Geodesic
Counting arithmetic lattices and surfaces
Mikhail Belolipetsky1 ; Tsachik Gelander2 ; Alexander Lubotzky3 ; Aner Shalev3
1 Department of Mathematical Sciences<br/>Durham University<br/>South Road<br/>Durham, DH1 3LE<br/>United Kingdom<br/>and <br/>Sobolev Institute of Mathematics<br/>Koptyuga 4<br/>630090 Novosibirsk<br/>Russia
2 Einstein Institute of Mathematics<br/>The Hebrew University of Jerusalem<br/>91904 Jerusalem<br/>Israel
3 The Hebrew University of Jerusalem<br/>Einstein Institute of Mathematics<br/>91904 Jerusalem<br/>Israel
Annals of mathematics, Tome 172 (2010) no. 3, pp. 2197-2221

Voir la notice de l'article provenant de la source Annals of Mathematics website

We give estimates on the number $\operatorname{AL}_H(x)$ of conjugacy classes of arithmetic lattices $\Gamma$ 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 $H=\operatorname{PSL}(2,\mathbb{R})$ where we show that \[ \lim_{x\to\infty}\frac{\log \operatorname{AL}_H(x)}{x\log x}=\frac{1}{2\pi}. \] The proofs use several different techniques: geometric (bounding the number of generators of $\Gamma$ as a function of its covolume), number theoretic (bounding the number of maximal such $\Gamma$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $\Gamma$).

DOI : 10.4007/annals.2010.172.2197

Mikhail Belolipetsky 1 ; Tsachik Gelander 2 ; Alexander Lubotzky 3 ; Aner Shalev 3

1 Department of Mathematical Sciences<br/>Durham University<br/>South Road<br/>Durham, DH1 3LE<br/>United Kingdom<br/>and <br/>Sobolev Institute of Mathematics<br/>Koptyuga 4<br/>630090 Novosibirsk<br/>Russia
2 Einstein Institute of Mathematics<br/>The Hebrew University of Jerusalem<br/>91904 Jerusalem<br/>Israel
3 The Hebrew University of Jerusalem<br/>Einstein Institute of Mathematics<br/>91904 Jerusalem<br/>Israel 
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Mikhail  Belolipetsky; Tsachik Gelander; Alexander Lubotzky; Aner Shalev. Counting arithmetic lattices and surfaces. Annals of mathematics, Tome 172 (2010) no. 3, pp. 2197-2221. doi: 10.4007/annals.2010.172.2197

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