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

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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
##### Publication
Received: 19 June 2012
Accepted: 11 January 2014
Published: 2 October 2014
Proposed: Martin R Bridson
Seconded: Danny Calegari, Benson Farb
##### Authors
 Sebastian Hensel Department of Mathematics University of Chicago 5734 S University Ave. Chicago, IL 60637 USA Damian Osajda Instytut Matematyczny Uniwersytet Wrocławski pl. Grunwaldzki 2/4 50-384 Wrocław Poland and Universität Wien Fakultät für Mathematik Oskar-Morgenstern-Platz 1 1090 Wien Austria Piotr Przytycki McGill University The Department of Mathematics and Statistics Burnside Hall, Room 1005 805 Sherbrooke Street West Montreal, QC, H3A 0B9 Canada and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warsaw Poland