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

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 $\mathrm{CAT}(0)$ space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin–Osajda–Przytycki (2022). 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.