Minimal amenable subshift with full mean topological dimension

Let G be an infinite countable amenable group and P a polyhedron with the topological dimension dim ⁡ ( P ) < ∞ . We construct a minimal subshift ( X , G ) of ( P G , G ) such that its mean topological dimension is equal to dim ⁡ ( P ) . This result answers the question of Dou (2017 Discrete Contin. Dyn. Syst. 37 1411–24). Moreover, it extends the work of Jin and Qiao (2023 arXiv:2102.10339) for Z -action.