Inverse semigroup from metrics on doubles, III: Commutativity and (in)finiteness of idempotents

Vladimir Manuilov

Vol. 328 (2024), No. 2, 325–338

DOI: 10.2140/pjm.2024.328.325

Abstract

We have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we study the property of idempotents in $M(X)$ of being finite or infinite, which is similar to this property for projections in $C^{\ast }$-algebras. We show that if $X$ is a free group then the unit of $M(X)$ is infinite, while if $X$ is a free abelian group then it is finite. As a by-product, we show that the inverse semigroup $M(X)$ is not a quasiisometry invariant. We also show that $M(X)$ is commutative if it is Clifford, and give a geometric description of spaces $X$ for which $M(X)$ is commutative.

Keywords

inverse semigroup, metric, finite projection

Mathematical Subject Classification

Primary: 20M18, 51F30

Milestones

Received: 15 January 2024

Accepted: 15 March 2024

Published: 30 April 2024