Automata and Square Complexes

连接拉丁方与双可逆自动机的组合结构

Combinatorial structures connecting Latin squares and bireversible automata

从组合视角探索字母变换器等自动机理论，将其转移集视为正交阵列类似物，讨论相关组合编码，用组合结构刻画各类自动机，提出字母变换器同构概念，推广对称性并保持双可逆自动机类别。

Cross-wired lamplighter groups and linearity of automata groups

We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machine and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine and give a presentation of the corresponding automata group. We show the automata group is a cross-wired lamplighter group and does not embed in the wreath product of a finite group with a torsion free group. For a subfamily of such finite step two nilpotent groups, we prove that their associated automata groups are linear.

弱特殊方形复形的乘积子复形与交集复形

Product subcomplexes and intersection complexes of weakly special square complexes

本文为弱特殊方形复形定义乘积子复形与交集复形，证明其与万有覆盖的覆叠商典范同构，并建立拟等距诱导交集复形同构的定理。

Automaton groups and complete square complexes

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…

Square complexes and simplicial nonpositive curvature

We prove that each nonpositively curved square VH-complex can be turned functorially into a locally 6-large simplicial complex of the same homotopy type. It follows that any group acting geometrically on a CAT(0) square VH-complex is systolic. In particular the product of two finitely generated free groups is systolic, which answers a question of Daniel Wise. On the other hand, we exhibit an example of a compact non-VH nonpositively curved square complex, whose fundamental group is neither systolic, nor even virtually systolic.

Groups acting on CAT(0) square complexes

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or contains a free group of rank two. In addition we discuss when a group generated by two hyperbolic isometries contains a free group of rank two and when two points in the ideal boundary of a CAT(0) 2-complex at Tits distance $π$ apart are the endpoints of a geodesic in the 2-comp…

关于CAT(0)三角-平方群的双自动性

On the biautomaticity of CAT(0) triangle-square groups

本研究针对CAT(0)三角-平方群的双自动性问题，构造两个反例说明Gersten-Short测地线无法用于证明相关群的双自动性，同时证明了其中一个复形基本群的双自动性。

Torsion groups do not act on $2$-dimensional $\mathrm{CAT}(0)$ complexes

We show, under mild hypotheses, that if each element of a finitely generated group acting on a $2$-dimensional $\mathrm{CAT}(0)$ complex has a fixed point, then there is a global fixed point. In particular all actions of finitely generated torsion groups on such complexes have global fixed points. The proofs rely on Masur's theorem on periodic trajectories in rational billiards, and Ballmann-Brin's methods for finding closed geodesics in $2$-dimensional locally $\mathrm{CAT}(0)$ complexes. As another ingredient we prove that the image of an immersed loop in a graph of girth $2π$ with length no…

弱特殊方形复形的拟等距不变量

Quasi-isometry invariants of weakly special square complexes

本文定义弱特殊方形复形万有覆盖的交复形作为拟等距不变量，用于研究二维右角Artin群和平面图2-辫群的拟等距分类，并证明两类群中各有无穷多个与右角Artin群拟等距或不等距。

作用于具有一致指数增长的 CAT(0) 立方复形的群

Groups acting on CAT(0) cube complexes with uniform exponential growth

研究作用于 CAT(0) 立方复形的群的一致指数增长，证明无全局不动点作用于方复形的群有一致指数增长或稳定欧几里得子复形，推广前人工作并应用于 Higman 群和无三角形 Artin 群子群。

Boundary action of automaton groups without singular points and Wang tilings

We study automaton groups without singular points, that is, points in the boundary for which the map that associates to each point its stabilizer, is not continuous. This is motivated by the problem of finding examples of infinite bireversible automaton groups with all trivial stabilizers in the boundary, raised by Grigorchuk and Savchuk. We show that, in general, the set of singular points has measure zero. Then we focus our attention on several classes of automata. We characterize those contracting automata generating groups without singular points, and apply this characterization to the Bas…

Abelian subgroup structure of square complex groups and arithmetic of quaternions

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.

On the recognition problem for virtually special cube complexes

We address the question of whether the property of being virtually special (in the sense of Haglund and Wise) is algorithmically decidable for finite, non-positively curved cube complexes. Our main theorem shows that it cannot be decided locally, i.e. by examining one hyperplane at a time. Specifically, we prove that there does not exist an algorithm that, given a compact non-positively squared 2-complex X and a hyperplane H in X can decide whether or not there is a finite-sheeted cover of X in which no lift of H self-osculates.

超平面复形的同伦类型与右角Artin群的应用

Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups

研究通过分析准中位图中超平面复形的同伦类型，证明了右角Artin群的准等距性与特定复形的同伦等价性。