6 problems

Let $M$ be a closed Riemannian manifold whose fundamental group satisfies $(T_n)_{L^1}$, and suppose that $M$ is $(n-1)$-connected. The $(T_n)_{L^1}$ waist conjecture. Does the fam…

Let $G$ be a simple Lie group of higher rank $d\ge2$ with finite center, and let $X=G/K$ be its associated symmetric space. The higher-rank waist conjecture. The family of finite-v…

Let $\Gamma$ be a group with property $(T_n)_{L^1}$, meaning the $L^1$-coefficient form of higher property T used in the source. The $(T_n)_{L^1}$ Zimmer conjecture. Every diffeomo…

Let $\Gamma$ be a group with property $[T_n]$, and let $M$ be a von Neumann algebra on which $\Gamma$ acts by automorphisms preserving a faithful normal state. The invariant-state…

Let $G$ be an $S$-semisimple group of rank $r$ and let $\Gamma<u001b[0mG$ be an irreducible lattice, meaning that its projection to each simple factor of $G$ is dense. Let $V$ be a…

Property $(T_n)$ and property $(\operatorname{FA}_n)$ are higher analogues of Kazhdan's property $(T)$ and the fixed-point property for actions on suitable non-positively curved sp…