Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Growth of relatively hyperbolic groups


Author: Xiangdong Xie
Journal: Proc. Amer. Math. Soc. 135 (2007), 695-704
MSC (2000): Primary 20F65
DOI: https://doi.org/10.1090/S0002-9939-06-08537-6
Published electronically: September 15, 2006
MathSciNet review: 2262865
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a finitely generated group that is hyperbolic relative to a collection of proper subgroups either is virtually cyclic or has uniform exponential growth.


References [Enhancements On Off] (What's this?)

  • [AN] R. Alperin, G. Noskov, Nonvanishing of algebraic entropy for geometrically finite groups of isometries of Hadamard manifolds, preprint(2004).
  • [B] B. Bowditch, Relatively hyperbolic groups, preprint(1999).
  • [D] C. Drutu, Quasi-isometric rigidity of groups, preprint(2004).
  • [EMO] A. Eskin, S. Mozes, H. Oh, Uniform exponential growth for linear groups, Int. Math. Res. Not. 2002, no. 31, 1675-1683. MR 1916428 (2003g:20074)
  • [F] B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810-840. MR 1650094 (99j:20043)
  • [G] M. Gromov, Hyperbolic Groups, in `` Essays in Group Theory" (ed. S. Gersten) M.S.R.I. Publications No.8, Springer-Verlag(1987) 75-263. MR 0919829 (89e:20070)
  • [GD] R. Grigorchuk, P. de la Harpe, One-relator groups of exponential growth have uniformly exponential growth, translation in Math. Notes 69 (2001), no. 3-4, 575-577. MR 1846003 (2002b:20041)
  • [GhD] E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics 83.
  • [HK] G. Hruska, B. Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005), 1501-1538. MR 2175151
  • [K] M. Koubi, Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 5, 1441-1453. MR 1662255 (99m:20080)
  • [O1] D. Osin, Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100pp. MR 2182268
  • [O2] D. Osin, The entropy of solvable groups, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 907-918. MR 1992670 (2004f:20065)
  • [O3] D. Osin, Weakly amenable groups, Contemp. Math., 298 (2002), 105-113.
  • [W] J. Wilson, On exponential growth and uniformly exponential growth for groups, Invent. Math. 155 (2004), no. 2, 287-303. MR 2031429 (2004k:20085)
  • [Y] A. Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004), 41-89. MR 2039323 (2005e:20064)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20F65

Retrieve articles in all journals with MSC (2000): 20F65


Additional Information

Xiangdong Xie
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221
Address at time of publication: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
Email: xxie@math.uc.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08537-6
Keywords: Exponential growth, uniform exponential growth, relatively hyperbolic groups, geometrically finite groups.
Received by editor(s): April 10, 2005
Received by editor(s) in revised form: October 18, 2005
Published electronically: September 15, 2006
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society