**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.