Global conformal assouad dimension in the heisenberg group

We study global conformal Assouad dimension in the Heisenberg group Hⁿ. For each α ∈ (0) ∪ [1, 2n + 2], there is a bounded set in Hⁿ with Assouad dimension α whose Assouad dimension cannot be lowered by any quasiconformal map of Hⁿ. On the other hand, for any set S in Hⁿ with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets F(S), taken over all quasiconformal maps F of Hⁿ, equals zero. We also consider dilatation-dependent bounds for quasiconformal distortion of Assouad dimension. The proofs use recent advances in self-similar fractal geometry and tilings in Hⁿ and regularity of the Carnot–Carathéodory distance from smooth hypersurfaces.