AI 中文总结
该论文引入凯莱树伪轨道跟踪性质，通过相关准则刻画虚拟自由性等，证明强拓扑Rokhlin性质可传递到有限指数超群，回答了虚拟循环情形问题，确定该性质是有限生成虚拟自由群康托尔作用的通用性质。
AI 中文摘要
我们引入凯莱树伪轨道跟踪性质（Cayley-tree POTP），该性质通过仅在凯莱图的生成树方向上施加有限生成群作用的伪轨道方程得到。对于零维作用，我们通过归一化替换路径上的等连续性刻画该性质；对于子转移，该准则以公共左周期子群的右陪集空间形式表达。这些准则刻画了虚拟自由性，对于可公度子群对，确定了陪集全转移的凯莱树伪轨道跟踪性质对应相对准树几何与有限Bass–Serre分解。对于无限指数VFP对，陪集全转移的凯莱树伪轨道跟踪性质等价于虚拟上同调余维数1，尽管普通伪轨道跟踪性质对所有此类转移均成立。最后，我们证明强拓扑Rokhlin性质可传递到有限指数超群，因此每个有限生成虚拟自由群都具有该性质，回答了Doucha提出的虚拟循环情形问题，且凯莱树伪轨道跟踪性质是其康托尔作用的通用性质。
英文摘要
We introduce Cayley-tree POTP, obtained by imposing the pseudo-orbit equations of a finitely generated group action only along a spanning tree of a Cayley graph. For zero-dimensional actions, we characterize this property by equicontinuity along normalized replacement paths; for subshifts, the criterion is expressed in the right-coset space of the common left-period subgroup. These criteria characterize virtual freeness and, for commensurated subgroup pairs, identifies Cayley-tree POTP of the coset full shift with relative quasi-tree geometry and a finite Bass--Serre decomposition. For infinite-index VFP pairs, Cayley-tree POTP of the coset full shift is equivalent to virtual cohomological codimension one, although ordinary POTP holds for every such shift. Finally, we prove that the strong topological Rokhlin property passes to finite-index overgroups. Consequently every finitely generated virtually free group has this property, answering the virtually cyclic case posed by Doucha, and Cayley-tree POTP is generic for its Cantor actions.
