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Geometriae Dedicata

, Volume 125, Issue 1, pp 63–74 | Cite as

Critical exponents for groups of isometries

  • Richard Sharp
Original Paper

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.

Keywords

CAT(−1)space Riemannian manifold Negative curvature Group of isometrics Critical exponent Amenable group 

Mathematics Subject Classifications (2000)

20F67 20F69 37C35 60B99 

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References

  1. Beardon, A.: The exponent of convergence of Poincaré series. Proc. Lond. Math. Soc. 18, 461–483 (1968)zbMATHCrossRefGoogle Scholar
  2. Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv. 56,581–598(1981)zbMATHCrossRefGoogle Scholar
  3. Brooks, R.: The bottom of the spectrum of a Riemannian covering. J. Reine Angew. Math. 357,101–114(1985)zbMATHGoogle Scholar
  4. Cohen, J.: Cogrowth and amenability in discrete groups. J. Funct. Anal. 48,301–309(1982)zbMATHCrossRefGoogle Scholar
  5. Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Progress. in Mathematics, vol. 83. Birkhauser, Boston (1990)Google Scholar
  6. Grigorchuk, R.: Symmetrical Random Walks on Discrete Groups. Multicomponent Random Systems, Adv. Probab. Related Topics, vol. 6, 285–325. Dekker, New York (1980)Google Scholar
  7. Hirsch, M., Thurston, W.: Foliated bundles, invariant measures and flat manifolds. Ann. Math. 101,369–390(1975)CrossRefGoogle Scholar
  8. Kaimanovich, V., Vershik, A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11,457–490(1983)zbMATHGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. Ledrappier, F., Sarig, O.: Invariant measures for the horocycle flow on periodic hyperbolic surfaces. to appear, Israel J. MathGoogle Scholar
  11. 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)Google Scholar
  12. Ortner, R., Woess, W.: Non-backtracking random walks and cogrowth of graphs. to appear Can. J. MathGoogle Scholar
  13. Pollicott, M., Sharp, R.: Comparison theorems and orbit counting in hyperbolic geometry. Trans. Amr. Math. Soc. 350,473–499(1998)zbMATHCrossRefGoogle Scholar
  14. 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. 229–240Google Scholar
  15. Rees, M.: Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergodic Theory Dyn. Syst. 1,107–133(1981)zbMATHGoogle Scholar
  16. 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)Google Scholar
  17. Woess, W.: Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41,363–370(1983)zbMATHGoogle Scholar
  18. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics,vol 138. Cambridge University Press Cambridge (2000)zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterU.K.

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