≥ k − 1, we then have that Proof. Fix j ∈ N with j ≥ k − 1. By Lemma 3.13, we then have that V j is L ϕ•π 1 -invariant. Let us first verify that without loss of generality we may assume that L ϕ ½ = ½. Otherwise, by Theorem 2.6, there exists a C (max {k − 1, 1})-measurable function h : Σ → R + with L ϕ (h) = e P(ϕ) h. For ϕ := ϕ + logh − logh • σ − P (ϕ), we then have that Lφ ½ = ½, P (φ) = 0 and, for each g ∈ G, Further, we have thatφ is asymptotically symmetric with respect to Ψ, by Remark 3.15. It remains to show that V j is Lφ •π 1 -invariant and that Since h is C (max {k − 1, 1})-measurable, we have that V j is M h•π 1 -invariant and, by the definition of the Perron-Frobenius operator, we obtain that We conclude that V j is Lφ •π 1 -invariant and that Lφ •π 1 V j and e −P(ϕ) L ϕ•π 1 V j have the same spectrum. The latter fact gives the equality in (3.13). Hence, we may assume without loss of generality that L ϕ ½ = ½.
By Lemma 3.16, we have that L ϕ•π 1 V j is asymptotically self-adjoint. Since the closed linear sub- Remark. Note that, in particular, under the assumptions of the previous corollary we have that ρ L ϕ•π 1 V j is independent of j ∈ N for all j ≥ k − 1.

Random Walks on Graphs and Amenability.
In this section we relate the Perron-Frobenius operator to the transition operator of a certain random walk on a graph. We start by introducing the following graphs.
Definition 3.18. For each j ∈ N 0 , the j-step graph of (Σ × G, σ ⋊ Ψ) consists of the vertex set Σ j × G where two vertices (ω, g) , (ω ′ , g ′ ) ∈ Σ j × G are connected by an edge in X j if and only if We use X j (Σ × G, σ ⋊ Ψ) or simply X j to denote this graph.
Provided that Ψ (Σ * ) = G, we have that each j-step graph of (Σ × G, σ ⋊ Ψ) is connected. Next lemma shows that each of these graphs is roughly isometric to the Cayley graph of G with respect to Ψ (I) ∪ Ψ (I) −1 denoted by X G, Ψ (I) ∪ Ψ (I) −1 . For a similar argument, see [OW07].
Proof. By identifying Σ 0 × G with G, we clearly have that X 0 is isometric to X G, Ψ (I) ∪ Ψ (I) −1 .
Suppose now that j ∈ N. We show that the map π 2 : Σ j × G → G, given by π 2 (ω, g) := g, for all (ω, g) ∈ Σ j × G, defines a rough isometry between the metric spaces Σ j × G, d j and (G, d), where d j denotes the graph metric on X j and d denotes the graph metric on X G, Ψ (I) ∪ Ψ (I) −1 .
Clearly, we have that π 2 is surjective. Further, by the definition of the edge set of X j , we have that if two vertices (ω, g) , (ω ′ , g ′ ) ∈ Σ j × G are connected by an edge in X j , then g and g ′ are connected by an edge in X G, Ψ (I) ∪ Ψ (I) −1 . Hence, for all (ω, g) , .
It remains to show that there exist constants A, B > 0 such that for all (ω, g) , (ω ′ , g ′ ) ∈ Σ j × G, First note that by our assumptions, there exists a finite set F ⊂ Σ * with the following properties.
Proposition 3.20. Suppose that Ψ (Σ * ) = G. Let ϕ : Σ → R be C (k)-measurable for some k ∈ N 0 , such that L ϕ ½ = ½. The following holds for all j ∈ N with j ≥ k − 1. For the bounded linear with equality if and only if G is amenable. In particular, we have that with equality if and only if G is amenable.
Proof. Fix j ∈ N with j ≥ k − 1. We first observe that for each f ∈ V j we have that E (U ( f ) |C ( j)) is the unique element in V j , such that (E (U ( f ) |C ( j)) , g) = (U ( f ) , g) for all g ∈ V j . Since (U ( f ) , g) = f , L ϕ•π 1 (g) and V j is L ϕ•π 1 -invariant by Lemma 3.13, we conclude that E (U (·) |C ( j)) is the ad- Because U is an isometry by Lemma 3.5 (1), we conclude that L ϕ•π 1 V j = E (U (·) |C ( j)) = 1.
