Separability in Coxeter groups

Question

I am looking for a reference for the following statement:

Theorem. Let $\Gamma$ be a finite labelled graph and $C(\Gamma)$ the corresponding Coxeter group. For every $\Xi \subset V(\Gamma)$, the standard parabolic subgroup $\langle \Xi \rangle$ is separable in $C(\Gamma)$.

Just in case: a subgroup $H \leq G$ is separable if, for every $g \in G \backslash H$, there exists a finite-index subgroup $K \leq G$ containing $H$ but not $g$. This amounts to saying that $H$ is closed in $G$ with respect to the profinite topology. I know how to prove the theorem, but probably it is already available somewhere. Does anyone know a precise reference?

I am also curious about the same statement for double cosets of standard parabolic subgroups:

Question. Let $\Gamma$ be a finite labelled graph and $C(\Gamma)$ the corresponding Coxeter group. For all $\Phi,\Psi \subset V(\Gamma)$, is the double coset $\langle \Phi \rangle \langle \Psi \rangle$ closed in $C(\Gamma)$ with respect to the profinite topology?

I have a few strategies in mind to attack the question, but I would like to check if something is already available in the literature.

Answer 1

Since I did not find any reference, I included the results in my recent preprint (in greater generality): Rotation groups virtually embed into right-angled rotation groups. I record below the sketch of the argument in the specific case of Coxeter groups.

Theorem. Let $\Gamma$ be a finite labelled graph and let $C(\Gamma)$ denote the corresponding Coxeter group. For all $\Phi,\Psi \subset V(\Gamma)$, the double coset $\langle \Phi \rangle \langle \Psi \rangle$ is separable in $C(\Gamma)$.

Lemma. For every $\Phi \subset V(\Gamma)$, $\langle \Phi \rangle$ is separable in $C(\Gamma)$.

Proof. In the Cayley graph of $C(\Gamma)$, $\langle \Phi \rangle$ is convex. Let $\mathcal{J}$ denote the collection of the walls tangent to $\langle \Phi \rangle$, i.e. disjoint from $\langle \Phi \rangle$ but not separated from $\langle \Phi \rangle$ by another wall. Let $\mathrm{Rot}$ denote the subgroup of $C(\Gamma)$ generated by the reflections along the walls in $\mathcal{J}$. It acts on $C(\Gamma)$ with $\langle \Phi \rangle$ as a fundamental domain. Then $\langle \mathrm{Rot}, \Phi \rangle$ yields a finite-index subgroup that splits as $\mathrm{Rot} \rtimes \langle \Phi \rangle$. Thus, $\langle \Phi \rangle$ is a virtual retract, and a fortiori separable since $C(\Gamma)$ is residually finite. $\square$

Lemma. For all walls $A$ and $B$, $\mathrm{Cross}(A,B):= \{ g \in C(\Gamma) \mid gA \text{ transverse to } B \}$ is separable in $C(\Gamma)$.

Proof. Let $a$ (resp. $b$) denote the reflection along $A$ (resp. $B$). Then, given an element $g \in C(\Gamma)$, $gA$ and $B$ are transverse iff $gag^{-1}$ and $b$ generate a finite dihedral group; otherwise, they span an infinite dihedral group. Therefore, fixing a number $N \geq 1$ divisible by all the labels from our graph $\Gamma$, we can write $\mathrm{Cross}(A,B)= \phi^{-1}(1)$ where $$\phi : \left\{ \begin{array}{ccc} C(\Gamma) & \to & C(\Gamma) \\ g & \mapsto & (gag^{-1}b)^N \end{array} \right..$$ Since $\phi$ is continuous with respect to the profinite topology and that $C(\Gamma)$ is residually finite, we conclude that $\mathrm{Cross}(A,B)$ is separable in $C(\Gamma)$. $\square$

Proof of the theorem. If $\Phi = \Psi$, then the desired conclusion follows from our first lemma; so we assume that $\Phi \neq \Psi$. Construct a new labelled graph $\Gamma^+$ from $\Gamma$ by adding two new vertices $u_\Phi$ and $u_\Psi$; by connecting $u_\Phi$ (resp. $u_\Psi$) to all the vertices in $\Phi$ (resp. $\Psi$) with edges labelled by $2$; and by connecting $u_\Phi$ and $u_\Psi$ with an edge labelled by $2$. Let $J_\Phi$ (resp. $J_\Psi$) denote the wall in $C(\Gamma^+)$ crossing the edge $[1,u_\Phi]$ (resp. $[1,u_\Psi]$). It turns out that $$\mathrm{Cross}(J_\Psi,J_\Phi)= \langle \Phi \rangle \langle \Psi \rangle \langle u_\Phi, u_\Psi \rangle.$$ Since $\mathrm{Cross}(J_\Psi,J_\Phi)$ is separable in $C(\Gamma^+)$ and since $C(\Gamma)$ is separable in $C(\Gamma^+)$ (once identified with the subgroup $\langle V(\Gamma) \rangle$), we conclude that $$\mathrm{Cross}(J_\Psi,J_\Phi) \cap C(\Gamma) = \langle \Phi \rangle \langle \Psi \rangle$$ is separable in $C(\Gamma)$. $\square$

Answer 2

Finally, there is a reference for the separability of standard parabolic subgroups in Coxeter groups. Olga Varghese sent me Cooper, Long, and Reid's article Infinite Coxeter groups are virtually indicable (1998). The argument is rather nice.

Lemma. Let $G$ be a residually finite group. For every automorphism $\varphi \in \mathrm{Aut}(G)$, the fixator $\mathrm{Fix}(\varphi):=\{g \in G \mid \varphi(g)=g \}$ is separable in $G$.

Proof. Let $g \in G$ be such that $\varphi(g) \neq g$. Because $G$ is residually finite, there exists a finite quotient $\pi : G \twoheadrightarrow F$ such that $\pi( \varphi(g)g^{-1}) \neq 1$. The pre-image of the diagonal subgroup $\{(f,f) \mid f \in F\} \leq F \times F$ under $$\left\{ \begin{array}{ccc} G & \to & F \times F \\ g & \mapsto & (\pi(g), \pi(\varphi(g))) \end{array} \right.$$ yields a finite-index subgroup of $G$ that contains $\mathrm{Fix}(g)$ but not $g$. $\square$

Corollary. Standard parabolic subgroups in Coxeter groups are separable.

Proof. Let $W$ be a Coxeter group and fix a standard parabolic subgroup $H \leq W$. The amalgamated product $W \ast_H W$ is again a Coxeter group (it suffices to write down a presentation). In particular, it is residually finite. There is an obvious automorphism $\iota : W \ast_H W \to W \ast_H W$ that switches the two factors $W$. One easily verify that $\mathrm{Fix}(\iota)=H$, so our lemma applies and shows that $H$ is separable in $W$. $\square$