Growth of etale groupoids and simple algebras

TL;DR

This paper explores the growth properties of étale groupoids and their convolution algebras, leading to the construction of simple finitely generated algebras with specified Gelfand-Kirillov dimensions and growth rates.

Contribution

It introduces new methods to relate étale groupoid growth to algebraic properties, constructing simple algebras with arbitrary Gelfand-Kirillov dimensions and quadratic growth.

Findings

Constructed simple finitely generated algebras with Gelfand-Kirillov dimension ≥ 2

Developed simple finitely generated algebras with quadratic growth over any field

Linked étale groupoid growth to algebraic complexity measures

Abstract

We study growth and complexity of \'etale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension $\ge 2$ and simple finitely generated algebras of quadratic growth over arbitrary fields.