Geodesic
Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups
Yeong Chyuan Chung1 orcid ; Diego Martínez2 orcid ; Nóra Szakács3 orcid
1Jilin University, Changchun, P. R. China
2University of Münster, Germany
3University of Manchester, UK
Groups, geometry, and dynamics, Tome 19 (2025) no. 4, pp. 1499-1543

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DOI

Given any quasi-countable, in particular, any countable inverse semigroup S, we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S. This allows us to unambiguously define the uniform Roe algebra of S, which we prove can be realized as a canonical crossed product of l∞(S) and S. We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally L-finite, and equivalently sparse as a metric space.
DOI : 10.4171/ggd/814
Classification : 20M18, 46L89, 46L05
Mots-clés : inverse semigroup, proper metric, right invariant metric, locally finite, asymptotic dimension

Yeong Chyuan Chung  1   ; Diego Martínez  2   ; Nóra Szakács  3

1 Jilin University, Changchun, P. R. China
2 University of Münster, Germany
3 University of Manchester, UK 
Yeong Chyuan Chung; Diego Martínez; Nóra Szakács. Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups. Groups, geometry, and dynamics, Tome 19 (2025) no. 4, pp. 1499-1543. doi: 10.4171/ggd/814
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     title = {Quasi-countable inverse semigroups as metric spaces, and the uniform {Roe} algebras of locally finite inverse semigroups},
     journal = {Groups, geometry, and dynamics},
     pages = {1499--1543},
     year = {2025},
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