6 problems

Let an automaton group have degree $2$, meaning that every element $g$ has activity $a_n(g)=O(n^2)$, and let the associated degree-$2$ mother groups be the specific universal autom…

Let $\widetilde{\mathcal{H}}_n$ denote the class of elements considered in the paper, and let $A\in\widetilde{\mathcal{H}}_n$ have infinite order. The exponential core growth conje…

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…

Let $G$ be a finite automaton group, and let $G_{\mathcal O}$ denote the orbit group associated with an orbit $\mathcal O$ of $G$. Orbit-group finiteness conjecture. For each finit…

Let $G$ be a polynomial automaton group of degree at most $2$, and let a random walk on $G$ have a step distribution with bounded support. The bounded-support zero-entropy conjectu…

Let $d$ be the degree of a mother group, and consider random walks whose step distribution is supported on a finite generating set. The mother-group zero-entropy conjecture. The mo…