On congruence subgroups of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916109767147520 |
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| author | Nyberg-Brodda, Carl-Fredrik |
| author_facet | Nyberg-Brodda, Carl-Fredrik |
| contents | We study the freeness problem for subgroups of $\operatorname{SL}_2(\mathbb{C})$ generated by two parabolic matrices. 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 $Δ_q$ generated by the two matrices \[ A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \text{ and } Q_q = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}. \] We introduce the conjecture that $Δ_{r/p} = \overlineΓ_1^{(p)}(r)$, the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ 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 $Δ_{r/p}$: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index $J_2(r)$ in $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the Jordan totient function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11258 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On congruence subgroups of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements Nyberg-Brodda, Carl-Fredrik Group Theory Number Theory 30F35, 30F40 (Primary), 20E05, 11J70 (Secondary) We study the freeness problem for subgroups of $\operatorname{SL}_2(\mathbb{C})$ generated by two parabolic matrices. 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 $Δ_q$ generated by the two matrices \[ A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \text{ and } Q_q = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}. \] We introduce the conjecture that $Δ_{r/p} = \overlineΓ_1^{(p)}(r)$, the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ 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 $Δ_{r/p}$: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index $J_2(r)$ in $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the Jordan totient function. |
| title | On congruence subgroups of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements |
| topic | Group Theory Number Theory 30F35, 30F40 (Primary), 20E05, 11J70 (Secondary) |
| url | https://arxiv.org/abs/2312.11258 |