Spherical growth of reciprocal classes in the Hecke Groups

2024/11/01 by Debattam Das, Das, Debattam, Krishnendu Gongopadhyay +1 · 2 citations
Chemistry · Materials Science · Mathematics · #05E16 #37C35 #FOS: Mathematics #Geometric Topology (math.GT) #Graph theory and applications #Group Theory (math.GR) #History and advancements in chemistry #Primary 20H10 #Quasicrystal Structures and Properties #Secondary 11F06

paper · pdf · doi:10.48550/arxiv.2411.00739

openalex publication_date 2024/11/01 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

Let Γ_p denote the Hecke group where p=2r, r>0. Let N_l denote the set of conjugacy classes of reciprocal elements of word length l in Γ_p. We prove that for l → ∞, |N_l| = O(\lfloor \tfracl+12 \rfloor^s-1 ρ^\lfloor \tfracl+12 \rfloor ), where \mathcal O is the `big O', ρ∈ [√(2), 2] is the unique positive real root of p(x) = x^r+1 - 2∑_j=1^r-1 x^r-j - 1, and s is the maximal multiplicity among the roots of p(x). Our method relies on the free product structure of the Hecke group Γ_p, a combinatorial counting function, and recurrence relations derived from cyclically reduced representatives. We also derive that the growth rate of the primitive reciprocal classes of word length l is in agreement with that of N_l. This work generalizes previous results for odd p and provides an explicit asymptotic bound for all Hecke groups.

Cited by

• Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups
• Counting and equidistribution of strongly reversible closed geodesics in negative curvature

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