Title:

Embeddings of infinite groups into Banach spaces

In this thesis we build on the theory concerning the metric geometry of relatively hyperbolic and mapping class groups, especially with respect to the difficulty of embedding such groups into Banach spaces. In Chapter 3 (joint with Alessandro Sisto) we construct simple embeddings of closed graph manifold groups into a product of three metric trees, answering positively a conjecture of Smirnov concerning the AssouadNagata dimension of such spaces. Consequently, we obtain optimal embeddings of such spaces into certain Banach spaces. The ideas here have been extended to other closed threemanifolds and to higher dimensional analogues of graph manifolds. In Chapter 4 we give an explicit method of embedding relatively hyperbolic groups into certain Banach spaces, which yields optimal bounds on the compression exponent of such groups relative to their peripheral subgroups. From this we deduce that the fundamental group of every closed threemanifold has Hilbert compression exponent one. In Chapter 5 we prove that relatively hyperbolic spaces with a treegraded quasiisometry representative can be characterised by a relative version of Manning's bottleneck property. This applies to the BestvinaBrombergFujiwara quasitrees of spaces, yielding an embedding of each mapping class group of a closed surface into a finite product of simplicial trees. From this we obtain explicit embeddings of mapping class groups into certain Banach spaces and deduce that these groups have finite AssouadNagata dimension. It also applies to relatively hyperbolic groups, proving that such groups have finite AssouadNagata dimension if and only if each peripheral subgroup does.
