## 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 $$G/H$$.

This paper is available as a pdf (296kB) file.

 Wednesday, September 3, 2014