Abstract
Let Γ be a convex co-compact group of isometries of a CAT(−1) space X and let Γ0 be a normal subgroup of Γ. We show that, provided Γ is a free group, a sufficient condition for Γ and Γ0 to have the same critical exponent is that Γ / Γ0 is amenable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beardon, A.: The exponent of convergence of Poincaré series. Proc. Lond. Math. Soc. 18, 461–483 (1968)
Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv. 56,581–598(1981)
Brooks, R.: The bottom of the spectrum of a Riemannian covering. J. Reine Angew. Math. 357,101–114(1985)
Cohen, J.: Cogrowth and amenability in discrete groups. J. Funct. Anal. 48,301–309(1982)
Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Progress. in Mathematics, vol. 83. Birkhauser, Boston (1990)
Grigorchuk, R.: Symmetrical Random Walks on Discrete Groups. Multicomponent Random Systems, Adv. Probab. Related Topics, vol. 6, 285–325. Dekker, New York (1980)
Hirsch, M., Thurston, W.: Foliated bundles, invariant measures and flat manifolds. Ann. Math. 101,369–390(1975)
Kaimanovich, V., Vershik, A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11,457–490(1983)
Lalley, S.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math. 163,1–55(1989)
Ledrappier, F., Sarig, O.: Invariant measures for the horocycle flow on periodic hyperbolic surfaces. to appear, Israel J. Math
Ol’shanskii, A.Yu.: On a geometric method in the combinatorial group theory. In: Proceedings of the International Congress of Mathematicians, vol. 1, pp. 415–424(Warsaw, 1983). PWN Warsaw (1984)
Ortner, R., Woess, W.: Non-backtracking random walks and cogrowth of graphs. to appear Can. J. Math
Pollicott, M., Sharp, R.: Comparison theorems and orbit counting in hyperbolic geometry. Trans. Amr. Math. Soc. 350,473–499(1998)
Pollicott, M., Sharp, R.: Poincaré Series and Comparison Theorems for Variable Negative Curvature Topology, Ergodic Theory, Real Algebraic Geometry. Amer. Math. Soc. Transl. Ser. 2,vol. 202. American Mathematical Society Providence, RI 2001,pp. 229–240
Rees, M.: Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergodic Theory Dyn. Syst. 1,107–133(1981)
Roblin, T.: Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative. Israel J. Math. 147,333–357(2005)
Woess, W.: Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41,363–370(1983)
Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics,vol 138. Cambridge University Press Cambridge (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sharp, R. Critical exponents for groups of isometries. Geom Dedicata 125, 63–74 (2007). https://doi.org/10.1007/s10711-007-9137-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-007-9137-9