8 problems

Ballmann–Buyalo's conjecture. If $X$ contains a geodesic of rank $1$, then it also contains a $G$-periodic geodesic of rank $1$.

Let $X$ be a locally compact geodesically complete $\operatorname{CAT}(0)$ space, and let $G$ be a discrete group acting properly and cocompactly on $X$ by isometries. Assume that…

Let $G$ be a simple Lie group with real rank at least $2$. For a countable group $\Gamma$, let $A_i$ consist of elements whose centralizer contains a free abelian subgroup of rank…

Let $X$ be a proper, geodesically complete $\operatorname{CAT}(0)$ space under a geometric action. A rank one axis is an axis in $X$ that does not bound a flat half-plane. Rank Rig…

Let $(N,g)$ be a finite-volume Riemannian manifold satisfying property (P), meaning that every geodesic in its universal covering has rank at least two. Local symmetry conjecture.…

Incompressible-covering conjecture. If $G$ has higher rank, then

Strong-collapse rank-one conjecture. The boundary $\partial_\infty X$ is strongly collapsible if and only if $G$ has rank one.

Closing lemma. If $\Lambda G=\partial X$ and