Volume 18, issue 4 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 6, 2483–2999
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
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(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).

Keywords
arc complex, sphere complex, disc complex, Nielsen realisation, dismantlability
Mathematical Subject Classification 2010
Primary: 20F65
References
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