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

TL;DR

This paper explores the properties of inverse semigroups derived from metrics on doubled spaces, focusing on the finiteness of idempotents, their relation to the underlying space's algebraic structure, and conditions for commutativity.

Contribution

It characterizes when the inverse semigroup's idempotents are finite or infinite and describes geometric conditions for commutativity, extending understanding of metric space invariants.

Findings

Unit of M(X) is infinite for free groups.

Unit of M(X) is finite for free abelian groups.

M(X) is not a quasi-isometry invariant.

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*-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 quasi-isometry invariant. More examples of finite and infinite idempotents are provided. We also give a geometric description of spaces, for which their inverse semigroup $M(X)$ is commutative.