  When decomposing a finite semigroup into a wreath product of groups and
aperiodic semigroups, complexity measures the minimal number of groups that are
needed. Determining an algorithm to compute complexity has been an open problem
for almost 60 years. The main result of this paper proves decidability of
Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata. This is
achieved by showing the lower bounds in work by Henckell, Rhodes and Steinberg
from 2012 is sharp using profinite methods and results of McCammond from 1991
and 2001.
