Testing spherical transitivity in iterated wreath products of cyclic groups

Abstract

We give a partial solution a question of Grigorchuk, Nekrashevych, Sushchanskii and Šunik by giving an algorithm to test whether a finite state element of an infinite iterated (permutational) wreath product $\hat{G}=\mathbb{Z}/k\mathbb{Z}\wr \mathbb{Z}/k\mathbb{Z}\wr \mathbb{Z}/k\mathbb{Z}\wr >...$ of cyclic groups of order $n$ acts spherically transitively. We can also decide whether two finite state spherically transitive elements of $\hat{G}$ are conjugate. For general infinite iterated wreath products, an algorithm is presented to determine whether two finite state automorphisms have the same image in the abelianization.