5 problems

Davis–Okun conjecture. If the $i$th $L^2$-Betti number of $G$ is nonzero, then

Action-dimension upper-bound conjecture. A result similar to (ii) should hold in both cases 1) and 2).

Let $\Gamma$ be a group with geometric dimension $\operatorname{gdim}(\Gamma)$, acting properly and cocompactly on a locally finite, thick building. Action dimension bound for loca…

Let an $S$-arithmetic group act on the product of a symmetric space and a Euclidean building. Denote the dimensions of these factors by the dimension of the symmetric space and the…

Let $\pi$ be a group acting properly on a contractible manifold $M$. Action Dimension Conjecture. The $\ell^2$-Betti numbers of $\pi$ vanish above the middle dimension of $M$. This…