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

We introduce the notion of (strong) subexponential growth for étale 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 $p\in [1,\mathrm{\infty })$ 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 étale groupoid associated with a shift of infinite type which has strong subexponential growth, but not polynomial.