  Minimal Cantor systems of finite topological rank (that can be represented by
a Bratteli-Vershik diagram with a uniformly bounded number of vertices per
level) are known to have dynamical rigidity properties. We establish that such
systems, when they are expansive, define the same class of systems, up to
topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic
subshifts. This is done establishing necessary and sufficient conditions for a
minimal subshift to be of finite topological rank. As an application, we show
that minimal subshifts with non-superlinear complexity (like all classical zero
entropy examples) have finite topological rank. Conversely, we analyze the
complexity of ${\mathcal S}$-adic subshifts and provide sufficient conditions
for a finite topological rank subshift to have a non-superlinear complexity.
This includes minimal Cantor systems given by Bratteli-Vershik representations
whose tower levels have proportional heights and the so called left to right
${\mathcal S}$-adic subshifts. We also exhibit that finite topological rank
does not imply non-superlinear complexity. In the particular case of
topological rank 2 subshifts, we prove their complexity is always subquadratic
along a subsequence and their automorphism group is trivial.
