Realisation and dismantlability

TL;DR

This paper investigates the fixed point properties of various groups acting on geometric complexes related to surfaces and free groups, establishing realisation theorems and classifying spaces.

Contribution

It proves fixed point results for finite subgroups and realisation theorems, including the Nielsen Realisation Problem, for groups acting on arc, disc, and sphere graphs.

Findings

Finite subgroups fix filling or simple cliques in the graphs.

Infinite subgroups have either empty or contractible fixed point sets.

Spines of complexes serve as classifying spaces for proper group actions.

Abstract

We study dismantling properties of the arc, disc and sphere graphs. We prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out(F_n) fixes a filling (resp. simple) clique in the appropriate graph. We deduce realisation theorems, in particular the Nielsen Realisation Problem in the case of a nonempty set of punctures. We also prove that infinite H have either empty or contractible fixed point sets in the corresponding complexes. Furthermore, we show that their spines are classifying spaces for proper actions for mapping class groups and Out(F_n).