发表机构
McGill University; Carleton University（麦吉尔大学; 卡尔顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过图的\textit{G}-不变树分解，给出可数群在局部有限双曲图上的作用诱导格罗莫夫边界超有限轨道等价关系的条件，并证明逆命题在附加条件下成立。
AI 中文摘要
我们研究可数群在连通局部有限双曲图上的作用，依据图的树分解，该作用能否在图的格罗莫夫边界上诱导超有限轨道等价关系的条件。我们证明，对于配备可数群\textit{G}作用的连通局部有限双曲图\textit{X}，若(\textit{T}, \beta)是\textit{X}的\textit{G}-不变树分解，满足对每个\textit{t}∈\textit{V}(\textit{T})，每个袋诱导\textit{X}的连通子图\textit{X_t}，每个黏合集有限，且\textit{T}的边仅存在有限个\textit{G}-轨道，则当\textit{G}在\textit{∂T}上的轨道等价关系超有限，且所有袋稳定子在\textit{∂X_t}上的轨道等价关系均超有限时，\textit{G}在格罗莫夫边界\textit{∂X}上作用的轨道等价关系超有限。我们还证明，若(\textit{T}, \beta)满足每个黏合集至少区分\textit{X}的两个端点这一附加性质，则逆命题也成立。
英文摘要
We study conditions for a countable group acting on a connected locally finite hyperbolic graph to induce a hyperfinite orbit equivalence relation on the Gromov boundary of the graph in terms of tree-decompositions of the graph. We prove that for a connected locally finite hyperbolic graph $X$ equipped with an action of a countable group $G$, if $(T, β)$ is a $G$-invariant tree-decomposition of $X$ such that each bag induces a connected subgraph $X_t$ of $X$ for each $t \in V(T)$, each adhesion set is finite and such that there are only finitely many $G$-orbits of edges of $T$, then the orbit equivalence relation of $G$ acting on the Gromov boundary $\partial X$ is hyperfinite provided the orbit equivalence relation of $G$ acting on $\partial T$ is hyperfinite and the orbit equivalence relations of the bag stabilizers acting on $\partial X_t$ are all hyperfinite. We show that the converse also holds if $(T, β)$ satisfies the additional property that each adhesion set distinguishes at least two ends of $X$.
Comments19 pages, no figures. Added an AI statement to this version. Comments welcome
