Polynomial growth and property ${\mathrm{RD}}_p$ for étale groupoids with applications to $K$-theory

Abstract

We investigate property ${\mathrm{RD}}_p$ for étale groupoids and apply it to the computation of the $K$-theory of reduced groupoid $L^p$-operator algebras. In particular, under the assumption of polynomial growth, we show that the $K$-theory groups for a reduced groupoid $L^p$-operator algebra are independent of $p\in (1,\infty )$. We apply the results to coarse groupoids and graph groupoids.