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arXiv · 0811.2482

Counting arithmetic lattices and surfaces

Mikhail Belolipetsky·Tsachik Gelander·Alex Lubotzky·Aner Shalev
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Abstract

We give estimates on the number $AL_H(x)$ of arithmetic lattices $Γ$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $Γ$ as a function of its covolume), number theoretic (bounding the number of maximal such $Γ$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $Γ$).

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BibTeXRIS

Mikhail Belolipetsky, Tsachik Gelander, Alex Lubotzky, Aner Shalev. 2010-04-22. Counting arithmetic lattices and surfaces. https://arxiv.org/abs/0811.2482

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