Convex co-compact groups with one dimensional boundary faces

TL;DR

This paper characterizes convex co-compact subgroups of the projective linear group based on the boundary face dimensions, linking geometric boundary properties to algebraic hyperbolicity conditions.

Contribution

It establishes a precise criterion connecting the boundary face dimension to the group's relative hyperbolicity and introduces the coarse Hilbert dimension for subgroup classification.

Findings

Groups with boundary faces of dimension at most one are relatively hyperbolic.

Introduces coarse Hilbert dimension to characterize hyperbolic properties.

Provides a geometric criterion for hyperbolicity based on boundary face dimensions.

Abstract

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face in the ideal boundary has dimension at most one. We also introduce the "coarse Hilbert dimension" of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two.