  We present a technique to lift some tilings of the discrete hyperbolic plane
-- tilings defined by a 1D substitution -- into a zero entropy subshift of
finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for
$n\geq2$. For well chosen hyperbolic tilings, this SFT is also aperiodic and
minimal. As an application we construct a strongly aperiodic SFT on $BS(1,n)$
with a hierarchical structure, which is an analogue of Robinson's construction
on $\mathbb{Z}^2$ or Goodman-Strauss's on $\mathbb{H}_2$.
