A right-angled Coxeter group (RACG) is a reflection group in which the mirrors are either parallel or at right-angles to one another. A RACG \(W_\Gamma\) can be described by a presentation graph \(\Gamma\). Graph properties of \(\Gamma\) can encode large-scale geometric features of \(W_\Gamma\).

Quasi-isometries between hyperbolic groups induce quasi-symmetric homeomorphisms between their Gromov boundaries. This metric statement gives a stronger QI invariant than the topological type of the boundary alone. A well-known quasi-symmetry invariant of metric spaces is conformal dimension. It is always greater than or equal to the topological dimension of the space. When it is strictly greater it indicates that the space has a fractal-like shape. For hyperbolic manifold groups, the boundary is a sphere whose topological and conformal dimensions agree, so the boundary is a nice round sphere.

We investigate to what extent this result generalizes to RACGs that are close to being manifold groups, and find a very different behavior. Focusing on the 3-dimensional case, a RACG \(W_\Gamma\) is virtually a hyperbolic manifold group when \(\Gamma\) is (the 1--skeleton of) a flag-no-square triangulation of a 2-sphere. It is a hyperbolic pseudo-manifold when \(\Gamma\) is a flag-no-square triangulation of a surface of positive genus. In this case \(\partial W_\Gamma\) is the Pontryagin sphere, which is a 2-dimensional space that can be imagined as an infinite genus surface with handles everywhere. By giving conditions on \(\Gamma\) that imply high conformal dimension of \(\partial W_\Gamma\), we show that there are RACGs with Pontryagin sphere boundary of arbitrarily high conformal dimension. So, in contrast to the manifold case, already in the pseudo-manifold case the boundary can be very fractalish.

This is joint work with Dani, Schreve, and Stark.