Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T09:52:02.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures on the space of horofunctions of a word hyperbolic group

Published online by Cambridge University Press:  21 May 2009

LEWIS BOWEN
LEWIS BOWEN*
Affiliation:
Mathematics Department, University of Hawaii, Mānoa, Hawaii (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a natural equivalence relation on the space ℋ0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen–Marcus theorem. Furthermore, if η is such a measure and G acts on a probability space (X,μ) by measure-preserving transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on ℋ0×X. This is comparable to a special case of the Howe–Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 97 - 129
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, S.. Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups. Topology 33(4) (1994), 765–783.CrossRefGoogle Scholar
[2]Aldous, D. and Lyons, R.. Processes on unimodular random networks. Electron. J. Probab. 12 (2007), 1454–1508.CrossRefGoogle Scholar
[3]Benjamini, I., Lyons, R., Peres, Y. and Schramm, O.. Group-invariant percolation on graphs. Geom. Funct. Anal. 9(1) (1999), 29–66.CrossRefGoogle Scholar
[4]Bowen, R. and Marcus, B.. Unique ergodicity for horocycle foliations. Israel J. Math. 26(1) (1977), 43–67.CrossRefGoogle Scholar
[5]Burger, M. and Monod, N.. Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12(2) (2002), 219–280.CrossRefGoogle Scholar
[6]Bufetov, A. I.. Convergence of spherical averages for actions of free groups. Ann. of Math. (2) 155(3) (2002), 929–944.CrossRefGoogle Scholar
[7]Coornaert, M.. Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. Math. 159(2) (1993), 241–270.CrossRefGoogle Scholar
[8]Coornaert, M. and Papadopoulos, A.. Horofunctions and symbolic dynamics on Gromov hyperbolic groups. Glasg. Math. J. 43(3) (2001), 425–456.CrossRefGoogle Scholar
[9]Coornaert, M. and Papadopoulos, A.. Symbolic coding for the geodesic flow associated to a word hyperbolic group. Manuscr. Math. 109(4) (2002), 465–492.CrossRefGoogle Scholar
[10]Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P.. Word Processing in Groups. Jones and Bartlett, Boston, MA, 1992, xii+330 pp.CrossRefGoogle Scholar
[11]Feldman, J. and Moore, C. C.. Ergodic equivalence relations and von Neumann algebras I. Trans. Am. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
[12]Fujiwara, K. and Nevo, A.. Maximal and pointwise ergodic theorems for word-hyperbolic groups. Ergod. Th. Dyn. Sys. 18(4) (1998), 843–858.CrossRefGoogle Scholar
[13] E. Ghys, P. de la Harpe (eds.) Sur les Groupes Hyperboliques d’apres Mikhael Gromov (Progress in Mathematics, 83). Birkhauser, Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988.Google Scholar
[14]Gorodnik, A. and Nevo, A.. The ergodic theory of lattice subgroups. math.DS/0605596.Google Scholar
[15]Gromov, M.. Hyperbolic Groups. Essays in Group Theory (Mathematical Sciences Research Institute Publication, 8). Springer, New York, 1987, pp. 75–263.CrossRefGoogle Scholar
[16]Guivarc’h, Y.. Généralisation d’un théorème de von Neumann. C. R. Acad. Sci. Paris Ser. A–B 268 (1969), A1020–A1023.Google Scholar
[17]Häggström, O.. Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25(3) (1997), 1423–1436.CrossRefGoogle Scholar
[18]Kaimanovich, V. A.. The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152(3) (2000), 659–692.CrossRefGoogle Scholar
[19]Kaimanovich, V. A.. Double ergodicity of the Poisson boundary and applications to bounded cohomology. Geom. Funct. Anal. 13(4) (2003), 852–861.CrossRefGoogle Scholar
[20]Monod, N.. Continuous Bounded Cohomology of Locally Compact Groups (Lecture Notes in Mathematics, 1758). Springer, Berlin, 2001, x+214 pp.CrossRefGoogle Scholar
[21]Nevo, A.. Pointwise Ergodic Theorems for Actions of Groups (Handbook of Dynamical Systems, 1B). Elsevier B. V., Amsterdam, 2006, pp. 871–982.Google Scholar
[22]Nevo, A.. On the ball averaging problem in ergodic theory. Preprint.Google Scholar
[23]Nevo, A. and Stein, E. M.. A generalization of Birkhoff’s pointwise ergodic theorem. Acta Math. 173(1) (1994), 135–154.CrossRefGoogle Scholar