Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincaré inequality

For every integer $d\geq 3$ and real number $Q\in \left(d,\frac{d(d-1)+2}{2}\right]$, we construct a $Q$-Ahlfors regular metric measure manifold $(M,\mathbf{d},μ)$ of topological dimension $d$ and analytic dimension $2$ that satisfies a Poincaré inequality. For every $Q>\frac{d(d-1)+2}{2}$, the cons