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

TL;DR

The paper proves that certain groups with the Liouville property acting on finite-dimensional CAT(0)-spaces must have a fixed point, advancing understanding in geometric group theory and addressing longstanding conjectures.

Contribution

It establishes fixed point properties for Liouville groups acting on finite-dimensional CAT(0)-spaces, partially confirming a conjecture and extending results to branch and simple groups.

Findings

Liouville groups with no infinite linear representations have fixed points on finite-dimensional CAT(0)-spaces

Applies to Grigorchuk's groups and other branch groups with intermediate growth

Provides partial answers to longstanding questions in geometric group theory

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.