Groups of finite type: classification and structural properties

Abstract

Groups of finite type (also called finitely constrained groups), introduced by Grigorchuk, are known to be the closure of regular branch groups. This article explores many of their properties. Firstly, we prove that being finitely generated, just-infinite and strongly complete are equivalent in a vast family of groups of finite type. As a consequence, we prove that the closure of the Hanoi towers group on 3 pegs is just-infinite although the group itself is not. Secondly, we improve the algorithm given by Bondarenko and Samoilovych in [9], to compute all the groups of finite type of a given depth and acting on a given tree. We use this to find the groups of finite type acting on the ternary tree with depth 2 and 3. Thirdly, we give a sufficient condition for a group generated by a finite automaton of Mealy type to have as closure a group of finite type. This allows us to identify groups of finite type as the closure of explicit groups generated by a finite automaton. Lastly, we give an algorithm to prove whether two groups of finite type are isomorphic. With this result, we classify groups of finite type up to isomorphism in the binary tree for depths 2, 3 and 4 and in the ternary tree for depths 2 and 3.