32 problems

A first-order sentence is a sentence in the language of groups, and a few-relator random quotient is a random quotient of a free group by a small number of relators. Knight's conje…

A random-group model is a probabilistic model producing finitely generated groups. Gromov's generic-group conjecture. There should exist a model for random groups in which a generi…

Let $\mathcal{G}'(r)$ denote the random model of two-generator one-relator groups whose defining relation is a commutator, and let $p_r$ be the probability that a group…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let the closure of a subgroup generated by random elements be considered in the profinite topolo…

Let $\mathcal{U}$ be a non-principal ultrafilter. For each $n$, let $(X_n,\mu_n)$ be the uniform random action of $F_d$ on an $n$-element set with normalized counting measure, and…

Let $0\leq d<\frac{1}{2}$, and let $\psi$ be a first-order sentence in the language of groups. Let $F_k$ be the free group of rank $k\geq 2$, and consider the density-$d$ random-gr…

Random-group graphical discreteness conjecture. Random groups in the few-relator model and Gromov's density model are graphically discrete asymptotically almost surely as the lengt…

Random Menger-boundary discreteness conjecture. We conjecture that, generically—for random groups—hyperbolic groups with Menger curve boundary are graphically discrete.

Surface curvature of random complexes in the density model. The probability that $\sigma_+(X)<0$ tends to $1$ as $l\to\infty$.

Irreducible curvature bounds of few-relator complexes. The probability that $\rho_+(X)=\kappa(X)$ tends to $1$ as $l\to\infty$. In particular, with high probability, $\rho_+(X)<0$…

Let $S$ be a finite set of generators, let $d\in(0,1)$ be a density, and consider a random group in the Gromov density model with density $d$ and relator length tending to infinity…

Weak convergence conjecture. On ${\mathcal P}_{\Gamma,q,\ell}$, the measures $nu_{q,\ell,n}$ converge weakly, as $n\to\infty$, to the measure $mu_{\Gamma,q}$ from the existence and…

A BMW group of degree $(m,n)$ is a group in the paper's class acting on a product of regular trees of degrees $m$ and $n$. Consider BMW groups of degree $(m,n)$ up to conjugacy, an…

A BMW group is a group acting on a product of two regular trees, and a BMW group of degree $(m,n)$ has tree factors of degrees $m$ and $n$. In the range $n>m^5$, consider the rando…

Let $d<1/16$, and let $\Gamma_\ell$ be a random group of density $d$ at length $\ell$. A first-order sentence is almost surely true in this model if the probability that it holds i…

Asymptotic threshold conjecture. The limit

Let $S_n$ be the symmetric group on $n$ points. Choose independent uniformly random permutations from $S_n$, and let the average number required to invariably generate a transitive…

Let $\Gamma$ be a finite group and let $Q$ be the global field $\mathbb{Q}$ or $\mathbb{F}_q(t)$ for a prime power $q$. For a Galois extension $k/Q$ with…

Random-group conjecture. With high probability,

Let $A(\Gamma)$ be a right-angled Artin group whose defining graph $\Gamma$ is connected and not a join, and let $H$ be a stable subgroup of $A(\Gamma)$. Let $(\mu_i)$ be a sequenc…

Equidistribution conjecture. As $K$ varies, ordered by $\operatorname{rDisc} K$, the $\Gamma$-groups

Discreteness probability conjecture. The probability that a group generated by two random parabolic elements is discrete is less than $0.232$.

Let $n$-vertex graphs be sampled from a random graph model with density greater than $1/n$ and bounded away from $1$, and let the associated right-angled Coxeter group be the group…

Let $d$ be a positive integer, and choose two permutations independently and uniformly at random from the symmetric group $\mathrm{S}_d$. Dixon's conjecture. The subgroup of…

Random-spine subgroup conjecture. For any fixed geodesic spine $K$, with overwhelming probability $G$ contains a subgroup commensurable with the orbifold fundamental group of some…