Counting arithmetic lattices and surfaces

Abstract

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 $\Gamma$).

Authors

Mikhail Belolipetsky

Department of Mathematical Sciences
Durham University
South Road
Durham, DH1 3LE
United Kingdom
and
Sobolev Institute of Mathematics
Koptyuga 4
630090 Novosibirsk
Russia

Tsachik Gelander

Einstein Institute of Mathematics
The Hebrew University of Jerusalem
91904 Jerusalem
Israel

Alexander Lubotzky

The Hebrew University of Jerusalem
Einstein Institute of Mathematics
91904 Jerusalem
Israel

Aner Shalev

The Hebrew University of Jerusalem
Einstein Institute of Mathematics
91904 Jerusalem
Israel