Tuesday, October 9
Alireza Golsefidy, Princeton University
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(Fq[1/t]) is the only lattice of minimum covolume in G(Fq((t))), where G is a Chevalley group of classical type or of type E6. 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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