5 problems

Large-cohomogeneity diameter conjecture. For every $\epsilon>0$ and all sufficiently large $k$ with $n>k$,

Diameter–cohomogeneity conjecture. The diameter of $S^n/G$ is increasing as the cohomogeneity of the group $G$ increases.

Let $(M,\partial M)$ be an $n$-manifold with contractible components that admits a geometric group action, and let $L^2H_i(M)$ denote its $L^2$-homology. Manifold cocompact action…

For a discrete group $G$, let $L^2H_i(G)$ denote the $L^2$-homology of a contractible $G$-space, and let $\operatorname{cadim}(G)$ be the least dimension of a contractible $G$-mani…

For a discrete group $G$, let $L^2H_i(G)$ denote the $L^2$-homology of a contractible $G$-space, and let $\operatorname{actdim}(G)$ be the least dimension of a contractible manifol…