Group actions and Helly's theorem

2008/06/10 by Benson Farb, Farb, Benson · 4 citations
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory #Representation Theory (math.RT) #math.GR #math.RT

paper · pdf · doi:10.48550/arxiv.0806.1692

17 pages, no figures

arxiv created 2008/06/10 · openalex publication_date 2008/06/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encode d in a property that we introduce called ``property \FA_r'', which reduces to Serre's property \FA when r=1. The method applies to S-arithmetic groups in higher \Q-rank, to simplex reflection groups (including some non-arithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example \SL_n(\Z[x₁,..., x_d]), n>2).

Citations

• Harmonic maps into singular spaces and p -adic superrigidity for lattices in groups of rank one

Cited by

• Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks
• Higher property T and below-rank phenomena of lattices
• Simple groups with strong fixed-point properties
• Helly-type theorems, CAT(0) spaces, and actions of automorphism groups of free groups

Related

• Group actions on injective spaces and Helly graphs
• Crystallographic Helly Groups
• Actions of higher rank groups on uniformly convex Banach spaces
• Helly-type theorems, CAT(0) spaces, and actions of automorphism groups of free groups
• Locally elliptic actions, torsion groups, and nonpositively curved spaces