22 problems

Let $n\in\mathbb{N}$. For an $n$-dimensional, closed, aspherical, orientable manifold $M$, write $\widetilde{M}$ for its universal cover, $b^{(2)}_i(\widetilde{M})$ for its $i$th…

A group is amenable if it has an invariant mean, and it satisfies Lück's dimension-flatness if, for every integer $j\geq 1$ and every $\mathbb Z\Gamma$-module $\mathcal M$, one has…

Matrix-kernel approximation conjecture.

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

Generalized Singer conjecture. All the $L^2$-Betti numbers of $\widetilde M$ vanish, possibly except in the middle dimension.

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…

Benjamini–Schramm FMSF conjecture. The graph $Gamma$ has uniqueness at $p_u$ if and only if its FMSF is connected. Moreover, this can occur only if

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 $\Gamma$ be a nonamenable countable group with positive first $\ell^2$-Betti number, $\beta^{_{(2)}}_{_1}(\Gamma)>0$, and let $\Gamma\curvearrowright X$ be a free ergodic p.m.p…

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…

Random-group conjecture. With high probability,

Let $\Gamma$ be a countable group with positive first $\ell^2$-Betti number, so that … Let $\Gamma\curvearrowright (X,\mu)$ be a free ergodic probability-measure-preserving action,…

Let $U_n^+$ be the free unitary quantum group and let $\hat U_n^+$ denote its discrete dual. Write $\beta_1^{(2)}(\hat U_n^+)$ for its first $\operatorname{\ell}^2$-Betti number. V…

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 $\mathcal{R}$ be a discrete, measure-preserving equivalence relation, let $c(\mathcal{R})$ denote its cost, and let $\beta^{(2)}_{1}(\mathcal{R})$ denote its first $l^{2}$-Bett…

Nonzero $\ell^2$-Betti number conjecture. Any group $\Gamma$ with at least one nonzero $\ell^2$-Betti number, $\beta^{(2)}_n(\Gamma)>0$, is $\mathcal{C}$-rigid.

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…

Let $\Gamma$ be a residually finite group satisfying appropriate finiteness properties, such as having a finite $K(\pi,1)$, and let … be a chain of normal subgroups of finite index…