Local spectral gap in the group of Euclidean isometries

We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in [BISG15]. This property has applications to strong ergodicity, the Banach-Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group o...

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Bibliographic Details
Published in:International mathematics research notices 2018-03
Main Authors: Boutonnet, Rémi, Ioana, Adrian
Format: Article
Language:English
Subjects:
Dynamical Systems
Group Theory
Mathematics
Online Access:Get full text
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Summary:We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in [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 Isom(R d) of orientation-preserving isometries of the euclidean space R d , for d ≥ 3. We prove that the translation action of a countable dense subgroup Γ on Isom(R d) 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 Isom(R d) and on work of Lindenstrauss and Varjú, [LV14].
ISSN:1073-7928
1687-0247