Hyperbolicity and quasiconformal maps on the affine-additive group

This thesis presents the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group. After analyzing the quasiconformal mapping theory associated to the affine-additive group, we define linear and radial stretch maps, and prove that they are minimizers of the mean quasiconformal distortion functional. For the proofs we use a method based on the notion of modulus of a curve family and the minimal stretching property (MSP) of the afore-mentioned maps. MSP relies on certain given curve families compatible with the respective geometric settings of the strech maps. Finally, by means of a Riemannian approximation scheme combined with Cartan’s formalism, we establish notions of mean and Gaussian curvature for surfaces embedded in the affine-additive group.

2025

dissertation

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