Helly-type theorems, CAT$(0)$ spaces, and actions of automorphism groups of free groups

TL;DR

This paper develops fixed-point theorems for groups acting on CAT(0) spaces using Helly-type theorems and subgroup configurations, providing bounds on the minimal dimension for fixed-point free actions, especially for automorphism groups of free groups.

Contribution

It introduces a bootstrapping technique and Helly-type theorems to establish fixed-point results and dimension bounds for group actions on CAT(0) spaces, with new bounds for automorphism groups of free groups.

Findings

Fixed points are guaranteed for certain group actions on CAT(0) spaces.

Lower bounds on the dimension for fixed-point free actions are established.

For automorphism groups of free groups, the minimal fixed-point dimension is at least loor 2n/3.

Abstract

We prove a variety of fixed-point theorems for groups acting on CAT$(0)$ spaces. Fixed points are obtained by a bootstrapping technique, whereby increasingly large subgroups are proved to have fixed points: specific configurations in the subgroup lattice of $\Gamma$ are exhibited and Helly-type theorems are developed to prove that the fixed-point sets of the subgroups in the configuration intersect. In this way, we obtain lower bounds on the smallest dimension ${\rm{FixDim}}(\Gamma)+1$ in which various groups of geometric interest can act on a complete CAT$(0)$ space without a global fixed point. For automorphism groups of free groups, we prove ${\rm{FixDim}}({\rm{Aut}}(F_n)) \ge \lfloor 2n/3\rfloor$.