Determining Fuchsian groups by their Finite quotients

Martin R. Bridson, Marston DE Conder and Alan W. Reid

Submitted, January 2013. Minor revisions, 15 Oct 2013

Let $\C(\G)$ be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of $\G$.
We investigate the extent to which $\C(\G)$ determines $\G$ when $\G$ is a group of geometric interest.
If $\Gamma_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $\Gamma_2$ is a lattice in any connected Lie group, then ${\cal C}(\Gamma_1) = {\cal C}(\Gamma_2)$ implies that $\Gamma_1 \cong \Gamma_2$.
If $F$ is a free group and $\G$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(\G)$ implies that $F\cong\G$.
If $\G_1$ in ${\rm{PSL}}(2,\Bbb C)$ and $\G_2$ in $G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group
with trivial centre and no compact factors, then $\C(\G_1)= \C(\G_2)$ implies that $G \cong {\rm{PSL}}(2,\Bbb C)$ and that $\G_2$ belongs to one of finitely many commensurability classes.
These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question.
But in the special case of two non-isomorphic triangle groups $\Delta_1\not\cong\Delta_2$, we give an explicit description of finite quotients that distinguish between them.

27 pages, no figures.