11 problems

Matrix-kernel approximation conjecture.

Rationality and integrality conjecture. For every degree $p$,

Let $Y$ be a finite connected $2$-dimensional $CW$-complex. A connected tower with target $Y$ is a finite sequence of covering maps and embeddings … Call such a tower non-positive…

Let $G$ be a finitely presented group and let $F$ be a field. Denote by $b^{(2)}_1(G;\mathcal N(G))$ the first $L^2$-Betti number with coefficients in the group von Neumann algebra…

Let $Y$ be a $2$-complex. Write $b^{(2)}_2(\widetilde Y;\mathcal N(\pi_1(Y)))$ for the second $L^2$-Betti number of its universal covering, with coefficients in the von Neumann alg…

Let $G$ be a virtually residually torsion-free nilpotent group. Its first $\ell^2$-Betti number is denoted by $\beta_1^{(2)}(G)$, and $G$ is algebraically fibred if it admits a sur…

Let $(M,\tau)$ be a tracial von Neumann algebra, and let $A$ be a weak$^{*}$-dense, finitely presented, algebraically sofic, unital $*$-subalgebra of $M$. Write…

Let $G$ be a group and let $\sigma:G\to\operatorname{GL}_k(\mathbb{C})$ be a homomorphism. Define $\widetilde{\sigma}:\mathbb{C}[G]\to\operatorname{Mat}_k(\mathbb{C}[G])$ by sendin…

Let $G$ be a torsionfree discrete group, and let $g_1,\dots,g_k$ be elements whose normal closure is $G$. The first $\ell^2$-Betti number of $G$ is denoted by $\beta_1^{(2)}(G)$. N…

Let $G$ be a torsion free discrete group, or a $p$-group for a prime $p$, and let $g_1,\dots,g_n$ be arbitrary elements of $G$. Denote by…

Let $G$ be a torsion free discrete group. Write ${\rm nrk}(G)$ for its normal rank, the least number of elements needed to normally generate $G$. Normal-rank bound. … The authors p…