4 problems

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

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 $G$ 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 $M_{k=j}^{(n)}$ 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…