11 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 an irreducible, geodesically complete, proper $\operatorname{CAT}(0)$ space, and let a geometric action on $X$ be one for which $X$ is neither a higher-rank symmetric sp…

Let $G$ be an infinite discrete group acting geometrically on an irreducible $\operatorname{CAT}(0)$ space $X$. A geodesic in $X$ is rank one when it does not bound a flat half-pla…

Conjecture de rigidité du rang. Alors soit $X$ est un espace symétrique de type non compact et de rang supérieur, soit $X$ est un immeuble affine de dimension $\b1geq2$, soit $G$ c…

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…

Rank Rigidity Conjecture. The following three conditions should be equivalent: $X$ admits a rank one axis; $\operatorname{diam}(\partial_T X)>\pi$; and the induced action on…

Higher hyperbolic rank rigidity conjecture. If $M$ has higher hyperbolic rank, then $M$ is isometric to a locally symmetric space of real rank $1$.

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