Counting arithmetic lattices and surfaces

Details Counting arithmetic lattices and surfaces Counting arithmetic lattices and surfaces

2008-11-15

arxiv.org/abs/0811.2482v2

Mathematics

Group Theory

20 pages, final version, to appear in Annals of Mathematics

Scientific paper

We give estimates on the number $AL_H(x)$ 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 $H=PSL(2,R)$ 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 $\Gamma$ 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 $\Gamma$).

Affiliated with

Also associated with

No associations

Rating

Counting arithmetic lattices and surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Counting arithmetic lattices and surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Counting arithmetic lattices and surfaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-39297