What type of study is this?

This is a Quantitative Study study.

October 3, 2025Open Access

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

Key Points

Lower bounds on dimension where groups can act without a global fixed point were established.
The study shows fixed-point sets for certain subgroups intersect under specific configurations.
A bootstrapping approach was used to prove fixed-point existence for larger subgroups.
For automorphism groups of free groups, the lower bound achieved was FixDim(Aut(F_n)) ≥ floor(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 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 FixDim () +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 FixDim (Aut (Fₙ) ) 2n/3.

AIに質問

Bookmark

View Full Paper

Cite This Study

Martin R. Bridson (Thu,) studied this question.

synapsesocial.com/papers/68e03501f0e39f13e7fa3950 https://doi.org/https://doi.org/10.48550/arxiv.2505.00943

AIに質問

Bookmark

View Full Paper