# Local spectral gap in the group of Euclidean isometries

## TL;DR

This paper demonstrates that certain translation actions of dense subgroups in Euclidean isometry groups exhibit the local spectral gap property, with implications for ergodic theory and geometric group theory.

## Contribution

It establishes the local spectral gap property for translation actions of dense subgroups in Euclidean isometry groups, linking it to spectral gap properties of rotation projections.

## Key findings

- Translation actions of dense subgroups have local spectral gap.
- The spectral gap of rotation projections implies local spectral gap.
- Proof uses amenability and prior work by Lindenstrauss and Varjú.

## 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 $d \geq 3$. 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 Varj\'u, \cite{LV14}.