Finite index subgroups of graph products

Abstract

We prove that every quasiconvex subgroup of a right-angled Coxeter group is an intersection of finite index subgroups. From this we deduce similar separability results for other types of groups, including graph products of finite groups and right-angled Artin groups.

Density of commensurators for uniform lattices of right-angled buildings

“…The "standard uniform lattice" 0 Ä G is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of 0 is dense in G. This result was also obtained by Haglund (2008). For our proof, we develop carefully a technique of "unfoldings" of complexes of groups.…”

“…This density theorem was proved independently and using different methods by Haglund [15,Theorem 4.30]. Although the paper [15] was submitted in 2004, it was not publicly available and was not known to us.…”

“…Although the paper [15] was submitted in 2004, it was not publicly available and was not known to us.…”

Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision

“…For (1), the key is that a quasi-convex subgroup H ⊂ G can be represented by a locally isometric immersion Z → X of finite special cube complexes. This follows from Haglund [Hag08] (the proof is a pleasant exercise in hyperbolic geometry: show that the intersection of half-spaces containing a given orbit is contained in a Hausdorff neighborhood of the orbit). Then (1) follows from Theorem 4.2 (it is not hard to see that virtual retracts in residually finite groups are separable).…”

“…), which is how Stallings [Sta83] proved the Marshall Hall theorem for free groups. See also [Hag08].…”