Torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point

2024/04/30 by Hiroyasu Izeki, Anders Karlsson, Izeki, Hiroyasu +1 · 1 citation
Mathematics · #20F65 #20K10 #58E20 #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2404.19273

openalex publication_date 2024/04/30 · openalex created_date 2024/05/03 · openalex updated_date 2026/07/28

Abstract

We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin, Osajda, and Przytycki. It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks.

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