13 problems

Let $G$ be a lattice in a higher-rank semisimple Lie group, and consider the asymptotic cones of $G$. Drutu–Mozes–Sapir conjecture. $G$ does not have cut-points in any of its asymp…

Let $\Gamma$ be a center-free irreducible higher-rank lattice. Stuck–Zimmer's IRS rigidity conjecture. For every ergodic probability-measure-preserving action of $\Gamma$ on a prob…

Let $\Gamma$ be a higher rank lattice acting continuously on the circle $S^1$. One-dimensional Zimmer rigidity conjecture. The action should factor through an action of a finite gr…

Let $\Gamma$ be an irreducible higher-rank lattice. The charfiniteness conjecture. The group $\Gamma$ is charfinite, in the sense of the character-rigidity terminology used in the…

Let $\Gamma$ be an irreducible higher-rank lattice, and let a URS mean a minimal subsystem of the space of subgroups of $\Gamma$ with the Chabauty topology. The URS rigidity conjec…

Let $\Gamma$ be an irreducible higher-rank lattice. Lubotzky's property $\tau$ conjecture. The group $\Gamma$ has property $\tau$. This is listed among consequences of the spectral…

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$ whose action does not extend to a $G$-action on any non-tr…

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$. Assume that the $\Gamma$-action on $V$ does not extend to…

Non-left-orderability conjecture. No irreducible lattice in $G$ has a left-invariant total order. Equivalently, no such lattice has a nontrivial, orientation-preserving action on t…

Let a higher rank lattice be a lattice in a simple Lie group with finite center and real rank at least $2$. Consider a continuous action of such a lattice on the circle…

Let $G$ be a higher-rank semisimple Lie group, let $\Gamma$ be a higher-rank lattice in $G$, and let $r(G)$ be the root-theoretic dimension defined in the source. Let $M$ be a mani…

Let $\Gamma\subset \operatorname{SL}(3,\mathbb {R})$ be a lattice, let $S$ be a closed connected surface, and let $\Gamma$ act on $S$ by $C^{1+\beta}$ diffeomorphisms. Assume that…

Let $\Gamma$ be an irreducible, higher-rank lattice. Consider a homomorphism from $\Gamma$ to either a mapping class group or the automorphism group of a finitely generated free gr…