Subgroup growth of lattices in semisimple Lie groups

2004/06/09 by Alexander Lubotzky, A. Lubotzky, Nikolay Nikolov +3
Mathematics · #20H05 #22E40 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #math.GR #msc:20H05 #msc:22E40

paper · pdf · doi:10.48550/arxiv.math/0406164

34 pages

arxiv created 2004/06/09 · openalex publication_date 2004/06/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the Generalized Riemann Hypothesis for number fields but we can state the following unconditional theorem: Let G be a simple Lie group of real rank at least 2, different than D₄(\bbc), and let Γ be any non-uniform lattice of G. Let s_n(Γ) denote the number of subgroups of index at most n in Γ. Then the limit lim_{n→ ∞} (log s_n(Γ))/((log n)²/ log log n) exists and equals a constant γ(G) which depends only on the Lie type of G and can be easily computed from its root system.

Related

• On the Asymptotic Number of Generators of High Rank Arithmetic Lattices
• Counting maximal arithmetic subgroups
• Arithmetic Groups Have Rational Representation Growth
• On the number of finite subgroups of a lattice
• Growth Estimates for Orbits of Self Adjoint Groups