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The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three squares, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete VH complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. In particular, this gives a nonresidually finite CAT(0) group isometric to F 2 ×F 2 . For the class of complete directed VH complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise on whether the main example from his PhD thesis is non-residually finite. As a by-product, we get finitely presented torsion-free simple groups which decompose into an amalgamated free product of free groups F 7 * F 49 F 7 .Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete VH complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F 2 and the first automaton representation of the free product C 3 * C 3 .

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT.0/ groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete V H complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. For the class of complete directed VH complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise. Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete VH complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F 2 .

The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥ cn for some constant c > 0. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3.In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods.