19 problems

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 $\Gamma$ be a finitely generated residually finite group satisfying a non-trivial group law. The rank gradient $RG(\Gamma)$ measures the asymptotic normalized growth of the min…

Property $R_\infty$ conjecture. Every finitely generated residually finite group that is not solvable-by-finite has property $R_\infty$. In the weaker version, every finitely gener…

Let $H$ be a finitely generated residually finite group. A group $G$ persistently contains $H$ if every finite-index subgroup $G'\subset G$ contains a subgroup isomorphic to $H$. P…

A finite-volume hyperbolic manifold is a quotient of real hyperbolic space by a group of isometries acting freely and properly discontinuously. RFRS hyperbolic virtual fibring conj…

Let $d e 0$, let $G$ be a finitely generated infinite residually finite group with uniformly $(1/d)$-almost flat quotients, and let $H$ be a finite-index subgroup admitting a quoti…

Let $G$ be a finitely generated residually finite group. For $d e 0$, write uniformly $(1/d)$-almost flat quotients and uniformly $(1/d)$-almost flat coset spaces for the correspon…

Let $G$ be a finitely generated residually finite group. Having uniformly almost flat quotients means satisfying the quotient condition defined in the paper, and having uniformly a…

Let $G$ be a finitely generated residually finite non-amenable group. Non-amenable property $R_\infty$ conjecture. The group $G$ has property $R_\infty$. This is presented as a wea…

Let $G$ be a finitely generated residually finite group. Fel'shtyn–Troitsky conjecture. The group $G$ either has property $R_\infty$ or is solvable-by-finite. The disjunction is no…

Let $\Gamma$ be a finitely generated residually finite group. An automorphism $\varphi$ of $\Gamma$ has infinite Reidemeister number when $R(\varphi)=\infty$; a group is $R_\infty$…

Let $G$ be a residually finite group and let $(G_n)$ be a nested sequence of finite-index normal subgroups with trivial intersection. The associated box space is the coarse disjoin…

Bounded-commutator local finiteness conjecture. The derived subgroup $G'$ is locally finite.

Zelmanov's conjecture. The group $G$ is locally finite.

Let $\Gamma$ be a finitely generated, residually finite group, and let $S_i$ for $i=1,2$ be finite generating sets with $\operatorname{Id}\notin S_i$ and $S_i^{-1}=S_i$. Set … Let…

Let $G$ be a finitely generated amenable residually finite group satisfying the ascending chain condition on closed subgroups. Virtual solvability conjecture. The group $G$ is virt…

Let $G$ be an infinite, residually finite, finitely generated group. It has the $S_{\infty}$-property if every automorphism of $G$ has infinitely many Reidemeister classes. Fel'sht…

Let $G=\langle x,y,t\mid txt^{-1}=xy,\ tyt^{-1}=yx\rangle$ be the hyperbolic ascending HNN extension of the free group on $x,y$. A group is linear if it embeds into a matrix group…

Let $\Gamma$ be a residually finite group satisfying appropriate finiteness properties, such as having a finite $K(\pi,1)$, and let … be a chain of normal subgroups of finite index…