An Algebraic View of Amenable Groupoids
Abstract
We introduce source-anchored algebraic amenability for Hausdorff ample groupoids, a Følner condition formulated in the Steinberg algebra and localized at compact open subsets of the unit space. The unanchored bisectional condition is exactly algebraic amenability of the Steinberg algebra, whereas the anchored version recovers ordinary amenability for one-unit groupoids and detects behavior invisible to the unanchored theory. We establish permanence under isomorphisms, open invariant reductions, disjoint unions, finite products, directed unions and AF approximations, while arbitrary subgroupoids need not inherit the property. For countable transitive discrete groupoids, infinite unit spaces are amenable through range escape, whereas finite unit spaces are amenable precisely when their isotropy groups are. This yields a characterization of hereditary algebraic amenability in terms of isotropy.
Related
- The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids
- Global and local triviality for extensions of ample groupoids: an algebraic approach
- Discretisation in algebraic K-theory of independent ample groupoids and their Steinberg algebras
- A Nica-amenable submonoid of a group which fails to embed in an amenable group
- Groups with rapid decay and trivial amenable radical are C*-simple
BibTeX
@misc{herrera2026algebraic,
title = {{An Algebraic View of Amenable Groupoids}},
author = {Karol Herrera and Andrés Rubiano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00789},
eprint = {2610.00789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00789},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Algebraic View of Amenable Groupoids
AU - Karol Herrera
AU - Andrés Rubiano
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00789
UR - https://arxiv.org/abs/2610.00789
AB - We introduce source-anchored algebraic amenability for Hausdorff ample groupoids, a Følner condition formulated in the Steinberg algebra and localized at compact open subsets of the unit space. The unanchored bisectional condition is exactly algebraic amenability of the Steinberg algebra, whereas the anchored version recovers ordinary amenability for one-unit groupoids and detects behavior invisible to the unanchored theory. We establish permanence under isomorphisms, open invariant reductions, disjoint unions, finite products, directed unions and AF approximations, while arbitrary subgroupoids need not inherit the property. For countable transitive discrete groupoids, infinite unit spaces are amenable through range escape, whereas finite unit spaces are amenable precisely when their isotropy groups are. This yields a characterization of hereditary algebraic amenability in terms of isotropy.
ER -