Groups of automorphisms of trees and their limit sets

@article{Hersonsky1997GroupsOA,
  title={Groups of automorphisms of trees and their limit sets},
  author={Sa'ar Hersonsky and John H. Hubbard},
  journal={Ergodic Theory and Dynamical Systems},
  year={1997},
  volume={17},
  pages={869-884}
}
Let T be a locally finite simplicial tree and let ! ⊂ Aut(T ) be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincare series associated with !, which is also the Hausdorff dimension of the limit set of !; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of !. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its… Expand

Figures from this paper

Noncommutative geometry on trees and buildings
We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case isExpand
On convex hulls and the quasiconvex subgroups of Fm×ℤn
TLDR
A method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn is explored and it is shown that with respect to the usual action of Fm×ℤn on Tree ×ℝ n ${\mathrm {Tree}\times \mathbb {R}^n}$, every quasiconvex subgroup of Fn is convex. Expand
Random Walks on Trees with Finitely Many Cone Types
This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adaptedExpand
Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of resultsExpand
Weighted cogrowth formula for free groups
We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by anExpand
GEOMETRY ON TREES AND BUILDINGS
The notion of a spectral triple, introduced by Connes (cf. [9], [7], [10]), provides a powerful generalization of Riemannian geometry to noncommutative spaces. It originates from the observationExpand
Ergodic theory of affine isometric actions on Hilbert spaces
The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, weExpand
No proper conjugation for quasiconvex cocompact groups of Gromov hyperbolic spaces
We prove that, if a quasiconvex cocompact subgroup of the isometry group of a Gromov hyperbolic space has a conjugation into itself, then it is onto itself.
Quasiconvex Subgroups of F_m x Z^n are Convex
In this paper we analyze the action of a quasiconvex subgroup of F_m x Z^n on the convex hull of its orbit and we show that this action is cocompact. Further, using our techniques, we obtain completeExpand
Growth and ergodicity of context-free languages
A language L over a finite alphabet E is called growth-sensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown thatExpand
...
1
2
...

References

SHOWING 1-10 OF 19 REFERENCES
Lattices in rank one Lie groups over local fields
AbstractWe prove that if $$G = \underline G (K)$$ is theK-rational points of aK-rank one semisimple group $$\underline G $$ over a non archimedean local fieldK, thenG has cocompact non-arithmeticExpand
Covering theory for graphs of groups
Abstract A tree action ( G, X ), consisting of a group G acting on a tree X , is encoded by a ‘quotient graph of groups’ A = G ⧹⧹ X . We introduce here the appropriate notion of morphism A → A′ = GExpand
Schottky Groups and Mumford Curves
Discontinuous groups.- Mumford curves via automorphic forms.- The geometry of mumford curves.- Totally split curves and universal coverings.- Analytic reductions of algebraic curves.- JacobianExpand
Ergodic theory, symbolic dynamics, and hyperbolic spaces
Alan F. Beardon: An introduction to hyperbolic geometry Michael Keane: Ergodic theory and subshifts of finite type Anthony Manning: Dynamics of geodesic and horocycle flows on surfaces of constantExpand
The density at infinity of a discrete group of hyperbolic motions
© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www.Expand
Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits
0. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Renewal theorems in symbolic dynamics . . . . . . . . . . . . 1. Background: Shifts, suspension flows, thermodynamicExpand
Actions quasi-convexes d'un groupe hyperbolique : flot géodésique
On etudie les actions isometriques quasi-convexes d'un groupe hyperbolique au sens de M. Gromov sur les espaces metriques simplement connexes a courbure strictement negative. A une telle action sontExpand
Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov.
0. Introduction. Considerons Γespace hyperbolique usuel H, equipe d'un point base XQ . Munissons le bord dH de H de la mέtrique visuelle obtenue en regardant dH a partir de XQ (la distance entre deuxExpand
Geometrical methods of symbolic coding
  • Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Eds T. Bedford, M. Keane and C. Series. Oxford University Press
  • 1991
Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees (London Math
  • Soc. Lecture Note Series 162). Cambridge University Press
  • 1991
...
1
2
...