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- Title
Local Spectral Gap in the Group of Euclidean Isometries.
- Authors
Boutonnet, Rémi; Ioana, Adrian
- 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 ].
- Subjects
COMPACT groups; MATHEMATICAL equivalence; ROTATIONAL motion
- Publication
IMRN: International Mathematics Research Notices, 2020, Vol 2020, Issue 2, p466
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rny029