Quasi-isometry classification of certain right-angled Coxeter groups

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