Quasi-isometry Classification of Certain Right-angled Coxeter Groups

We investigate the quasi-isometry classification of the right-angled Coxeter groups WΓ which are 1-ended and have triangle-free defining graph Γ. We begin by characterising those WΓ which split over 2-ended subgroups, and those which are cocompact Fuchsian, in terms of properties of Γ. This allows us to apply a theorem of Papasoglu [21] to distinguish several quasi-isometry classes. We then carry out a complete quasi-isometry classification of the hyperbolic WΓ with Γ a generalised Θ graph. For this we use Bowditch’s JSJ tree [8] and the quasi-isometries of “fattened trees” introduced by Behrstock–Neumann [5]. Combined with a commensurability classification due to Crisp–Paoluzzi [12], it follows that there are right-angled Coxeter groups which are quasi-isometric but not commensurable. Finally, we generalise the work of Crisp–Paoluzzi [12].