21 problems

A hyperbolic group is a group that is Gromov-hyperbolic with respect to a word metric; a group is residually finite if every nontrivial element survives in some finite quotient. Re…

Let $G$ be a torsion-free nilpotent group, and let $\delta$ be the constant in the previously established upper bound for the residual finiteness growth $\operatorname{RF}_G$. Exac…

Let $G$ be a nilpotent group with complex Mal'cev completion, and let $\operatorname{RF}_G$ denote its residual finiteness growth. Mal'cev-completion invariance conjecture. The fun…

Undecidability conjecture. The problem of deciding whether a given hyperbolic group has a finite quotient is undecidable. Equivalently, there exists a hyperbolic group which is not…

Let $M^n$ be a closed aspherical $n$-manifold with residually finite fundamental group. Let $\Gamma_k\vartriangleleft\pi_1(M^n)$ be any normal chain with $\bigcap_k\Gamma_k=1$. Lüc…

A hyperbolic manifold is arithmetic when its fundamental group is an arithmetic lattice. A subgroup is geometrically finite if its convex core has finite volume, and a lattice is G…

Magnus's conjecture. The free group $F_{2}$ is residually $S$ for every such set $S$.

Let $G$ be an algebraic group over a local field of positive characteristic, and let $U$ be a discrete unipotent subgroup of $G$. A group is residually finite if every nontrivial e…

Let $F_2=\langle x,y\rangle$ be the free group of rank two, and let $u(x,y),v(x,y)$ be normal generators of $F_2$. For a finite factor group $G$ of $F_2$, write $\bar x,\bar y$ and…

A laminar group is a group admitting the relevant laminar action, and a very full lamination is a lamination all of whose gaps are finite-sided ideal polygons. Calegari's conjectur…

A group is residually finite if every nontrivial element survives in some finite quotient. A group is hyperbolic in the sense of Gromov if its Cayley graph is Gromov-hyperbolic. Re…

Residual finiteness growth conjecture. For any $k>1$,

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 simple, simply connected algebraic group over $\mathbb{Q}$ of real rank $1$. The Serre conjecture. The congruence kernel of $G$ is infinite. This conjecture concerns t…

Residual finiteness conjecture for branched good stackings. Wise's proof that one-relator groups with torsion are residually finite should go through for groups with branched good…

Residual-finiteness conjecture. All of those non-linear groups fail to be residually finite, and are thus virtually simple.

Let $\Delta_S$ be the complete $\mathcal{VH}$ complex with four squares described in the paper, and let $\pi_1(\Delta_S)$ denote its fundamental group. Rattaggi's conjecture. The g…

A one-relator group with torsion is a group admitting a presentation with one defining relator that has finite order in the corresponding one-relator quotient. Baumslag's conjectur…

Involutory quandles are finitely residual. If $IQ(D)$ is non-trivial, then there is a finite non-trivial involutory quandle $Q$ and a surjective homomorphism

Non-residually finite hyperbolic group conjecture. There exists a hyperbolic group that is not residually finite.

A group $\Gamma$ is word-hyperbolic if it is a finitely generated group whose Cayley graph is Gromov-hyperbolic. Gromov's residual finiteness conjecture. Every word-hyperbolic grou…