Oxford

We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. We then present a technique for obtaining `good' embeddings of such spaces into $\ell^p$ spaces, and show how results of Bestvina-Bromberg-Fujiwara and Mackay-Sisto allow us to better understand the metric geometry of such groups.

We'll present a proof of the basic Burgess bound for short character sums, following the simplified presentation of Gallagher and Montgomery.

I will discuss similarities and differences between the geometry of
relatively hyperbolic groups and that of mapping class groups.
I will then discuss results about random walks on such groups that can
be proven using their common geometric features, namely the facts that
generic elements of (non-trivial) relatively hyperbolic groups are
hyperbolic, generic elements in mapping class groups are pseudo-Anosovs
and random paths of length $n$ stay $O(\log(n))$-close to geodesics in
(non-trivial) relatively hyperbolic groups and
$O(\sqrt{n}\log(n))$-close to geodesics in mapping class groups.