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Dynamical properties of profinite actions

Published online by Cambridge University Press:  16 November 2011

MIKLÓS ABÉRT  and
GÁBOR ELEK
MIKLÓS ABÉRT
Affiliation:
Alfred Renyi Institute of the Hungarian Academy of Sciences, 13–15 Realtanoda u., 1053 Budapest, Hungary (email: [email protected])
GÁBOR ELEK
Affiliation:
Alfred Renyi Institute of the Hungarian Academy of Sciences, 13–15 Realtanoda u., 1053 Budapest, Hungary (email: [email protected])
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Abstract

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 1805 - 1835
Copyright
Copyright © Cambridge University Press 2011

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