# Scotts Theorem for one ended Fuchsian groups

in Peter Scott's work "Subgroups of Surface groups are almost geometric" is proven that for a closed surface $S$ and any fin. gen. subgroup $U$ of $G:=\pi_1(S,x)$ there exists a finitely sheeted covering map $p: (\tilde{S},\tilde{x})\to (S,x)$ such that U is realized by a subsurface in $\tilde{S}$. This is Theorem 3.3 in his work.

Is there a generalization of this theorem for the case where $G$ is a one-ended Fuchsian group? In this case $S$ is a two-orbifold and $U\leq G=\pi_1(S,x)$. And the question is: Does there exist a finite sheeted cover $\tilde{S}$ such that the subgroup $U$ corresponds to a sub-two-orbifold of $\tilde{S}$. And if not, does anyone knows a counterexample?

Thanks.

• Maybe I am confused, but consider $G$ the $(2, 3, 7)$ triangle group, and $U$ is the cyclic group of order $7.$ Is there such a cover then? – Igor Rivin Mar 27 '15 at 17:20
• @Igor: Yes there is such a cover: $\tilde S$ is the "pillow" $(2,3,7)$ orbifold that double covers $S$ which is the "triangle" $(2,3,7)$ orbifold, the suborbifold of $\tilde S$ just being a neighborhood of the $\mathbb{Z}/7\mathbb{Z}$ point. – Lee Mosher Mar 27 '15 at 20:44
• @LeeMosher Right, I wasn't sure if that kind of thing counted... – Igor Rivin Mar 27 '15 at 20:51