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Journal Article

#### Ergodic theorems for coset spaces

##### External Resource

https://doi.org/10.1007/s11854-018-0032-0

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##### Fulltext (public)

arXiv:1408.6692.pdf

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##### Citation

Björklund, M., & Fish, A. (2018). Ergodic theorems for coset spaces.*
Journal d'Analyse Mathématique,* *135*(1), 85-122. doi:10.1007/s11854-018-0032-0.

Cite as: https://hdl.handle.net/21.11116/0000-0004-6208-5

##### Abstract

We study in this paper the validity of the mean ergodic theorem along \emph{left} F\o lner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left F\o lner sequence in $G$, we provide examples where the \emph{mean} ergodic theorem fails in quite dramatic ways. On the other hand, if $G$ does not admit any ICC quotients, e.g. if $G$ is virtually nilpotent, then we prove that the mean ergodic theorem does indeed hold along \emph{any} left F\o lner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a "sufficiently thin" subgroup, we prove that the mean ergodic theorem holds along any left F\o lner sequence 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 F\o lner sequence $(F_n)$ in $L$, there exists a sequence $(s_n)$ in $G$ such that the mean (but \emph{not} the weak) ergodic theorem fails for this representation along the sequence $(F_n s_n)$.

Finally, we provide examples of countable (not necessarily amenable) groups $G$ with proper, infinite-index subgroups $H$, so that the \emph{pointwise} ergodic theorem holds for averages along \emph{any} strictly increasing and nested sequence of finite subsets of the coset $G/H$.

Finally, we provide examples of countable (not necessarily amenable) groups $G$ with proper, infinite-index subgroups $H$, so that the \emph{pointwise} ergodic theorem holds for averages along \emph{any} strictly increasing and nested sequence of finite subsets of the coset $G/H$.