abstract. we extend matui’s notion of almost finiteness to general étale groupoids
and show that the reduced groupoid c∗ -algebras of minimal almost finite groupoids
have stable rank one. the proof follows a new strategy, which can be regarded as a local
version of the large subalgebra argument.
the following three are the main consequences of our result. (i) for any group of
(local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) any countable amenable
group admits a minimal action on the cantor set all whose minimal extensions form the
crossed product of stable rank one. (iii) for any amenable group, the crossed product of
the universal minimal action has stable rank one.