L3

We show how renormalisation methods similar to the ones used by physicists to make sense of Feynman integrals can be implemented to make sense of sums on infinite cones. On the basis of joint work with D. Manchon, we also discuss multiple zeta functions which can be seen as sums on a specific class of infinite cones.

Both Kazhdan and Haagerup properties turn out to be related to actions

of

groups on median spaces and on spaces with measured walls.

These relationships allows to study the connection between Kazhdan

property (T) and the fixed point property

for affine actions on $L^p$ spaces, on one hand.

On the other hand, they allow to discuss conjugacy classes of subgroups

with property (T) in Mapping Class Groups. The latter result

is due to the existence of a natural structure of measured walls

on the asymptotic cone of a Mapping Class Group.

The talk is on joint work with I. Chatterji and F. Haglund

(first part), and J. Behrstock and M. Sapir (second part).

Recent joint work with Greg McShane has answered the following question: Which curves can be short in a given cell of the decomposition of Teichmueller space? The answer involves a new combinatorial structure called "screens on fatgraphs" as we shall describe. The techniques of proof involve Fock's path-ordered product expansion of holonomies, Ptolemy transformations, and the triangle inequalities. This is a main step in giving a combinatorial description of the Deligne-Mumford compactification of moduli space which we shall also discuss as time permits.