Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

TL;DR

This paper introduces a unique way to equip quasi-countable inverse semigroups with metrics, defines their uniform Roe algebras, and characterizes those with asymptotic dimension zero, revealing new structural insights.

Contribution

It generalizes proper, right invariant metrics from groups to inverse semigroups and characterizes inverse semigroups with asymptotic dimension zero using uniform Roe algebras.

Findings

Inverse semigroups with asymptotic dimension 0 are exactly the locally finite inverse semigroups.

Uniform Roe algebras are strongly quasi-diagonal for these semigroups.

Having a finite uniform Roe algebra is characterized by local -finiteness and sparsity.

Abstract

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 $\ell^\infty(S)$ and $S$. We relate these metrics to the analogous metrics on Hausdorff \'{e}tale 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 $\mathcal L$-finite, and equivalently sparse as a metric space.