E. Swenson: $\pi$-convergence and rank Rigidity in CAT(0) spaces.

Abstract: Let $X$ be a proper CAT(0) space and $G$ be a group acting properly by isometries on $X$. The boundary of $X$, $\partial X$ is the set of equivalence classes of unit speed geodesic rays in $X$. (Unit speed geodesic rays $R,S:[0,\infty)\to X$ are equivalent if $d(R(t),S(t))$ is bounded.) Adding the boundary compactifies $X$ and the action of $G$ on $X$ extends to an action by homeomorphisms on $\partial X$.

In the case where $X$ is negatively curved (in any sense of the word), there is a simple dynamical description of the action of $G$ on $\partial X$ in terms of attracting and repelling points. The non-positively curved case was seen as being intractable, but by throwing in a Tits fudge factor of $/pi$ we reclaim the original description.

Ballmann conjectured that if the diameter of the Tits boundary of a CAT(0) group was more than $\pi$, then the diameter is infinite. Ballmann and Buyalo show this for $2\pi$. Using $\pi$-convergence, improve this to $3\pi/2$.