#### Volume 18, issue 4 (2014)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Realisation and dismantlability

### Sebastian Hensel, Damian Osajda and Piotr Przytycki

Geometry & Topology 18 (2014) 2079–2126
##### Abstract

We prove that a finite group acting on an infinite graph with dismantling properties fixes a clique. We prove that in the flag complex spanned on such a graph the fixed point set is contractible. We study dismantling properties of the arc, disc and sphere graphs. We apply our theory to prove that any finite subgroup $H$ of the mapping class group of a surface with punctures, the handlebody group, or $Out\left({F}_{n}\right)$ fixes a filling (respectively simple) clique in the appropriate graph. We deduce some 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 $Out\left({F}_{n}\right)$.

##### Keywords
arc complex, sphere complex, disc complex, Nielsen realisation, dismantlability
Primary: 20F65