Coarse geometry of metric measure spaces

Takayuki Okuda; Takashi Shioya

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arXiv · 2609.30700

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 $\ell^\infty$ 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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Takayuki Okuda, Takashi Shioya. 2026-09-25. Coarse geometry of metric measure spaces. https://arxiv.org/abs/2609.30700

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