发表机构
University of Torino; Vanderbilt University（都灵大学; 范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从两个视角刻画群的图乘积的轴双曲性，并确立纯共轭自同构的拓扑生成、闭性等性质，还给出阿贝尔群图乘积对应的典范拓扑半直积分解。
AI 中文摘要
我们从两个密切相关的视角研究任意基数定义图上的群的图乘积。首先，我们刻画轴双曲性：若定义图不可约、至少含两个顶点且具有有限星基，则每个抛物完全子群要么是虚循环群，要么是轴双曲群；对于顶点完全子群，有限星基条件也是必要的，特别地，图乘积本身是轴双曲群当且仅当该图具有有限星基且群不是虚循环群，等价于不是无限二面体群，我们还为可约定义图给出了相应分类。其次，受早期关于可数直角科克斯群工作的启发，我们研究逐点收敛拓扑下的纯共轭自同构，证明它们在拓扑上由有限支撑的逐因子自同构和部分共轭生成，并确立了闭性、稠密性、等号性及离散性结果；在星连通情形下，轴双曲性准则中出现的同一有限星基条件支配着闭性与离散性，对任意图而言，其自然细化是存在有限分支见证集。最后，对于阿贝尔群的图乘积，我们得到纯共轭自同构群的典范拓扑半直积分解。
英文摘要
We study graph products of groups over defining graphs of arbitrary cardinality from two closely related viewpoints. First, we characterize acylindrical hyperbolicity. If the defining graph is irreducible, has at least two vertices, and has a finite star base, then every parabolically full subgroup is either virtually cyclic or acylindrically hyperbolic. For vertex-full subgroups the finite star base condition is also necessary. In particular, the graph product itself is acylindrically hyperbolic exactly when the graph has a finite star base and the group is not virtually cyclic, or equivalently is not the infinite dihedral group. We also give the corresponding classification for reducible defining graphs. Second, motivated by our earlier work on countable right-angled Coxeter groups, we study pure conjugating automorphisms in the topology of pointwise convergence. We prove that they are topologically generated by finite-support factorwise automorphisms and partial conjugations, and establish closedness, density, exact-equality, and discreteness results. The same finite star base condition that appears in the acylindrical hyperbolicity criterion governs closedness and discreteness in the star-connected case; for arbitrary graphs, its natural refinement is the existence of a finite component witness set. Finally, for graph products of abelian groups we obtain a canonical topological semidirect decomposition of the pure conjugating automorphism group.
