What follows is a brief overview of my paper with Eric Swenson (BYU):

**[1] ***“A `transversal’ for minimal invariant sets in the boundary of a CAT(0) group” with Eric L. Swenson (Brigham Young University),* Trans. Amer. Math. Soc. **365** (2013), 3069-3095.

The rank rigidity problem for CAT(0) groups in its boundary form was introduced by Ballmann and Buyalo, and asks whether (1) an infinite group *G* acting properly and co-compactly on a CAT(0) space *X* whose visual boundary ∂*X* has Tits-diameter diam(∂*X*)>π must contain a rank one element; and (2) in the case that the visual boundary of *X* has diameter π, is this boundary a spherical building?

Ballman and Buyalo show that diam(∂*X*)>2π implies the presence of rank one elements in *G*. Later work by Swenson and Papasoglu improves this bound to 3π/2.

In our attempt at problems (1) and (2), we observe that the Ellis action of the Stone-Cech compactification β*G *of *G* on the cone boundary ∂*X *is a semigroup action by Lip-1 endomorphisms when ∂*X *is taken with the Tits metric, and derive some new structural results about minimal invariant sets of the *G*-action on the cone boundary.

In a nutshell, a pair of points *p,q* in ∂*X *is said to be * compressible*, if

∃ω∈β*G * dist(ω*p*,ω*q*)<dist(*p,q*) .

We are interested in the ** maximal G-incompressible subsets** of ∂

*X*, because of the intuitive idea that:

1. If G has rank one elements, then non-degenerate incompressible subsets should not exist;

2. If ∂*X *i*s *a building, then the maximal G-incompressible subsets are natural candidates for being *simplices* of the building structure.

The animated figure above illustrates a compression in the boundary of the product of two copies of a non-abelian free group.

So far, our approach allowed us to improve the Swenson-Papasoglu bound to a bound dependent on the geometric dimension of diam(∂*X*), and results in a characterization of Bieberbach groups as CAT(0) groups with no proximal pairs in their Tits boundary.

Press here for my presentation of this work to the Penn Geometry & Topology seminar back in 2012. Also take a look at recent work by Ricks building on ours, studying CAT(0) groups with 1-dimensional boundaries.