Fibred coarse embeddability of box spaces and proper isometric affine\n actions on L^p spaces

2015/08/20 by Sylvain Arnt, Arnt, Sylvain
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1508.05033

openalex publication_date 2015/08/20 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We show the necessary part of the following theorem : a finitely generated,\nresidually finite group has property PL^p (i.e. it admits a proper isometric\naffine action on some L^p space) if, and only if, one (or equivalently, all)\nof its box spaces admits a fibred coarse embedding into some L^p space. We\nalso prove that coarse embeddability of a box space of a group into a L^p\nspace implies property PL^p for this group.\n

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• There is no equivariant coarse embedding of L_p into ℓ_p
• Locally compact groups with every isometric action bounded or proper
• Hyperbolic groups admit proper affine isometric actions on l^p-spaces
• Spaces with labelled partitions and isometric affine actions on Banach spaces