Arithmetic lattices in unipotent algebraic groups

2018/04/13 by Khalid Bou-Rabee, Bou-Rabee, Khalid, Daniel Studenmund +1
Mathematics · #20E18 #20E26 #20G05 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20E18 #msc:20E26 #msc:20G05

paper · pdf · doi:10.48550/arxiv.1804.04973

10 pages: revised introduction, made minor corrections, and added references

arxiv created 2018/04/18 · arxiv updated 2018/04/19

Abstract

Fixing an arithmetic lattice Γ in an algebraic group G, the commensurability growth function assigns to each n the cardinality of the set of subgroups Δ with [Γ: Γ∩ Δ] [Δ: Γ∩ Δ] = n. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent groups" apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic ℤ-group with arithmetic lattice Γ, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in p^-s, where the degrees of the numerator and denominator are independent of p. This gives regularity results for the set of arithmetic lattices in G.

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