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Consider the subgroup $G_{\lambda}$ of $SL_2(\mathbb R)$ generated by $N_{\lambda} = \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ where $\lambda>0$. Then it's known by Hecke that $G_{\lambda}$ is discrete if and only if $\lambda \geq 2$ or $\lambda=2 \cos(\frac{\pi}{n})$ where $n \geq 3$ is an integer. They are named for Hecke, and are used by Hecke to study modular forms.

Is there is a generalization of this fact to more general groups? For example, consider a subgroup of $SL_n(\mathbb R)$ generated by several elementary matrices and some involutions, when is it discrete?

What about $p$-adic analogues?

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    $\begingroup$ Even when you consider two unipotent matrices in $SL(2,C)$ this problem does not have a good answer (it is undecidable in certain sense). mathoverflow.net/questions/109967/… $\endgroup$
    – Misha
    Commented Feb 8, 2019 at 20:35
  • $\begingroup$ Even the group generated by $\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}$ inside $\operatorname{SL}_2(\mathbb Q_p)$ is not discrete. (In $\operatorname{SL}_2(\mathbb F_p((t)))$ it's finite, although I don't know if your group $G_\lambda$ still is.) $\endgroup$
    – LSpice
    Commented Feb 9, 2019 at 0:16
  • $\begingroup$ @Misha Thank you for that answer.. How about other groups? For example $SL_3(\mathbb R)$? $\endgroup$
    – Zhiyu
    Commented Feb 11, 2019 at 20:58
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    $\begingroup$ @zzy: I do not know for sure about $SL(3,R)$. For "positive relatively Anosov" subgroups (a very special class of discrete subgroups) a description should be possible. But for general discrete subgroups I am very skeptical. However, not enough is known about discrete subgroups of $SL(3,R)$ at this point to prove anything definitive. $\endgroup$
    – Misha
    Commented Feb 11, 2019 at 21:24

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