Determining Fuchsian groups by their finite quotients

Abstract

Let $\C(Γ)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(Γ)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If ${\Gamma }_1$ is a lattice in $\mathrm{PSL}(2,\mathbb{R})$ and ${\Gamma }_2$ is a lattice in any connected Lie group, then $\C(Γ_1) = \C(Γ_2)$ 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 $\C(F)=\C(Γ)$ implies that $F\cong \Gamma$. If ${\Gamma }_1<\mathrm{PSL}(2,\mathbb{C})$ 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 $\C(Γ_1)= \C(Γ_2)$ implies that $G\cong \mathrm{PSL}(2,\mathbb{C})$ 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.

Minor edits. Version accepted by Israel J Math

References in corpus (3)

• Dense embeddings of surface groups
• Polygonal words in free groups
• Hyperbolic surface subgroups of one-ended doubles of free groups

Cited by in corpus (1)

• Virtual pro-$p$ properties of 3-manifold groups