Bounded cohomology of subgroups of mapping class groups

We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb-Kaimanovich-Masur rigidity theor

ISSN 1364-0380(on line)1465-3060(printed)69G eometry &T opology

G G G G G G G G G G G G G G G T T T

T T T T T T T T T T T T Volume 6(2002)69–89

Published:1March 2002Bounded cohomology of subgroups of

mapping class groups

Mladen Bestvina

Koji Fujiwara

Mathematics Department,University of Utah

155South 1400East,JWB 233

Salt Lake City,UT 84112,USA

and

Mathematics Institute,Tohoku University

Sendai,980-8578,Japan

Email:bestvina@math.utah.edu and fujiwara@math.tohoku.ac.jp

Abstract

We show that every subgroup of the mapping class group MCG (S )of a com-pact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology.As an application,we give another proof of the Farb–Kaimanovich–Masur rigidity theorem that states that MCG (S )does not con-tain a higher rank lattice as a subgroup.

AMS Classification numbers

Primary:57M07,57N05Secondary:57M99

Keywords Bounded cohomology,mapping class groups,hyperbolic groups Proposed:Joan Birman

Received:15December 2000Seconded:Dieter Kotschick,Steven Ferry

Revised:28February 2002c G eometry &T opology P ublications

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