26 problems

Let $\Gamma$ be an amenable group, let $\Sigma$ and $\Sigma'$ be two sofic approximations of $\Gamma$, and let $X$ be a uniformly bounded representation of $\Gamma$. Approximation-…

Let $2<p<\infty$, let $\Gamma$ be a countable discrete sofic group, and let $\Sigma$ be a sofic approximation of $\Gamma$. Equality conjecture. For every $n\in\mathbb N$, … This co…

Gournay's dimension-zero conjecture. One has

The graph-product conjecture. The free product $\ast_{v\in V}\Gamma_v$ is weakly sofic, and, more generally, $G_X$ is weakly sofic for every graph $X=(V,E)$ on $V$.

The graph-product conjecture. The free product $\ast_{v\in V}\Gamma_v$ is $\mathbb{K}$-linear sofic, and, more generally, $G_X$ is $\mathbb{K}$-linear sofic for every graph…

Minor-excluded soficity conjecture. Every such unimodular random rooted graph $(G,\rho)$ is sofic.

Let $G$ be a finitely generated group and let $A$ be a finite alphabet. Let $M\subset G$ be finite, let $S=A^{A^M}$, and let $s\in S^G$. Suppose that the NUCA map…

Ergodic commutant conjecture. Every sofic group $G$ admits an embedding $\pi:G\to\mathcal{S}$ such that the action

High-degree free-boundary state nonequilibrium conjecture. For every $\varepsilon>0$, for all sufficiently large $r$, if $J\geq(1+\varepsilon)J_{uniq}(r)$, then

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 group, let $K$ be a field, and let $\phi_1,\phi_2:K\to\mathbb{C}$ be embeddings of $K$ into $\mathbb{C}$. For a matrix $A\in\operatorname{Mat}_{n\times m}(K[G])$, writ…

A group is sofic when every finitely generated group can be homomorphically embedded into a metric ultraproduct of permutation groups. Let $\mathcal{S}=(S_n,\|\cdot\|_H)$, where…

Let $G$ be a finitely generated virtually nilpotent group, let $S$ be a finite generating set, and let $\mathcal{D}_{G,S}^{sof}$, $\mathcal{D}_{G,S}^{hyp}$,…

A Gromov hyperbolic group is a group satisfying the usual metric hyperbolicity condition; a group is residually finite if its nontrivial elements survive in finite quotients, and i…

Let $G$ be a sofic group. The sofic profile $athcal{D}_{G}^{sof}(n)$ and hyperlinear profile $athcal{D}_{G}^{hyp}(n)$ are compared using the relation $reccurlyeq$. The profile comp…

Weighted Aldous–Lyons soficity conjecture. There exists a sequence of $K$-weighted graphs ${G_n,p_n}_{n=1}^\infty$ such that $\alpha$ is the limit of ${G_n,p_n}_{n=1}^\infty$.

Sofic–Moufang conjecture. A group $G$ is sofic if and only if it is locally embeddable into finite Moufang loops.

Let $G$ be a group, let $N$ range over the normal subgroups of $G$, and let stability and weak stability refer to the notions for systems of relator words introduced in the paper.…

Let $\Gamma$ be an amenable group and let $Y\subseteq l^{p}(\Gamma)^{\oplus n}$ for some $n\in \mathbb N$. Let $\dim_{l^{p}}^{G}(Y,\Gamma)$ denote the $l^{p}$-dimension defined by…

Let $F$ be a field and $G$ a countable group. The group algebra $F(G)$ is directly finite, meaning that for any $x,y\in F(G)$, $xy=1$ implies $yx=1$. Direct finiteness conjecture.…

The left-ideal lower-bound conjecture. For the action of $\Gamma$ by left multiplication,

The Fourier lower-bound conjecture.

Let $\Gamma$ be an infinite sofic group, let $\Sigma$ be a sofic approximation, and let $V$ be a finite-dimensional representation of $\Gamma$. The finite-dimensional vanishing con…

Let $\Gamma$ be an $\mathcal{R}^{\omega}$-embeddable group, fix $2<p<\infty$, and consider the multiplication action of $\Gamma$ on $L^{p}(L(\Gamma),\tau_{\Gamma})^{\oplus n}$. Let…

A weakly sofic group is a subgroup of a metric ultraproduct of an arbitrary family of finite groups equipped with bi-invariant metrics. Glebsky–Rivera Martínez's weak soficity conj…