On congruence subgroups of SL₂(ℤ[(1)/(p)]) generated by two parabolic elements

2023/12/18 by Nyberg-Brodda, Carl-Fredrik
#11J70 (Secondary) #20E05 #30F35 #30F40 (Primary) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.11258

Abstract

We study the freeness problem for subgroups of SL₂(ℂ) generated by two parabolic matrices. For q = r/p ∈ ℚ ∩ (0,4), where p is prime and gcd(r,p)=1, we initiate the study of the algebraic structure of the group Δ_q generated by the two matrices A = \beginpmatrix 1 amp; 0
1 amp; 1 \endpmatrix, and Q_q = \beginpmatrix 1 amp; q
0 amp; 1 \endpmatrix. We introduce the conjecture that Δ_r/p = Γ₁^(p)(r), the congruence subgroup of SL₂(ℤ[(1)/(p)]) consisting of all matrices with upper right entry congruent to 0 mod r and diagonal entries congruent to 1 mod r. We prove this conjecture when r ≤ 4 and for some cases when r = 5. Furthermore, conditional on a strong form of Artin's conjecture on primitive roots, we also prove the conjecture when r ∈ \ p-1, p+1, (p+1)/2 \. In all these cases, this gives information about the algebraic structure of Δ_r/p: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index J₂(r) in SL₂(ℤ[(1)/(p)]), where J₂(r) denotes the Jordan totient function.

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