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Tuesday Seminar on Topology
Seminar information archive ～05/02｜Next seminar｜Future seminars 05/03～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
2012/02/21
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Masato Mimura (The University of Tokyo)
Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)
Masato Mimura (The University of Tokyo)
Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)
[ Abstract ]
Mapping class groups (MCG's), of compact oriented surfaces (possibly
with punctures), have many mysterious features: they behave not only
like higher rank lattices (namely, irreducible lattices in higher rank
algebraic groups); but also like rank one lattices. The following
theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one
phenomenon for MCG's: "every group homomorphism from higher rank
lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into
MCG's has finite image."
In this talk, we show a generalization of the superrigidity above, to
the case where higher rank lattices are replaced with some
(non-arithmetic) matrix groups over general rings. Our main example of
such groups is called the "universal lattice", that is, the special
linear group over commutative finitely generated polynomial rings over
integers, (such as SL(3,Z[x])). To prove this, we introduce the notion
of "property (TT)/T" for groups, which is a strengthening of Kazhdan's
property (T).
We will explain these properties and relations to ordinary and bounded
cohomology of groups (with twisted unitary coefficients); and outline
the proof of our result.
Mapping class groups (MCG's), of compact oriented surfaces (possibly
with punctures), have many mysterious features: they behave not only
like higher rank lattices (namely, irreducible lattices in higher rank
algebraic groups); but also like rank one lattices. The following
theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one
phenomenon for MCG's: "every group homomorphism from higher rank
lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into
MCG's has finite image."
In this talk, we show a generalization of the superrigidity above, to
the case where higher rank lattices are replaced with some
(non-arithmetic) matrix groups over general rings. Our main example of
such groups is called the "universal lattice", that is, the special
linear group over commutative finitely generated polynomial rings over
integers, (such as SL(3,Z[x])). To prove this, we introduce the notion
of "property (TT)/T" for groups, which is a strengthening of Kazhdan's
property (T).
We will explain these properties and relations to ordinary and bounded
cohomology of groups (with twisted unitary coefficients); and outline
the proof of our result.
