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Which pairs of conjugates of $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ generate $\operatorname{SL}(2,\mathbb{Z})$?
When do two distinct conjugates of $U := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ generate $\DeclareMathOperator\SL{SL}\SL(2,\mathbb{Z})$? The classic example is $U,L^{-1}$, where $L = \...
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Subgroup of $SL_2(O)$ with nice fundamental domain in complex upper half-plane
Let $O$ be the ring of $S$-integers in a real quadratic number field. Let $G$ be an $S$-arithmetic subgroup of $SL_2(O)$ whose intersection with $SL_2(\mathbb Z)$ is not of finite index in $SL_2(\...
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