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The large scale geometry of inverse semigroups and their maximal group images

Mark Kambites, Nóra Szakács

TL;DR

The paper studies how the geometry of inverse semigroups relates to their maximal group images, focusing on distortion of the natural map and when coarse properties like Property A transfer. It develops a comprehensive framework using standard metrics and Schützenberger graphs to analyze when distortion is bounded, providing a direct elementary proof of Property A lifting in $E$-unitary cases and characterizing bounded distortion via $F$-inverse properties for special inverse monoids. It also examines computability aspects, showing both recursively bounded distortion and limits of algorithmic decidability, and furnishes explicit constructions demonstrating unbounded distortion. The results have implications for semigroup theory and operator algebra constructions from inverse semigroups.

Abstract

The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative $C^*$-algebras. We study the relationship between the geometry of an inverse semigroup and that of its maximal group image, and in particular the geometric \textit{distortion} of the natural map from the former to the latter. This turns out to have both implications for semigroup theory and potential relevance for operator algebras associated to inverse semigroups. Along the way, we also answer a question of Lledó and Martínez by providing a more direct proof that an $E$-unitary inverse semigroup has Yu's Property A if its maximal group image does.

The large scale geometry of inverse semigroups and their maximal group images

TL;DR

The paper studies how the geometry of inverse semigroups relates to their maximal group images, focusing on distortion of the natural map and when coarse properties like Property A transfer. It develops a comprehensive framework using standard metrics and Schützenberger graphs to analyze when distortion is bounded, providing a direct elementary proof of Property A lifting in -unitary cases and characterizing bounded distortion via -inverse properties for special inverse monoids. It also examines computability aspects, showing both recursively bounded distortion and limits of algorithmic decidability, and furnishes explicit constructions demonstrating unbounded distortion. The results have implications for semigroup theory and operator algebra constructions from inverse semigroups.

Abstract

The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative -algebras. We study the relationship between the geometry of an inverse semigroup and that of its maximal group image, and in particular the geometric \textit{distortion} of the natural map from the former to the latter. This turns out to have both implications for semigroup theory and potential relevance for operator algebras associated to inverse semigroups. Along the way, we also answer a question of Lledó and Martínez by providing a more direct proof that an -unitary inverse semigroup has Yu's Property A if its maximal group image does.
Paper Structure (5 sections, 25 theorems, 37 equations)

This paper contains 5 sections, 25 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Inverse Semigroups and Their Geometry
  3. Property A in $E$-unitary inverse semigroups
  4. Distortion and (weak) $F$-Inverse properties
  5. Computable distortion

Key Result

Lemma 2.3

Let $S$ be an $E$-unitary inverse monoid generated by a set $X$, and $w \in (X \cup X^{-1})^\ast$. Let $G$ be the maximal group image of $S$, and $\Gamma(G)$ the Cayley graph of $G$ with respect to the generating set $X$. Then the Schützenberger graph $S\Gamma(w)$ is isomorphic to the smallest subgr

Theorems & Definitions (51)

  • Definition 2.1: standard metric on an inverse semigroup
  • Definition 2.2: proper length function on an inverse semigroup
  • Lemma 2.3
  • Proposition 3.1
  • Definition 3.2: Property A
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Definition 4.1: bounded group distortion
  • ...and 41 more