**Definition:** Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite index normal subgroup $G\_0$ in $G$ such that if $\pi$ is the projection of $G$ onto $G/G\_0$, then $$\pi(h)\not\in\pi (HgK).$$


[Here](http://www.sciencedirect.com/science/article/pii/002240499290019C) G. Niblo proved that 


**Theorem: A surface gorup is double coset separable.**


**Definition:** Let $\{\Gamma\_m\}\_{m\in\mathbb{N}}$ be a sequence of nested finite index normal subgroups of $\Gamma\_0=\Gamma$. We say that $\{\Gamma\_m\}\_{m\in\mathbb{N}}$ is a vanishing sequence if for all $\gamma ,\eta \in \Gamma$ and for any set $H$ invariant by left multiplication by $\gamma$ and right multiplication by $\eta$ whose projection in $<\eta> \setminus \Gamma /<\gamma>$ is finite, there exists $n\_0$ such that for all $n>n\_0$, $H\subset \Gamma\_n\cap<\eta>.<\gamma>$.


[Here Corollary 11.0.5](http://arxiv.org/abs/1212.5015) F. Labourie stated that


**As surface subgroups are double coset separable and countable, Vanishing sequence exists.** 


**Q) What is the proof of the above statement?**


PS: I can show double coset separability inpleis subgroup separability and residually finiteness, but I can't prove the above statement. 



Double coset separability and the existence of vanishing sequences for surface group
This is not correct (the stated conclusion implies $H \subset \langle \eta \rangle \cdot \langle \gamma \rangle$, which need not be true), so there must be a typographical error. Probably the conclusion should be $H \cap \Gamma\_n \subset \langle \eta \rangle \cdot \langle \gamma \rangle$. Since $\Gamma$ is countable, there are only countably many possibilities for the triple $(\eta, \gamma, H)$. Given $\eta\_m$, $\gamma\_m$ and $H\_m$ as described, double coset separability implies there is a finite-index normal subgroup $\Gamma\_m'$, such that $H\_m \cap \Gamma\_m' \subset \langle \eta\_m \rangle \cdot \langle \gamma\_m \rangle$. Let $\Gamma\_n = \bigcap\_{m=1}^n \Gamma\_m'$.


