The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and $\operatorname{CAT}(0)$ 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 $\mathcal{VH}$ complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. For the class of complete directed $\mathcal{VH}$ 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 $\mathcal{VH}$ 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 $F_2$.

中文翻译：

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自动机群和完全方形复合体

Wise 构造了有限呈现的小抵消群、自动群和 $\operatorname{CAT}(0)$ 群类中的非剩余有限群的第一个例子，作为完全平方复形的基本群 (简称 CSC）有十二个正方形。同时，Janzen 和 Wise 证明了最多三个、五个或七个正方形的 CSC 具有剩余有限的基本群。最小的开放案例是具有四个正方形的 CSC 和具有六个正方形的有向完全 $\mathcal{VH}$ 复合体。我们证明了 Janzen 和 Wise 研究的具有四个正方形的 CSC 具有非剩余有限基本群。对于完全有向 $\mathcal{VH}$ 复形类，我们证明了恰好有两个具有非剩余有限基本群的六个正方形的复形。尤其，这积极回答了 Wise 的问题。我们的方法依赖于 Glasner 和 Mozes 发现的正方形复形和自动机之间的联系，其中具有一个顶点的完整 $\mathcal{VH}$ 复形对应于双可逆自动机。我们证明了与具有两个状态的双可逆自动机相关联的平方复数或在生成无限自动机群的二进制字母表上具有非剩余有限基本群。我们用四个正方形描述与 CSC 相关的自动机群，并得到自由群 $F_2$ 的两个简单自动机表示。我们证明了与具有两个状态的双可逆自动机相关联的平方复数或在生成无限自动机群的二进制字母表上具有非剩余有限基本群。我们用四个正方形描述与 CSC 相关的自动机群，并得到自由群 $F_2$ 的两个简单自动机表示。我们证明了与具有两个状态的双可逆自动机相关联的平方复数或在生成无限自动机群的二进制字母表上具有非剩余有限基本群。我们用四个正方形描述与 CSC 相关的自动机群，并得到自由群 $F_2$ 的两个简单自动机表示。