Global conformal Assouad dimension in the Heisenberg group

Abstract

Abstract. We study global conformal Assouad dimension in the Heisenberg group H n . For each α ∈ {0} ∪ [1, 2n + 2], there is a bounded set in H n with Assouad dimension α whose Assouad dimension cannot be lowered by any quasiconformal map of H n . On the other hand, for any set S in H n with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets F (S), taken over all quasiconformal maps F of H n , 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 H n and regularity of the Carnot-Carathéodory distance from smooth hypersurfaces.

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