In order to prove the amenability dichotomy for ρ (E (U (·) |C ( j))) we aim to apply Theorem 2.20 to a transition matrix on the vertex set Σ j × G of the graph X j . Since ½ [ω]×{g} : (ω, g) ∈ Σ j × G is a basis of V j , we obtain a Hilbert space isomorphism between V j and ℓ 2 Σ j × G, ν j by setting ν j (ω, g) := µ ϕ × λ ([ω] × {g}) for every (ω, g) ∈ Σ j × G. Using this isomorphism and with respect to the canonical basis of ℓ 2 Σ j × G, ν j , we have that E (U (·) |C ( j)) is represented by the matrix P = (p ((ω, g) , (ω ′ , g ′ ))) given by Note that we have chosen the matrix P to act on the left. Summing over (ω ′ , g ′ ) ∈ Σ j × G in the previous line, we obtain that P is a transition matrix on Σ j × G. Using that µ ϕ × λ is (σ ⋊ Ψ)invariant by Lemma 3.3, one then deduces from (3.15) that ν j is P-invariant. Let us now verify that Theorem 2.20 is applicable to the transition matrix P acting on the vertex set Σ j × G of X j .
We are now in the position to apply Theorem 2.20 to the transition matrix P, which gives that ρ (P) = 1 if and only if X j is amenable. Since X j is roughly isometric to the Cayley graph of G with respect to Ψ (I) ∪ Ψ (I) −1 by Lemma 3.19, it follows from Theorem 2.15 that X j is amenable if and only if G is amenable (cf. Proposition 2.17) . Finally, since E (U (·) |C ( j)) and P are conjugated by an isomorphism of Hilbert spaces, we have ρ (E (U (·) |C ( j))) = ρ(P), which completes the proof.
Summarizing the outcomes of this section, we obtain the following main result.
Theorem 3.21. Suppose that Ψ (Σ * ) = G and let ϕ : Σ → R be C (k)-measurable for some k ∈ N 0 . The following holds for all j ∈ N with j ≥ k − 1. We have with equality in the second inequality if and only if G is amenable. Moreover, if ϕ is asymptotically symmetric with respect to Ψ, then and so, G is amenable if and only if P ϕ, Ψ −1 {id} ∩ Σ * = P (ϕ).
Proof. Fix j ∈ N with j ≥ k − 1, which implies that V j is L ϕ•π 1 -invariant by Lemma 3.13. As shown in the proof of Corollary 3.17 we may assume without loss of generality that L ϕ ½ = ½ and thus P (ϕ) = 0.
The first inequality in (3.16) follows from Corollary 3.7 applied to V = V j . The second inequality in (3.16) is an immediate consequence of the definition of the spectrum. The amenability dichotomy follows from Proposition 3.20. The equality log ρ L ϕ•π 1 = P (ϕ) follows from Lemma 3.5 (3).
In order to complete the proof, we now address (3.17) under the assumption that ϕ is asymptotically symmetric with respect to Ψ. By Corollary 3.17, we then have that sup g∈G P ϕ, Ψ −1 {g} ∩ Σ * = log ρ L ϕ•π 1 V j .
Using that Ψ (Σ * ) = G and item (3) of our standing assumptions, one easily verifies that the pressure P ϕ, Ψ −1 {g} ∩ Σ * is independent of g ∈ G, which completes the proof.
Proof. Using item (3) of our standing assumptions and that ϕ is asymptotically symmetric with respect to Ψ, one verifies that G ′ := Ψ(Σ * ) is a subgroup of G. Since G is amenable, it is wellknown that also G ′ is amenable (see e.g. [Woe00, Theorem 12.2 (c)]), and the corollary follows from Theorem 3.21.
