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PreprintarXiv licencearXiv

Coarse geometry of metric measure spaces

  • Takayuki Okuda
  • Takashi Shioya
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Abstract

Using ideas from optimal transport theory, we introduce a notion of measured coarse equivalence for metric measure spaces and define a corresponding variant of the uniformly finite homology of Block and Weinberger, called weighted ℓ∞ homology, for large-scale doubling metric measure spaces. We prove that this homology is invariant under measured coarse equivalence and that the vanishing of its zeroth homology is equivalent to weighted non-amenability. The proofs combine techniques from optimal transport theory and the disintegration of measures.

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Cite

BibTeXRIS
BibTeX
@misc{okuda2026coarse,
  title = {{Coarse geometry of metric measure spaces}},
  author = {Takayuki Okuda and Takashi Shioya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30700},
  eprint = {2609.30700},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30700},
}
RIS
TY  - UNPB
TI  - Coarse geometry of metric measure spaces
AU  - Takayuki Okuda
AU  - Takashi Shioya
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30700
UR  - https://arxiv.org/abs/2609.30700
AB  - Using ideas from optimal transport theory, we introduce a notion of measured coarse equivalence for metric measure spaces and define a corresponding variant of the uniformly finite homology of Block and Weinberger, called weighted ℓ^∞ homology, for large-scale doubling metric measure spaces. We prove that this homology is invariant under measured coarse equivalence and that the vanishing of its zeroth homology is equivalent to weighted non-amenability. The proofs combine techniques from optimal transport theory and the disintegration of measures.
ER  -