Dismantlability of weakly systolic complexes and applications

TL;DR

This paper proves a fixed point theorem for finite groups acting on weakly systolic complexes, using dismantlability, and derives new results on classifying spaces and conjugacy classes of finite subgroups.

Contribution

It introduces a dismantlability approach to weakly systolic complexes, establishing fixed point properties and characterizations that extend to systolic groups.

Findings

Weakly bridged graphs are dismantlable.

Fixed point theorem for finite group actions on weakly systolic complexes.

New results on classifying spaces and conjugacy classes of finite subgroups.

Abstract

The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups. The fixed point theorem is proved by using a graph-theoretical tool - dismantlability. In particular we show that 1-skeleta of weakly systolic complexes, i.e. weakly bridged graphs, are dismantlable. On the way we show numerous characterizations of weakly bridged graphs and weakly systolic complexes.