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Counting lattices in semi-simple Lie groups
Abstract
My lecture is based on results of [1] and [2]. In [1] we use an extension of the method due to Borel and Prasad to determine the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group. In [2] the results of [1] are combined with the previously known asymptotic of the number of subgroups in a given lattice in order to study the general lattice growth. We show that for many high-rank simple Lie groups (and conjecturally for all) the rate of growth of lattices of covolume at most $x$ is like $x^{\log x}$ and not $x^{\log x/ \log\log x}$ as it was conjectured before. We also prove that the
conjecture is still true (again for "most" groups) if one restricts to counting non-uniform lattices. A crucial ingredient of the argument in [2] is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.
I plan to give an overview of these recent results and discuss some ideas beyond the proofs.
[1] M. Belolipetsky (with an appendix by J. Ellenberg and A.
Venkatesh), Counting maximal arithmetic subgroups, arXiv:
math.GR/0501198.
[2] M. Belolipetsky, A. Lubotzky, Class field towers and subgroup
growth, work in progress.
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