Local Spectral Gap in the Group of Euclidean Isometries

• Published in: International mathematics research notices 2020-01, Vol.2020 (2), p.466-486
• Main Authors: Boutonnet, Rémi, Ioana, Adrian
• Format: Article
• Language: English
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rny029

Abstract We provide new examples of translation actions on locally compact groups with the “local spectral gap property” introduced in [5]. 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 $\operatorname{Isom}\left (\mathbb{R}^{d}\right )$ of orientation-preserving isometries of the Euclidean space $\mathbb{R}^{d}$, for d ≥ 3. We prove that the translation action of a countable dense subgroup Γ on Isom$\left (\mathbb R^{d}\right )$ has local spectral gap, whenever the translation action of the rotation projection of Γ on SO(d) has spectral gap. Our proof relies on the amenability of $\operatorname{Isom}\left (\mathbb{R}^{d}\right )$ and on work of Lindenstrauss and Varjú [12].