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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 non-semisimple 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.

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