Polynomial growth and property $R{D}_p$ for étale groupoids with applications to $K$-theory

We investigate property $R{D}_p$ for étale groupoids and apply it to $K$-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 $p\in (1,\mathrm{\infty })$. We apply the results to coarse groupoids and graph groupoids.