Determining Fuchsian groups by their finite quotients

Let $C(Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $Gamma$. We investigate the extent to which $C(Gamma)$ determines $Gamma$ when $Gamma$ 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 $C(Gamma_1) = C(Gamma_2)$ implies that $Gamma_1$ is isomorphic to $Gamma_2$. If $F$ is a free group and $Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $C(F)=C(Gamma)$ implies that $FcongGamma$. If $Gamma_1<{rm{PSL}}(2,Bbb C)$ and $Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $C(Gamma_1)= C(Gamma_2)$ implies that $G cong {rm{PSL}}(2,Bbb C)$ and that $Gamma_2$ 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, we give an explicit description of finite quotients that distinguish between them.

Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $rge 2$, and that a version of the Freiheitssatz holds for parafree groups.