Furstenberg entropy realizations for virtually free groups and lamplighter groups
Let (G,µ) be a discrete group with a generating probability measure. Nevo showed that if G has Kazhdan's property (T), then there exists ɛ > 0 such that the Furstenberg entropy of any (G,µ)-stationary ergodic space is either 0 or larger than ɛ. Virtually free groups, such as SL_2(ℤ), do not have property (T), and neither do their extensions, such as surface groups. For virtually free groups, we construct stationary actions with arbitrarily small, positive entropy. The construction involves building and lifting spaces of lamplighter groups. For some classical lamplighter gropus, these spaces realize a dense set of entropies between 0 and the Poisson boundary entropy.
© 2015 Hebrew University Magnes Press. Received February 21, 2013 and in revised form April 15, 2013. First Online: 01 July 2015. Y. Hartman is supported by the European Research Council, grant 239885. O. Tamuz is supported by ISF grant 1300/08, and is a recipient of the Google Europe Fellowship in Social Computing. This research is supported in part by this Google Fellowship. We are grateful to Yuri Lima for many useful discussions. We also thank Uri Bader and Amos Nevo for motivating conversations, and Lewis Bowen and Vadim Kaimanovich for commenting on the first draft of this paper. Finally, we thank the referee for many insightful comments and suggestions.
Submitted - 1210.5897.pdf