20 problems

Exponential core growth conjecture. All strongly synchronizing transducers which generate an automaton group of infinite order have exponential core growth rate.

Steinberg's conjecture. There is an automaton group whose word problem is ${\mathsf{PSPACE}}$-complete.

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…

Aleshin's automaton is a certain three-state automaton over a binary alphabet; its states are automorphisms of the rooted binary tree and generate a subgroup of the corresponding a…

Let ${\mathcal A}$ be the Aleshin automaton shown in Figure 1 (bottom right), and let $G$ be the group generated by the transformations associated with its states. Aleshin automato…

Equality conjecture. For all relevant $n$, $p$, and $q$, one has

Exponential core growth conjecture. The core growth rate of $A$ is exponential. Moreover, for every $i\in\mathbb{N}$,

Core growth conjecture. Any invertible strongly synchronizing automaton which generates an infinite group has exponential core growth rate; moreover, the size of the $i$th power of…

Let $X$ be a finite set, let $X^*$ be the rooted tree given by the free monoid on $X$, and let $G\leq\operatorname{Aut}(X^*)$ be a generalized basilica group: namely, $G$ admits a…

Let $A$ be the so-called Aleshin–Vorobets automaton, and let $G(A)$ denote the group generated by the permutations induced by its states on the rooted tree. Aleshin–Vorobets automa…

Let $G$ be a group generated by a finite state automaton. Bartholdi's conjecture. The group $G$ is either finitely presented or densely related. This conjecture concerns the relati…

Let $F$ be a finite non-abelian group, and let $\mathcal G(\mathcal C(F))$ denote the automata group associated to the Cayley machine of $F$. Kambites–Silva–Steinberg conjecture. T…

Let $F$ be a finite non-abelian group, and let $\mathcal G(\mathcal C(F))$ denote the automata group resulting from the Cayley machine of $F$. Silva–Steinberg conjecture. The group…

A group is elementary amenable if it belongs to the smallest class of groups containing finite and abelian groups and closed under taking subgroups, quotients, extensions, and dire…

Let $M_{k=j}^{(n)}$ be the transition operator indexed by the non-negative integer $j$, let $\operatorname{Sp}(M_{k=j}^{(n)})$ denote its spectrum, and let $m^{(n;j)}(\lambda)$ be…

Let $M_{k=j}^{(n)}$ be the transition operator indexed by the non-negative integer $j$, and let $\operatorname{Sp}(M_{k=j}^{(n)})$ denote its spectrum. Write…

Let $p$ and $q$ be positive integers, and let $\langle\aut{B}_{p,q}\rangle$ denote the group generated by the displayed $p$-letter, $q$-state bireversible Mealy automaton…

Let $p>2$ and $q>2$ when considering the displayed family, together with the specializations for $p=2$ or $q=2$. Let $\langle\aut{M}_{p,q}\rangle$ denote the group generated by the…

Let $\mathcal A$ be an invertible Mealy automaton with an arbitrary stateset. Free-semigroup extension conjecture. Proposition should extend to $\mathcal A$, so that its conclusion…

Let $\mathcal F_1$ be the automaton whose three states generate the group $G(\mathcal F_1)$. Sidki's conjecture. The group $G(\mathcal F_1)$ is a free group on its three generators…