Local spectral gap in the group of Euclidean isometries

Abstract

We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in \cite{BISG15}. This property has applications to strong ergodicity, the Banach-Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group $\text{Isom}({\mathbb{R}}^d)$ of orientation-preserving isometries of the euclidean space ${\mathbb{R}}^d$, for $d\ge 3$. We prove that the translation action of a countable dense subgroup $\Gamma$ on Isom$({\mathbb{R}}^d)$ has local spectral gap, whenever the translation action of the rotation projection of $\Gamma$ on $\text{SO}(d)$ has spectral gap. Our proof relies on the amenability of $\text{Isom}({\mathbb{R}}^d)$ and on work of Lindenstrauss and Varjú, \cite{LV14}.

arXiv comment14 pages