K-theory invariance of $L^p$-operator algebras associated with \'etale groupoids of strong subexponential growth

TL;DR

This paper introduces the concept of strong subexponential growth for étale groupoids and proves that their associated $L^p$-operator algebras have invariant K-theory across all $p$, with examples illustrating the theory.

Contribution

It establishes K-theory invariance for $L^p$-operator algebras of étale groupoids with strong subexponential growth, including new examples with non-polynomial growth.

Findings

K-theory groups are independent of p for groupoids with strong subexponential growth

Introduction of strong subexponential growth for étale groupoids

Existence of examples with strong subexponential but non-polynomial growth

Abstract

We introduce the notion of (strong) subexponential growth for \'etale groupoids and study its basic properties. In particular, we show that the K-groups of the associated groupoid $L^p$-operator algebras are independent of $p \in [1,\infty)$ whenever the groupoid has strong subexponential growth. Several examples are discussed. Most significantly, we apply classical tools from analytic number theory to exhibit an example of an \'etale groupoid associated with a shift of infinite type which has strong subexponential growth, but not polynomial.