# Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated?

A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb{R})$.

• Every fuchsian group is either finitely generated or is isomorphic to a countably infinite free product of cyclic groups. – Misha Nov 23 '15 at 13:18
• @Misha Sounds great! What happens for Kleinian groups? – Pablo Nov 23 '15 at 13:41
• mathoverflow.net/questions/223658/engulfing-kleinian-groups/…, the same example. A suggestion: It is good that you have so many questions, but perhaps you should think a bit more about the answers you already got before asking more questions at the rate you do. – Misha Nov 24 '15 at 18:17
• @Misha Thank you so much for answering and reanswering my questions! Thank you also for your suggestion. I will pay more attention to the answers I already got. – Pablo Nov 25 '15 at 13:05