Commensurators and classifying spaces with virtually cyclic stabilizers

Abstract

By examining commensurators of virtually cyclic groups, we show that for each natural number $n$, any locally finite-by-virtually cyclic group of cardinality ${\aleph }_n$ admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension $n+3$. As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality ${\aleph }_n$ admits a finite dimensional classifying space with virtually cyclic stabilizers.