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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.

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  • $\begingroup$ 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? $\endgroup$
    – Igor Rivin
    Commented Mar 27, 2015 at 17:20
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    $\begingroup$ @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. $\endgroup$
    – Lee Mosher
    Commented Mar 27, 2015 at 20:44
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    $\begingroup$ @LeeMosher Right, I wasn't sure if that kind of thing counted... $\endgroup$
    – Igor Rivin
    Commented Mar 27, 2015 at 20:51

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Scott explicitly claims that Fuchsian groups are LERF on the first page of his paper. Lemma 1.4 shows that LERF is equivalent to the "geometric condition". He assumes the cover is regular, but I think the proof goes through for a regular orbifold cover. Thus Fuchsian groups also have the "geometric condition". Finally, orbifolds (with finitely generated orbifold fundamental group) have compact cores, completing the proof.

NB: There is an erratum for this paper, published in 1985.

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