Remark 3.23. It is not difficult to extend Corollary 3.22 to arbitrary Hölder continuous potentials by approximating a Hölder continuous potential by a C (k)-measurable potential and then letting k tend to infinity. One obtains that, for an amenable group G and for an asymptotically symmetric Hölder continuous potential ϕ, we have P ϕ, Ψ −1 {id} ∩ Σ * = P (ϕ
For a normal subgroup N of F d , we let Ψ N : I * → F d /N denote the unique semigroup homomorphism such that Ψ N (g) = g mod N for all g ∈ I. Clearly, we have that Since the assertions in Theorem 1.1 and Proposition 1.3 are clearly satisfied in the case that N = {id}, we will from now on assume that N = {id}. Using that N is a normal subgroup of F d and d ≥ 2, one easily verifies that there exists a finite set F ⊂ Σ * ∩ Ψ −1 N {id} with the following property: Note that (4.2) implies that the group-extended Markov system (Σ × (F d /N) , σ ⋊ Ψ N ) satisfies item (3) of our standing assumptions at the beginning of Section 4. Hence, the results of Section 3 are applicable to the C (1)-measurable potential ϕ : Σ → R, given by ϕ |[g] = log (c Φ (g)) for all g ∈ I.
Proof of Theorem 1.1 . Our aim is to apply Theorem 3.21 to the group-extended Markov system (Σ × (F d /N) , σ ⋊ Ψ N ) and the C (1)-measurable potential sϕ : Σ → R, for each s ∈ R. By (4.1) and (4.2), we are left to show that sϕ is asymptotically symmetric with respect to Ψ N . Since Φ is symmetric we have that c Φ (ω) = c Φ (κ (ω)), for all ω ∈ Σ * . Hence, for all s ∈ R, n ∈ N and g ∈ F d /N, we have that which proves that sϕ is asymptotically symmetric with respect to Ψ N . We are now in the position to apply Theorem 3.21, which gives that amenability of F d /N is equivalent to Since δ (N, Φ) is equal to the unique zero of s → P sϕ, Ψ −1 N {id} ∩ Σ * and δ (F d , Φ) is equal to the unique zero of s → P (sϕ) by Fact 2.3, we conclude that The proof is complete.
For the proof of Theorem 1.2 we need the following lemma.
Proof of Theorem 1.2. By Theorem 1.1, the assertion is clearly true if F d /N is amenable. We address the remaining case that F d /N is non-amenable. Suppose for a contradiction that the claim is wrong. By Lemma 4.1, we obtain that For notational convenience, we set G := F d /N throughout this proof.
Consider the non-negative matrix P ∈ R (I×G)×(I×G) , given by . By the assertions in (4.1) and (4.2), we have that P is irreducible in the sense that, for all x, y ∈ I × G there exists n ∈ N such that p (n) (x, y) > 0. Using the irreducibility of P and that card (I) = 2d < ∞, we deduce from (4.4) and Lemma 4.1 that P is R-recurrent with R = 1 in the sense of Vere-Jones ( [VJ62], see also Seneta [Sen06, Definition 6.4]). That is, P satisfies the following properties. It also follows from [Sen06, Theorem 6.2] that the vector h in (4.6) is unique up to a constant multiple. Next, we define the non-negative matrix P h ∈ R (I×G)×(I×G) , which is for all x, y ∈ I × G given by It follows from (4.6) that P h is a transition matrix on I × G. Further, we deduce from (4.5) that P h is 1-recurrent.
In order to derive a contradiction, we consider P h as a random walk on the graph X 1 associated to the group-extended Markov system (Σ × G, σ ⋊ Ψ N ) (see Definition 3.18), and we investigate the automorphisms of X 1 . Let Aut (X 1 ) denote the group of self-isometries of (X 1 , d X 1 ), where d X 1 denotes the graph metric on X 1 . Note that each element g ∈ G gives rise to an automorphism γ g ∈ Aut (X 1 ), which is given by γ g (i, τ) := (i, gτ), for each (i, τ) ∈ I × G. The next step is to verify that also γ g ∈ Aut (X 1 , P h ), where we have set Aut (X 1 , P h ) := {γ ∈ Aut (X 1 ) : P h (x, y) = P h (γx, γy) , for all x, y ∈ I × G} .
Since P has the property that p (x, y) = p (γ g (x) , γ g (y)), for all x, y ∈ I × G and g ∈ G, it follows that the vector h g ∈ R I×G , given by h g (i, τ) := h (i, gτ), (i, τ) ∈ I × G, satisfies h g P = h g as well. Since the function h in (4.6) is unique up to a constant multiple, we conclude that there exists a homomorphism r : G → R + such thath g = r (g) h, for each g ∈ G. Consequently, we have p h (x, y) = p h (γ g (x) , γ g (y)) for all x, y ∈ I × G and g ∈ G. Hence, γ g ∈ Aut (X 1 , P h ) for each g ∈ G. Since card(I) < ∞, we deduce that Aut (X 1 , P h )) acts with finitely many orbits on X 1 .
