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Two questions on mapping class groups. (English) Zbl 1239.57004

Two main results are proved. The first one is about central extensions by \(\mathbb Z\) of the mapping class group \(M_g\) of the closed orientable surface of genus \(g\), namely those groups \(G\) admitting a central element \(a \in Z(G)\) of infinite order such that \(G/ \langle a \rangle \cong M_g\). It is proved that \(G\) is residually finite (the same conclusion holds for central extensions by \(\mathbb Z\) of the once punctured mapping class group \(M_g^1\) for \(g \geq 4\)).
The proof relies on the asymptotic faithfulness of the \(SU(2)\) TQFT representation of mapping class groups. This implies that there is a sequence of homomorphisms from \(M_g\) to some projective unitary group whose kernels have trivial intersection (this is true for \(g \geq 3\) while the case \(g = 2\) can be treated similarly by considering the \(SU(n)\) TQFT representations).
These representations lift to unitary representations of a suitable non-trivial central extension of \(M_g\), which are still asymptotically faithful. This implies that this central extension (and so also the universal one) is residually finite. Now it is simple to conclude the proof, because a non-trivial central extension of \(M_g\) by \(\mathbb Z\) embeds in the universal central extension.
The second main result gives an extimate on the maximal number \(N_g\) such that every homomorphism \(M_g \to PU(N_g)\) has finite image. Namely, the author shows that \(N_g \geq \sqrt{g + 1}\) and gives also an exponential upper bound for \(N_g\) (which depends on the Fibonacci numbers).
The proof depends on a result of [M. R. Bridson, On the dimension of CAT(0) spaces where mapping class groups act, arXiv:0908.0690] which implies that \(M_g\) has property \(FA_g\) (meaning that isometric actions of \(M_g\) on any CAT(0) \(n\)-complex have a fixed point).
The author derives the lower bound from the following: if \(\Gamma\) is a finitely generated group with property \(FA_{n^2 - 1}\) then any homomorphism \(\Gamma \to SL(n, \mathbb C)\) has finite image. This is proved by (mostly) algebraic arguments.
As a corollary every homomorphism \(M_g \to SL([\sqrt{g + 1}], \mathbb C)\) has finite image.

MSC:

57M07 Topological methods in group theory
20F36 Braid groups; Artin groups
20F38 Other groups related to topology or analysis
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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