446 problems

For every locally compact second-countable group $G$, every invariant random subgroup $\nu$ of $G$ is co-sofic; equivalently, $\nu$ is a weak- limit of invariant random subgroups o…

For every genus $g$ and every finitely generated subgroup $G\leq \mathrm{Mod}(S_g)$, where $S_g$ is a closed oriented surface, the finite-index Frattini subgroup…

For every $n\ge 2$, every cocompact lattice $\Gamma<\mathrm{Isom}(\mathbb H^n)$, and every finitely supported admissible probability measure $\mu$ on $\Gamma$, the hitting measure…

Let $A_\Gamma$ be an Artin group and let $W_\Gamma$ be its associated Coxeter group, obtained from $A_\Gamma$ by imposing $s_i^2=1$ for every standard generator $s_i$. For a group…

Let $\Gamma<\mathrm{PSL}_2(\mathbb{R})$ be a cocompact Fuchsian group, and let $\mu$ be a finitely supported non-degenerate probability measure on $\Gamma$. If $\nu_\mu$ denotes th…

For every finitely generated commutative monoid $M$ and every bounded-to-one Borel action $M\curvearrowright X$, the action has finite Borel asymptotic dimension:…

For each integer $n\geq 3$, let $G=\mathrm{SL}_n(\mathbb{Z})$ and, for a finite generating set $S\subseteq G$, let $\kappa(G,S)$ denote its Kazhdan constant. Is…

Let $\mathscr{S}$ be the class of all non-discrete, compactly generated, totally disconnected locally compact groups that are topologically simple. Is the quotient of $\mathscr{S}$…

For every finitely generated, torsion-free, non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, the group von Neumann algebra $L(\Gamma)$ is isomorphic to a free…

For every $1<p<\infty$, every measure space $(\Omega,\mu)$, and every real Banach space $X$ that is either a closed subspace of $L_p(\Omega,\mu;\mathbb{R})$ or a quotient of such a…

Let $X$ be a Hadamard space and let $r=\operatorname{asrk}(X)$ be its asymptotic rank. For every integer $k\ge r$, there exists a constant $C=C(X,k)>0$ such that every integral $k$…

Drift conjecture. The drift function satisfies

Atiyah conjecture. The von Neumann dimension of the kernel of this map is an integer.

Baum–Connes conjecture. The assembly map $\mu_{r}$ is an isomorphism for $\ast=0,1$.

Let $G$ be a Gromov hyperbolic group whose boundary at infinity is homeomorphic to the two-sphere $\mathbb{S}^2$. A group is virtually Kleinian if, up to a finite-index subgroup, i…

Let $G$ be a Gromov hyperbolic group, and consider affine actions of $G$ on a Hilbert space whose linear parts are uniformly bounded representations. An affine action is proper whe…

Ballmann–Buyalo's conjecture. If $X$ contains a geodesic of rank $1$, then it also contains a $G$-periodic geodesic of rank $1$.

Minimal-polynomial conjecture. A well-defined tree pair for $\textbf{F}_{\beta}$ can be derived from $P(x)$ only if $P(x)$ is the minimal polynomial having root $\beta$ among polyn…

Coarse Menger conjecture. There is a function $g:\mathbb{N}^2\to\mathbb{N}$ such that for every $k,d\in\mathbb{N}$, every graph or geodesic metric space $G$, and every two sets…

Let $G=\langle X\rangle$ be a group generated by a finite set $X$, with growth function … Write $f\prec g$ for the growth comparison used in the source: there is a constant $C>0$ s…

Let $G$ be a finitely generated group. The domino problem asks whether a given finite set of Wang tiles admits a tiling of $G$. Ballier–Stein conjecture. $G$ has undecidable domino…

Let $G$ be a finitely generated group and let $H$ be a subgroup. An $H$-almost invariant subset $A$ is $H$-proper if it is not equivalent to either the empty set or $G$ modulo fini…

Let $G$) be a finitely generated group and let $H$ be a subgroup. An $H$-almost invariant subset $A$ is $H$-proper if it is not equivalent to either the empty set or $G$ modulo fin…

Let $\Gamma<\mathrm{SL}_N(\mathbb{R})$ be a discrete subgroup, and let the random walk on $\Gamma$ have finite support. Its hitting measure is a measure on the boundary at infinity…

Let $G$ be a finitely generated group. A strongly aperiodic subshift of finite type (SFT) is an SFT with no periodic configurations. The strongly aperiodic SFT conjecture. $G$ admi…