Ergodic theorems for coset spaces

Michael Bjorklund, Alexander Fish

Abstract

We study in this paper the validity of the mean ergodic theorem along left Følner sequences in a countable amenable group \(G\). Although the weak ergodic theorem always holds along any left Følner sequence in \(G\), we provide examples where the mean ergodic theorem fails in quite dramatic ways. On the other hand, when \(G\) does not admit any ICC quotients, e.g. if \(G\) is virtually nilpotent, we prove that the mean ergodic theorem does indeed hold along any left Følner sequence. In the case when a unitary representation of any countable amenable group \(G\) is induced from a "sufficiently thin" subgroup, we prove that the mean ergodic theorem holds along any left Følner sequence in \(G\) for this representation. Furthermore, we show that every countable (infinite) amenable group \(L\) embeds into a countable group \(G\) which admits a unitary representation with the property that for any left Følner sequence \((F_n)\) in \(L\), there exists a sequence \((s_n)\) in \(G\) such that the mean (but not the weak) ergodic theorem fails for this representation along the sequence \((F_ns_n)\). Finally, we provide examples of countable (not necessarily amenable) groups \(G\) with proper, infinite-index subgroups \(H\), so that the pointwise ergodic theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset \(G/H\).