In the terminology of [Woe00] this is to say that (X 1 , P h ) is a quasi-transitive recurrent random walk.
By [Woe00, Theorem 5.13] we then have that X 1 is a generalized lattice of dimension one or two.
In particular, we have that X 1 has polynomial growth with degree one or two ([Woe00, Proposition 3.9]). Since X 1 is roughly isometric to the Cayley graph of G by Lemma 3.19, we conclude that also G has polynomial growth (see e.g. [Woe00, Lemma 3.13]). This contradicts the well-known fact that each non-amenable group has exponential growth. The proof is complete.
Remark. The construction of the matrix P h and the verification of its invariance properties is analogous to the discussion of the h-process in [Woe00, Proof of Theorem 7.8] and goes back to the work of Guivarc'h ([Gui80, page 85]) on random walks on groups. However, note that in our case P is in general not stochastic.
Proof of Proposition 1.3. In order to investigate the radial limit sets of N, we introduce an induced GDMSΦ, whose edge set consists of first return loops in the Cayley graph of F d /N. We definẽ Φ := V, (X v ) v∈V ,Ẽ,ĩ,t, φ ω ω∈Ẽ ,Ã as follows. The edge setẼ andĩ,t :Ẽ → V are given bỹ the matrixÃ = (ã (ω, ω ′ )) ∈ {0, 1}Ẽ ×Ẽ satisfiesã (ω, ω ′ ) = 1 if and only if a ω |ω| , ω ′ 1 = 1, and the family φ ω ω∈Ẽ is given byφ ω := φ ω , ω ∈Ẽ. One immediately verifies thatΦ is a conformal GDMS. Note that there are canonical embeddings from ΣΦ into Σ Φ and from Σ * Φ into Σ * Φ , which we will both indicate by omitting the tilde, that isω → ω. For the coding maps πΦ : ΣΦ → J Φ and π Φ : Σ Φ → J (Φ) we have πΦ (ω) = π Φ (ω), for eachω ∈ ΣΦ. The following relations between the limit set ofΦ and the radial limit sets of N are straightforward to prove. We have that Note that the right-hand side in the latter chain of inclusions can be written as a countable union of images of J Φ under Lipschitz continuous maps. Since Lipschitz continuous maps do not increase Hausdorff dimension and since Hausdorff dimension is stable under countable unions, we obtain Since the incidence matrix ofΦ is finitely irreducible by property (4.2), the generalised Bowen's formula (Theorem 2.9) implies that dim H J * Φ = dim H J Φ , so equality holds in (4.7).
The final step is to show that dim H J Φ = δ (N, Φ). By Theorem 2.9 and Fact 2.3, we have Since the elementsω ∈ Σ * Φ are in one-to-one correspondence with ω ∈ C N , where C N is given by and using that Sω ζΦ = S ω ζ Φ for allω ∈ Σ * Φ , we conclude that Finally, since the map from C N onto N, given by ω which completes the proof.

KLEINIAN GROUPS
In this section we give a more detailed discussion of Kleinian groups and how these relate to the concept of a GDMS. In particular, in Proposition 5.6 we will give the motivation for our definition of the radial limit set in the context of a GDMS associated to the free group (see Definition 2.10).
In the following we let G ⊂ Con (m) denote a non-elementary, torsion-free Kleinian group acting properly discontinuously on the (m + 1)-dimensional hyperbolic space D m+1 , where Con (m) denotes the set of orientation preserving conformal automorphisms of D m+1 . The limit set L (G) of G is the set of accumulation points with respect to the Euclidean topology on R m+1 of the G-orbit of some arbitrary point in D m+1 , that is, for each z ∈ D m+1 we have that where the closure is taken with respect to the Euclidean topology on R m+1 . Clearly, L (G) is a subset of S. For more details on Kleinian groups and their limit sets, we refer to [Bea95,Mas88,Nic89,MT98,Str06].
Let us recall the definition of the following important subsets of L(G), namely the radial and the  (0), c) .
