Year: 2026
Type: preprint
Abstract: 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 $X$.
Source: arXiv (Cornell University)
Authors: Karpinski, Chris, Miraftab, Bobby
Institutions: McGill University, Carleton University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203666100