Local spectral gap in the group of Euclidean isometries

2017/02/21 by Rémi Boutonnet, Boutonnet, Rémi, Adrian Ioana +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #math.DS #math.GR

paper · pdf · doi:10.48550/arxiv.1702.06323

14 pages

arxiv created 2018/02/13 · arxiv updated 2018/02/14

Abstract

We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in \citeBISG15. 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(\mathbb 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, \citeLV14.

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