### Conferences - Seminars

27NOV

2014

Thursday 27 November 2014
13:00
- 14:00
MA A 3 31

## Buildings of type \tilde{A}_n and the growth of the natural cocycle for the Steinberg representation

**By** Thibaut Dumont (EPFL)

In this talk, I will try to present some basics about Euclidean buildings and explain the topic of my Ph.D. thesis. Buildings are very complicated structures; to simplify the picture, we will keep in mind a building of type \tilde{A}_n. Those buildings are modelled on the Euclidean space \mathbb{R}^n tessellated by equilateral n-tetrahedrons. I will briefly explain how one gets a building from a (B,N)-pair of a group. The most famous such example is the building of Bruhat and Tits associated to the group SL_{n+1}(F) where F is a non-archimedean local field, say \mathbb{Q}_p. Such groups have a unitarisable representation "St", the so-called the Steinberg representation, which is of cohomological interest, and a natural cocycle "Vol". The latter is a geometrically defined map that sends an n-tuple of points in the building to a vector in St. The aim of my thesis is to compute the growth of the norm of these vectors as the points get far away from each other inside the building. If time permits, I will present the case n=1, in which case the building is the (p+1)-regular tree T and the cocycle Vol is the Busemann cocycle of T. Here, the square of the norm grows at most linearly with the distance between two points.

**Organization** MATHGEOM-EGG

**Contact** Nicolas Monod

**Accessibility** Informed public

**Admittance **Free