# Coarse Geometry and Inverse Semigroups

## Document Type

Presentation Abstract

## Presentation Date

11-25-2019

## Abstract

In this talk we will discuss, mainly, two seemingly disconnected notions in mathematics: coarse geometry and inverse semigroups.

Geometry often studies certain objects (such as sets or manifolds) equipped with a distance function. For instance, one classical problem would be to classify every compact manifold up to diffeomorphism. Coarse geometry shifts the point of view, and defines two sets to be *coarse equivalent* if they *look the same from far away*. In this way, for instance, a point and a sphere are indistinguishable from each other. Coarse geometry then studies properties that remain invariant under this weak equivalence relation, that is, properties of the space that only *appear at infinity*.

On the other hand, an inverse semigroup is a natural generalization of the notion of group, and is closely related to the idea of groupoid. Starting with one of these objects we will introduce how to construct a metric space, in the same fashion as the Cayley graph construction in the context of groups. We will then study its coarse structure, in particular its property A and its amenability. Time permitting, we will also relate these properties to analogue properties in some operator algebras.

## Recommended Citation

Martinez, Diego, "Coarse Geometry and Inverse Semigroups" (2019). *Colloquia of the Department of Mathematical Sciences*. 573.

https://scholarworks.umt.edu/mathcolloquia/573

## Additional Details

Monday, November 25, 2019 at 3:00 p.m. in Math 103

Refreshments at 4:00 p.m. in Math Lounge 109