Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T04:12:02.083Z Has data issue: false hasContentIssue false
  • English
  • Français

A subgroup formula for f-invariant entropy

Published online by Cambridge University Press:  30 November 2012

BRANDON SEWARD
BRANDON SEWARD*
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by L. Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. We show that the ratio of these entropies equals the index of the subgroup. This generalizes a well-known formula for the Kolmogorov–Sinai entropy of amenable group actions. We then extend the definition of f-invariant entropy to actions of finitely generated virtually free groups. We also obtain a numerical virtual measure conjugacy invariant for actions of finitely generated virtually free groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 263 - 298
Copyright
Copyright © 2012 Cambridge University Press 

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]Bollobás, B.. Modern Graph Theory. Springer, New York, 1998.Google Scholar
[2]Bowen, L.. A new measure conjugacy invariant for actions of free groups. Ann. of Math. (2) 171(2) (2010), 1387–1400.Google Scholar
[3]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217–245.Google Scholar
[4]Bowen, L.. The ergodic theory of free group actions: entropy and the f-invariant. Groups Geom. Dyn. 4(3) (2010), 419–432.Google Scholar
[5]Bowen, L.. Nonabelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula. Ergod. Th. & Dynam. Sys. 30(6) (2010), 1629–1663.Google Scholar
[6]Bowen, L.. Weak isomorphisms between Bernoulli shifts. Israel J. Math. 183(1) (2011), 93–102.Google Scholar
[7]Bowen, L.. Sofic entropy and amenable groups. Ergod. Th. & Dynam. Sys. 32(2) (2012), 427–466.Google Scholar
[8]Bowen, L. and Gutman, Y.. A Juzvinskii addition theorem for finitely generated free group actions.Ergod. Th. & Dynam. Sys. to appear, doi:10.1017/etds.2012.126.Google Scholar
[9]Chung, N. P.. The variational principle of topological pressures for actions of sofic groups, Ergod. Th. & Dynam. Sys. to appear, Preprint, http://arxiv.org/abs/1110.0699.Google Scholar
[10]Danilenko, A. I.. Entropy theory from the orbital point of view. Monatsh. Math. 134 (2001), 121–141.Google Scholar
[11]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.Google Scholar
[12]Hall, M.. Coset representations in free groups. Trans. Amer. Math. Soc. 67(2) (1949), 421–432.CrossRefGoogle Scholar
[13]de la Harpe, P.. Topics in Geometric Group Theory, 1st edn. University of Chicago Press, Chicago, 2000.Google Scholar
[14]Karrass, A., Pietrowski, A. and Solitar, D.. Finite and infinite cyclic extensions of free groups. J. Austral. Math. Soc. 16 (1973), 458–466.Google Scholar
[15]Kerr, D.. Sofic measure entropy via finite partitions, Groups Geom. Dyn. to appear, Preprint, http://arxiv.org/abs/1111.1345.Google Scholar
[16]Kerr, D. and Li, H.. Soficity, amenability, and dynamical entropy, Amer. J. Math., to appear.Google Scholar
[17]Kerr, D. and Li, H.. Entropy and the variational principle for actions of sofic groups. Invent. Math. 186 (2011), 501–558.Google Scholar
[18]Kerr, D. and Li, H.. Bernoulli actions and infinite entropy. Groups Geom. Dyn. 5 (2011), 663–672.Google Scholar
[19]Kolmogorov, A. N.. New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces. Dokl. Akad. Nauk 119(5) (1958), 861–864 (in Russian).Google Scholar
[20]Kolmogorov, A. N.. Entropy per unit time as a metric invariant for automorphisms. Dokl. Akad. Nauk 124 (1959), 754–755 (in Russian).Google Scholar
[21]Lyndon, R. and Schupp, P.. Combinatorial Group Theory. Springer, New York, 1977.Google Scholar
[22]Magnus, W., Karrass, A. and Solitar, D.. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, revised edition. Dover, New York, 1976.Google Scholar
[23]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.Google Scholar
[24]Ornstein, D.. Two Bernoulli shifts with infinite entropy are isomorphic. Adv. Math. 5 (1970), 339–348.Google Scholar
[25]Zhang, G. H.. Local variational principle concerning entropy of a sofic group action. Preprint, http://arxiv.org/abs/1109.3244.Google Scholar
[26]Zhou, X. and Chen, E.. The variational principle of local pressure for actions of sofic group. J. Funct. Anal. 262(4) (2012), 1954–1985.Google Scholar