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

TL;DR

This work develops fixed-point theorems for groups acting on complete CAT$(0)$ spaces by combining Helly-type arguments with subgroup-configuration methods. The central advance is the Ample Duplication Criterion, together with the $\Delta_n$-type and bootstrapping tools (product and wreath-product lemmas) that derive fixed points for large groups from fixed points of smaller subgroups. Concrete outcomes include lower bounds on the fixed-point dimension, notably ${\rm FixDim}({\rm Aut}(F_n)) \ge \lfloor 2n/3\rfloor$, and parallel results for ${\rm SAut}(F_n)$, ${\rm SL}(n,\mathbb{Z})$, mapping class groups, and braid groups, illustrating a unifying framework for geometric group actions. The methods have broad potential to illuminate how algebraic structure constrains action types on finite-dimensional CAT$(0)$ spaces, with implications for lattices, automorphism groups, and geometric topology.

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 $Γ$ 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 ${\rm{FixDim}}(Γ)+1$ 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 ${\rm{FixDim}}({\rm{Aut}}(F_n)) \ge \lfloor 2n/3\rfloor$.

Theorems & Definitions (102)

• Theorem A
• Theorem B
• Proposition A: Product Lemma with Torsion
• Proposition B
• Proposition C: Bootstrap Lemma
• Theorem C: Ample Duplication Criterion
• Remark 1
• Proposition 1.1
• Corollary 1.2
• Proposition 1.3