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 
@article{10_4007_annals_2010_172_2197,
     author = {Mikhail  Belolipetsky and Tsachik Gelander and Alexander Lubotzky and Aner Shalev},
     title = {Counting arithmetic lattices and surfaces},
     journal = {Annals of mathematics},
     pages = {2197--2221},
     publisher = {mathdoc},
     volume = {172},
     number = {3},
     year = {2010},
     doi = {10.4007/annals.2010.172.2197},
     mrnumber = {2726109
},
     zbl = {1214.22002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.2197/}
}
TY  - JOUR
AU  - Mikhail  Belolipetsky
AU  - Tsachik Gelander
AU  - Alexander Lubotzky
AU  - Aner Shalev
TI  - Counting arithmetic lattices and surfaces
JO  - Annals of mathematics
PY  - 2010
SP  - 2197
EP  - 2221
VL  - 172
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.2197/
DO  - 10.4007/annals.2010.172.2197
LA  - en
ID  - 10_4007_annals_2010_172_2197
ER  - 
%0 Journal Article
%A Mikhail  Belolipetsky
%A Tsachik Gelander
%A Alexander Lubotzky
%A Aner Shalev
%T Counting arithmetic lattices and surfaces
%J Annals of mathematics
%D 2010
%P 2197-2221
%V 172
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.2197/
%R 10.4007/annals.2010.172.2197
%G en
%F 10_4007_annals_2010_172_2197
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. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.2197/

Cité par Sources :