Jeffrey Brock |
Asymptotics
of Weil-Peterson geodesics on Teichmuller space and the geometry of
hyperbolic 3-manifolds Course outline (4 lectures): We will discuss a fruitful but complex analogy between the internal geometry of hyperbolic 3-manifolds and geodesics in the Weil-Petersson metric on Teichmu¼ller space and its quotient moduli space. Features of this connection include comparisons between convex core volume and Weil-Petersson distance, volume spectra for fibered 3-manifolds and length spectra of moduli space, ending laminations for geodesic rays, and bounded geometry and bounded combinatorics. Each feature points to a deeper connection, but each discussion will conclude a problematizing divergence between behaviors on each side of the discussion. We will conclude with a discussion of open problems in the field and suggestions for how we should reconcile these perspectives. |

Hamish Short |
Limit
groups Course outline (4 lectures): This course aims to give an introduction to limit groups, a class of finitely presented groups introduced by Kharlampovich-Miasnikov and Sela in their work on the Tarski conjecture. The class includes finitely generated free and free abelian groups, compact surface groups and many others. There are several different ways to define limit groups, and I shall try to describe four different points of view -- fully residually free groups, stable kernals, limits of marked free groups, tower groups and others. I shall try to establish many of the elementary properties of this extremely interesting class. Most of the methods come from "geometric group theory", and the aim is to be elementary: the most difficult results will not be proved. I shall suppose some familiarity with free groups, finite presentations, Cayley graphs. |

Juan Souto |
Mapping
class groups Course outline (4 lectures): The goal of these lectures is to describe some rigidity features present in the mapping class group $\Map(X)$ of a surface $X$; this is the group of isotopy classes of diffeomorphisms of $X$. After discussing a few basic facts on $\Map(X)$ I will concentrate on two different instances of rigidity: differential topological rigidity and rigidity of homomorphisms between mapping class groups. Differential topological rigidity: I will give a proof of Morita's non-lifting theorem which asserts that the homomorphism $\Diff(X)\to\Map(X)$ does not have an right inverse. In more concrete terms, Morita's theorem says that there is no coherent way of representing isotopy classes of diffeomorphisms of $X$ by actual diffeomorphisms. Another interpretation of Morita's theorem is that the mapping class group does not act smoothly on $X$ in the "homotopically desirable way". It is a theorem of Markovic that there is also no such action by homeomorphisms. At first one may think that this just reflects the fact that diffeomorphisms and homeomorphisms of surfaces are more or less the same thing. However, they are also more or less the same thing for 3-manifolds and there one finds a rather curious phenomenon: while the mapping class group admits a natural action on the unit tangent bundle of $X$, a 3-manifold, this action is not homotopic to a smooth action. Homomorphisms between mapping class groups. Mapping class groups share many properties with lattices in higher rank Lie groups. It is therefore an interesting problem to figure out to which extent does Margulis's superrigidity theorem hold for homomorphisms between mapping class groups. In rather empty terms, recall that Margulis's superrigidity asserts that every homomorphism between two lattices is "one of the obvious ones". I will discuss some recent, very partial results, which can be taken as evidence towards the conjecture that superrigidity holds indeed for homomorphisms between mapping class groups of surfaces of sufficiently large genus. |

The programme and list of participants will be available online.

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