Are there characterizations of subgroups of a special linear group SL$(n, \mathbb{Z})$?
Since SL$(n, \mathbb{Z})$ has infinite order, it would be enough if I know how to generate subgroups of SL$(n, \mathbb{Z}_p)$. Here, $n$ is rather large.
2026-03-26 04:32:25.1774499545
Subgroups of special linear group SL$(n, \mathbb{Z})$
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There is a characterization of finite-index subgroups of $SL(n,\mathbb{Z})$ as follows:
Theorem (Mennicke) For $n\ge 3$ every finite-index subgroup of $SL(n,\mathbb{Z})$ is a congruence subgroup.
This is not true for $SL(2,\mathbb{Z})$, where we have a veritable zoo of finite-index subgroups, as $SL(2,\mathbb{Z})\cong C_2\ast_{C_2}C_3$ contains a free subgroup of finite index.