# generalization of the discreteness of Hecke groups to general reductive groups

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?

• 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/… – Misha Feb 8 at 20:35
• 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.) – LSpice Feb 9 at 0:16
• @Misha Thank you for that answer.. How about other groups? For example $SL_3(\mathbb R)$? – zzy Feb 11 at 20:58
• @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. – Misha Feb 11 at 21:24