Counting primes, groups, and manifolds.

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.

authors

Goldfeld D,Lubotzky A,Nikolov N,Pyber L

doi

10.1073/pnas.0404571101

keywords:

subject

Has Abstract

pub_date

2004-09-14 00:00:00

pages

13428-30

issue

37

eissn

0027-8424

issn

1091-6490

pii

0404571101

journal_volume

101

pub_type

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