43 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-…

Gournay's dimension-zero conjecture. One has

The Fourier lower-bound conjecture.

A group is sofic if it admits finite approximations in the sense described above. Non-sofic groups conjecture. There exist groups that are not sofic. No example is currently known,…

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…

Let $K$ be a finitely generated field of characteristic zero, let $G$ be a group, and let $M$ be a $KG$-module. Each embedding of $K$ into $\mathbb{C}$ induces values…

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…

Authors' conjecture. The answers to Questions (i) and (ii) are positive, namely

The Poisson equivalence-relation conjecture. The group $G$ is sofic if and only if the measured equivalence relation $\mathcal{R}$ is sofic. This is presented as a locally compact…

Let $G$ be a connected Lie group with property (T). For a complete finite-volume Riemannian manifold $M$, its Cheeger constant is … where the infimum is over compact smooth submani…

Let $\Gamma$ be a discrete group, let $A\in M_n(\mathbb{Z}\Gamma)$ be an $n\times n$ matrix over the group ring, and let $\ell(A)$ be the corresponding operator on…

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 $\Gamma\curvearrowright (X,\mu)$ be a strongly ergodic distal action, and let its $0$-dimensional sofic homology be the degree-zero sofic homology associated to the action. The…

Let $\mathbb{K}$ be a compact abelian group, let $\Gamma$ be a countable group, and let $\Gamma$ act on the compact abelian group $\mathbb{K}^\Gamma/\mathbb{K}$ by the shift action…