Celebratio Mathematica — Freedman

Topology in dimension 3

Works connected to V. Slava Krushkal

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M. H. Freedman and V. S. Krushkal: “Notes on ends of hyperbolic 3-manifolds” in Thirteenth annual workshop in geometric topology (Colorado Springs, CO, June 13–15, 1996). The Colorado College, 1997. Informal publication of The Colorado College, 1997.

article M. Freedman and V. Krushkal: “On the asymptotics of quantum \( \mathit{SU}(2) \) representations of mapping class groups,” Forum Math. 18 : 2 (2006), pp. 293–304. MR 2218422 Zbl 1120.57014

We investigate the rigidity and asymptotic properties of quantum \( \mathit{SU}(2) \) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum \( \mathit{SU}(2) \) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan’s property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of \( SL(2,\mathbb{Z}) \) on \( L^2(\mathbb{R}) \). As a consequence we obtain a curious analytic fact about the Fourier transform on the real line which may not have been previously observed.