14 problems

A finitely generated group $G$ is B-scale-invariant if it has a nested sequence of finite-index subgroups, each isomorphic to $G$, whose intersection is finite. If the intersection…

Let $\mathsf{X}=\{0,1\}$, and let $a,b,c$ be the three states of the automaton defined by … … … for all $w\in\mathsf{X}^*$. Sidki's conjecture. The group generated by $a,b,c$ is a…

Let $G$ be a finitely presented self-similar metabelian group, and let $\mathbb{Z} \wr \mathbb{Z}$ denote the wreath product of two copies of the infinite cyclic group. The authors…

Let $G\leq \operatorname{Aut}(\mathcal{T}_d)$ be a contracting self-similar group, and let $V_dG$ be its associated Rӧver–Nekrashevych group. Nekrashevych's conjecture. The group…

Noseda–Snopce conjecture. The group $H$ is not self-similar.

Decidability equivalence. The following are equivalent:

Let $H_q$ be the Thue–Morse group, realized as the iterated monodromy group of the degree-$q$ branched covering described using the post-critical set…

Classification conjecture. $G$ is strongly hereditarily self-similar of index $p$ if and only if $G$ is isomorphic to

Lower central series rank conjecture. The ranks satisfy

A contracting self-similar group is a self-similar group whose action has a finite nucleus controlling all sufficiently deep sections. Nekrashevych's amenability conjecture. Every…

Let $f$ be a topological polynomial, and let $\mathfrak G_f$ denote the bimodule associated with $f$. Nekrashevych's conjecture. The bimodule $\mathfrak G_f$ is sub-hyperbolic. The…

Let $G$ be a self-similar group as described in the source, and let ${\mathscr L}^\Bbbk(G)$ be its associated Lie algebra over the field $\Bbbk$. Let ${\mathscr L}(G)$ be the self-…

Let $G$ be a self-similar group as described in the source, let ${\mathscr A}(G)$ be its thinned algebra with augmentation ideal $\varpi$, and define its associated graded algebra…

Let $G$ be a group, and let its associated Lie algebra be defined from the relevant group-theoretic filtration. Torsion-group conjecture. If $G$ is a torsion group, then its associ…