Ergodic theorems for coset spaces

Journal article, 2018

We study in this paper the validity of the Mean Ergodic Theorem along left Folner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Folner sequence in G, we provide examples where the 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 the Mean Ergodic Theorem holds along any left Folner 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 show that the Mean Ergodic Theorem holds for the induced representation along any left Folner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Folner sequence (F (n) ) in L, there exists a sequence (s (n) ) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (F (n) s (n) ) in G. 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.