

Seminar of the Department of Algebra
November 1, 2005, Moscow, Steklov Mathematical Institute, Room 540 (8 Gubkina)






New properties of lattices in Lie groups
V. P. Platonov^{} 
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Abstract:
I will represent some final results in the series of papers published together with F. Grunewald (1997–2004). Let $G$ be a Lie group with finetely many components and $H$ be a lattice in $G$ (it means $H$ is discrete and $G/H$ has a finite volume). Let $D$ be a finite extension of $H$. The following two problems were open for more than 40 years:
1) Is $D$ a lattice?
2) Is it true that $D$ has only finitely many conjugacy classes of finite subgroups?
It was a very surprising result that the question 1) has a negative answer. After that we found a criterion when $D$ is a lattice. The proof is difficult and is based on our new results about rigidity for lattices in nonsemisimple groups. This criterion allowed us to solve the problem 2) positively. We used additionally some natural actions of finite groups on Teichmuller spaces and on CAT(0) spaces.
As a corollary, we have obtained a solution of the problem of Borel–Serre, formulated in 1964.

