Automaton groups and complete square complexes

The first example of a non-residually finite group in the classes finitely presented small-cancelation groups, automatic and $\\operatorname{CAT}(0)$ groups was constructed by Wise as fundamental complete square complex (CSC for short) with twelve squares. At same time, Janzen proved that CSCs at most three, five or seven squares have residually group. smallest open cases were four directed $\\mathcal{VH}$ complexes six We prove CSC studied has For class complexes, we there are exactly two having In particular, this positively answers to question Wise. Our approach relies on connection between automata discovered Glasner Mozes, where one vertex correspond bireversible automata. associated automaton states over binary alphabet generating an infinite describe get simple representations free $F_2$.