The smallest Mealy automaton generating an indicable regular branch group
Adam Woryna · Forum Mathematicum · 2023
Abstract We construct a two-state Mealy automaton A over the three-letter alphabet generating a regular branch group G ( A ) {G(A)} , which surjects onto the infinite cyclic group. Some algebraic and geometric properties of the group G ( A ) {G(A)} are derived. In particular, this group has a nearly finitary subgroup of index two, is amenable, just non-solvable, has exponential growth, and its action on the corresponding regular rooted tree is self-replicating, contracting, but it does not have the congruence subgroup property. We also derive in detail an ascending finite L-presentation for the group G ( A ) {G(A)} .