On a free group of transformations defined by an automaton

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We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3.

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Amenability of linear-activity automaton groups (2009, background): “There are exponential activity growth automaton groups that are isomorphic to the free group (see Glasner and Mozes, 2005; Vorobets and Vorobets, 2007).”
Asymptotic aspects of Schreier graphs and Hanoi Towers groups (2006, background): “The growth of Γ is exponential for the automata generating the lamplighter group [13] and Baumslag-Solitar solvable groups [4], the Bellaterra automaton [15] generating the free product of three cyclic groups of order 2, and the Aleshin automaton [1,18] generating the free group of rank 3.”