Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag–Solitar groups BS(1,n)

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 groups BS(1,n) for n≥2. 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 ℤ 2 or Goodman–Strauss's on ℍ 2 .