TY - JOUR

T1 - Counting arithmetic lattices and surfaces

AU - Belolipetsky, Mikhail

AU - Gelander, Tsachik

AU - Lubotzky, Alexander

AU - Shalev, Aner

PY - 2010

Y1 - 2010

N2 - We give estimates on the number ALH(x) of conjugacy classes 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{double-struck}) where we show that 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).

AB - We give estimates on the number ALH(x) of conjugacy classes 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{double-struck}) where we show that 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).

UR - http://www.scopus.com/inward/record.url?scp=77957893427&partnerID=8YFLogxK

U2 - 10.4007/annals.2010.172.2197

DO - 10.4007/annals.2010.172.2197

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AN - SCOPUS:77957893427

SN - 0003-486X

VL - 172

SP - 2197

EP - 2221

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -