## Topology & Geometric Group Theory Seminar

## Fall 2007

### 1:30 - 2:30, Malott 253

Tuesday, October 9

**Alireza Golsefidy**, Princeton University

*Lattices with small covolume*

In this talk, I discuss lattices with "small" covolume in almost
simple algebraic groups over non-Archimedean fields. In the case of
characteristic p, I will quickly recall my result, saying that up to
isomorphism G(F_{q}[1/t]) is the only lattice of minimum
covolume in G(F_{q}((t))), where G is a Chevalley group of
classical type or of type E_{6}. Then I give a partial answer
to Lubotzky's question by showing that in "most" of the cases in
characteristic p, a lattice of minimum covolume is non-uniform.

I will also give a very short proof of the Siegel-Klingen theorem
using covolume of lattices.

In the characteristic zero case, in a joint work
with **A. Mohammadi**, we study discrete transitive actions on the
Bruhat-Tits building, and prove that there is no lattice in PGL(n,K)
which acts transitively on the vertices of the Bruhat-Tits building if
n > 8, give a list of 14 lattices which are the only potential such
examples for 9 > n > 4, and show that at least one of them in
dimension 5 actually acts transitively.

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