If they are coxeter group, they will be generated by some order-2 elements, with eigenvalues $\pm 1$ and characteristic polynomial $(x-1)^m(x+1)^n$. And I don't know how to continue.
2026-03-25 09:26:52.1774430812
Is $SL_n(\mathbb Z)$ or $\Gamma_1(N)$ a coxeter group?
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By this paper Coxeter groups are virtually inside right angled Artin groups which are (bi)automatic. Hence Coxeter groups cannot contain infinite nilpotent non-virtually abelian groups. On the other hand infinite $SL(n, .)$ with $n\ge 3$ and its finite index subgroups contain non-virtually abelian infinite nilpotent groups of uni-upper-triangular matriced. Therefore no finite index subgroup of an infinite $SL(n, .)$, $n\ge 3$, can embed into a Coxeter group.