Abstract:
:Let Lambda=SL(2)(Z) be the modular group and let c(n)(Lambda) be the number of congruence subgroups of Lambda of index at most n. We prove that lim(n--> infinity )(log c(n)(Lambda)/((log n)(2)/log log n))=(3-2(sqrt)2)/4. The proof is based on the Bombieri-Vinogradov "Riemann hypothesis on the average" and on the solution of a new type of extremal problem in combinatorial number theory. Similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple Lie groups. If G is such a Lie group and Gamma is an irreducible lattice of G it turns out that the subgroup growth of Gamma is independent of the lattice and depends only on the Lie type of the direct factors of G. It can be calculated easily from the root system. The most general case of this result relies on the Generalized Riemann Hypothesis, but many special cases are unconditional. The proofs use techniques from number theory, algebraic groups, finite group theory, and combinatorics.
journal_name
Proc Natl Acad Sci U S Aauthors
Goldfeld D,Lubotzky A,Nikolov N,Pyber Ldoi
10.1073/pnas.0404571101keywords:
subject
Has Abstractpub_date
2004-09-14 00:00:00pages
13428-30issue
37eissn
0027-8424issn
1091-6490pii
0404571101journal_volume
101pub_type
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