数理解析研究所講究録 1223 巻 2001 年 90-92

90

Bounded cohomology of subgroups of
mapping class groups
東北大学 藤原耕二 (Koji Fujiwara)

Math Institute, Tohoku University
Igive atalk on ajoint work with Mladen Bestvina [3].
When is adiscrete group, aquasi-homornorphism onsuch that$G$$C_{r}$

is afunction$h:Garrow \mathrm{R}$.

$\Delta(h):=\sup_{\gamma_{1},\gamma_{2}\in G}|h(\gamma_{1}\gamma_{2})-h(\gamma_{1})-h(\gamma_{2})|<\infty$

The number
is the defect of . We denote by $QH(G)$ thc vector space of all quasi-homomorphisms
modulo the subspace ofbounded functions, and by
the vector space of all quasi-homomorphisms
modulo the subspace of functions within uniform distance to ahomomorphism.
Let be acompact orientable surface of genus and punctures. We
consider the associated mapping class group Mod(S) of . This group
acts on the curve complex $X$ of defined by Harvey [7] and successfully used in the study of mapping class groups by Harer [6], [5]. For
our purposes, we will restrict to the 1-skeleton of Harvey’s complex, sothat $X$ is agraph whose vertices are isotopy classes of essential, nonparallel, nonperipheral, pairwise disjoint simple closed curves in (alsocalled curve systems) and two distinct vertices are joined by an edge if
the corresponding curve systems can be realized simultaneously by pairwise disjoint curves. In certain sporadic cases $X$ as defined above is0-dimensional(this happens when there are no curve systems consistingof two curves, i.e. when $g=0$ , $p\leq 4$ and when $g=1$ , $p\leq 1$ ). In
the theorem below these cases are excluded. The mapping class group
Mod(S) acts on $X$ by $f\cdot a=f(a)$ .
H. Masur and Y. Minsky proved the following remarkable result$\Delta(h)$$h$$Garrow \mathrm{R}$$\overline{QH}(G)$$Garrow \mathrm{R}$

$S$

$g$

$p$

$S$

$S$

$S$

91
Theorem 1 j9j The cur ve complex X is -hyperbolic. An element ofMod(S) acts hyperbolically on X and only if it is pseudO-Anosov$S$$i\ovalbox{\tt\small REJECT}$\yenUsing their result, we show the following theorem. H. Endo and D.
Kotschick [2] have shown using 4-manifold topology and Seiberg-Witten
invariants that $\overline{QH}(Mod(S))\neq 0$ when is hyperbolic.
$S$

Theorem 2[3] Let
be a subgroup of Mod(S) which is not virtually$\dim\overline{QH}(G)=\infty$
abelian. Then.
$G$

The following is aversion of superrigidity for mapping class groups. It
was conjectured by. Ivanov and proved by Kaimanovich and Masur[8] in the case when the image group contains independent pseud0-Anosov
homeomorphisms and it was extended to the general case by Farb andMasur [4] using the classification of subgroups of Mod(S) as above. Our
proof is different in that it does not use random walks on mapping classgroups, but instead uses the work of M. Burger and N. Monod [1] onbounded cohomology of lattices.
$\mathrm{N}.\mathrm{V}$Corollary 3[3] Let be an irreducible lattice in a connected semi-simple
Lie group with no compact factors and of $rank>1$ . Then every homOmorphism $\Gammaarrow Mod(S)$ has finite image.
$\Gamma$$G$$g_{\vee}’\not\equiv \mathrm{X}\mathrm{f}\mathrm{f}\mathrm{l}$[1] M. Burger and N. Monod. Bounded cohomology of lattices in higher
rank Lie groups. J. $Eur$. Math. Soc. (JEMS), 1(2) $:199-235$ ,1999.
[2] H. Endo and D. Kotschick. Bounded cohomology and non-uniform
perfection of mapping class groups.
Preprint, October 2000,
http:.math.ucdav is.
.
$//\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{t}$$\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{G}\mathrm{T}/0010300$[3] K. Fujiwara and M. Bestvina. Bounded cohomology of subgroups ofmapping class groups.
Preprint, December 2000,
http:.math.ucdavi$//\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{t}$$\mathrm{s}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{G}\mathrm{T}/0012115$[4] Benson Farb and Howard Masur. Superrigidity and mapping classgroups. Topology, 37(6):1169-1176,1998

92[5] John L. Harer. Stability of the homology of the mapping class groups
1985.
of orientable surfaces. Ann. of Math. (2), 121(2)$\ovalbox{\tt\small REJECT} 215-249,$[6] John L. Harer. The virtual cohomological dimension of the mappingclass group of an orientable surface. Invent. Math., 84(1):157-176,1986.
[7] W. J. Harvey. Boundary structure of the modular group. In Riemannsurfaces and related topics: Proceedings of the 1978 Stony Brook
Conference (State Univ. New York, Stony Brook, $N$. Y., 1978), pages
245-251, Princeton, N.J., 1981. Princeton Univ. Press.
[8] Vadim A. Kaimanovich and Howard Masur. The Poisson boundaryof the mapping class group. Invent. Math., 125(2):221-264, 1996.

[9] Howard A. Masur and Yair N. Minsky. Geometry of the complex ofcurves. I. Hyperbolicity. Invent. Math., 138(1):103-149, 1996.