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On congruence subgroups generated by two parabolic rational matrices

Part of: Diophantine approximation, transcendental number theory Structure and classification of infinite or finite groups Riemann surfaces

Published online by Cambridge University Press:  08 August 2025

CARL-FREDRIK NYBERG-BRODDA
CARL-FREDRIK NYBERG-BRODDA*
Affiliation:
June E. Huh Centre for Mathematical Challenges (KIAS), 85 Hoegi-ro, Dongdaemun-gu 02455 Seoul, Republic of Korea. e-mail: cfnb@kias.re.kr
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Abstract

We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$. For $q = r/p$ in $\mathbb{Q} \cap (0,4)$, where p is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by

\[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\]
We introduce the conjecture that $\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$, the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$ 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 $r \leq 4$ 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 $r \in \{ p-1, p+1, (p+1)/2 \}$. In all these cases, this gives information about the algebraic structure of $\Delta_{r/p}$: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index $J_2(r)$ in $\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$, where $J_2(r)$ denotes the Jordan totient function.

MSC classification

Primary: 30F35: Fuchsian groups and automorphic functions 30F40: Kleinian groups
Secondary: 20E05: Free nonabelian groups 11J70: Continued fractions and generalizations

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 20
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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