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The center of some braid groups and the Farrell cohomology of certain pure mapping class groups

Yu Qing Chen, Henry H Glover and Craig A Jensen

Algebraic & Geometric Topology 7 (2007) 1987–2006

arXiv: 0712.3334


In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the p–primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the p–primary part of the Farrell cohomology of a few pure mapping class groups.

braid group, mapping class group, normalizer, centralizer, cohomology of groups
Mathematical Subject Classification 2000
Primary: 20F36, 20J06
Received: 14 March 2007
Revised: 16 September 2007
Accepted: 5 October 2007
Published: 26 December 2007
Yu Qing Chen
Department of Mathematics and Statistics
Wright State University
OH 45435-0001
Henry H Glover
Department of Mathematics
Ohio State University
OH 43210-1174
Craig A Jensen
Department of Mathematics
University of New Orleans
New Orleans
LA 70148