A Kleinian group G is said to be geometrically finite if the action of G on D m+1 admits a fundamental polyhedron with finitely many sides. We denote by E G the set of points in D m+1 , which lie on a geodesic connecting any two limit points in L (G). The convex hull of E G , which we will denote by C G , is the minimal hyperbolic convex subset of D m+1 containing E G . G is called convex cocompact n . For each n ∈ {1, . . . d}, let g n ∈ Con (m) be the unique hyperbolic element such that g n D m+1 ∩ ∂ D −1 n = D m+1 ∩ ∂ D n , where ∂ D j n denotes the boundary of D j n with respect to the Euclidean metric on R m+1 . Then G := g 1 , . . . , g d is referred to as the Kleinian group of Schottky type generated by D.
Note that a Kleinian group of Schottky type G = g 1 , . . . , g d is algebraically a free group. The following construction of a particular GDMS associated to the free group g 1 , . . . , g d is canonical.
Definition 5.3. Let G = g 1 , . . . , g d be a Kleinian group of Schottky type generated by D. The canonical GDMS Φ G associated to G is the GDMS associated to the free group g 1 , . . . , g d which satisfies X g j n := D m+1 ∪ S m ∩ D j n , for each n ∈ {1, . . ., d} and j ∈ {−1, 1}, and for which the contractions φ (v,w) : X w → X v are given by φ (v,w) := v X w , for each (v, w) ∈ E.
For the following fact we refer to [MU03, Theorem 5.1.6].
Fact 5.4. For a Kleinian group of Schottky type G we have that L(G) = J (Φ G ).
Remark 5.5. We remark that without our assumption on G that diam (D n ) = diam D −1 n , for each n ∈ {1, . . . , d} in Definition 5.2, the generators of the associated GDMS Φ G may fail to be contractions.
However, in that case, by taking sufficiently high iterates of the generators, we can pass to a finite index subgroup of G, for which there exists a set D as in Definition 5.2.
The following brief discussion of the geometry of a Kleinian group of Schottky type G contains nothing that is not well known, however, the reader might like to recall a few of its details. Let Φ G denote the canonical GDMS associated to G. Recall that for the half-spaces H v := z ∈ D m+1 : d (z, 0) < d (z, v (0)) , for each v ∈ V, we have that the set F := v∈V H v is referred to as a Dirichlet fundamental domain for G. That F is a fundamental domain for G means that F is an open set which satisfies the conditions g∈G g F ∩ D m+1 = D m+1 and g (F) ∩ h (F) = ∅ for all g, h ∈ G with g = h.
We also make use of the fact that a Kleinian group of Schottky type G is convex cocompact. This follows from a theorem due to Beardon and Maskit ([BM74], [Str06, Theorem 2]), since G is geometrically finite and L (G) contains no parabolic fixed points (cf. [Rat06,Theorem 12.27]). Clearly, if G is convex cocompact, then there exists R G > 0 such that (0) , R G ) , for all g ∈ G.
In particular, we have that L ur (G) = L r (G) = L (G).
Using the fact that G acts properly discontinuously on D m+1 and that G is convex cocompact, one easily verifies that for each r > 0 there exists a finite set Γ ⊂ G such that (5.2) B (0, r) ∩C G ⊂ γ∈Γ γF.
The next proposition provides the main motivation for our definition of the (uniformly) radial limit set of a normal subgroup N of F d with respect to a GDMS associated to F d .
Proposition 5.6. Let G be a Kleinian group of Schottky type and let Φ G denote the canonical GDMS associated to G. For every non-trivial normal subgroup N of G, we have that L r (N) = Λ r (N, Φ G ) and L ur (N) = Λ ur (N, Φ G ) .
Finally, let us demonstrate that L r (N) ⊂ Λ r (N, Φ G ). To that end, pick an arbitrary ξ ∈ L r (N) and let ω = (v k , w k ) k∈N ∈ Σ Φ G with π Φ G (ω) = ξ be given. Then, by definition of L r (N), there exists c > 0 and a sequence (h k ) k∈N of pairwise distinct elements in N such that s ξ ∩ B (h k (0) , c) = ∅, for all k ∈ N. Using (5.2) we deduce that there exists a finite set Γ ⊂ G such that for all k ∈ N we have Since Γ is finite, there exist γ 0 ∈ Γ and sequences (n k ) k∈N and (l k ) k∈N tending to infinity such that s ξ ∩ B h n k (0) , c ∩ h n k γ 0 F = ∅ and v 1 v 2 · · · v l k = h n k γ 0 , for all k ∈ N. Hence, ξ ∈ Λ r (N, Φ G ).