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Realisation and dismantlability

Sebastian Hensel, Damian Osajda and Piotr Przytycki

Geometry & Topology 18 (2014) 2079–2126

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(Fn) 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(Fn).

arc complex, sphere complex, disc complex, Nielsen realisation, dismantlability
Mathematical Subject Classification 2010
Primary: 20F65
Received: 19 June 2012
Accepted: 11 January 2014
Published: 2 October 2014
Proposed: Martin R Bridson
Seconded: Danny Calegari, Benson Farb
Sebastian Hensel
Department of Mathematics
University of Chicago
5734 S University Ave.
Chicago, IL 60637
Damian Osajda
Instytut Matematyczny
Uniwersytet Wrocławski
pl. Grunwaldzki 2/4
50-384 Wrocław
and Universität Wien
Fakultät für Mathematik
Oskar-Morgenstern-Platz 1
1090 Wien
Piotr Przytycki
McGill University
The Department of Mathematics and Statistics
Burnside Hall, Room 1005
805 Sherbrooke Street West
Montreal, QC, H3A 0B9
and Institute of Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw