Hyperfiniteness of boundary actions via tree decompositions

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 ∈ 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 ∂ X is hyperfinite provided the orbit equivalence relation of G acting on ∂ T is hyperfinite and the orbit equivalence relations of the bag stabilizers acting on ∂ 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.

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BibTeX RIS

BibTeX

@misc{karpinski2026hyperfiniteness,
  title = {{Hyperfiniteness of boundary actions via tree decompositions}},
  author = {Chris Karpinski and Bobby Miraftab},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.14748},
  eprint = {2608.14748},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14748},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Hyperfiniteness of boundary actions via tree decompositions
AU  - Chris Karpinski
AU  - Bobby Miraftab
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2608.14748
UR  - https://arxiv.org/abs/2608.14748
AB  - 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 ∈ 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 ∂ X is hyperfinite provided the orbit equivalence relation of G acting on ∂ T is hyperfinite and the orbit equivalence relations of the bag stabilizers acting on ∂ 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.
ER  -

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