Bubani, Elia (2025). Hyperbolicity and quasiconformal maps on the affine-additive group. (Thesis). Universität Bern, Bern
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Abstract
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.
| Item Type: | Thesis |
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| Dissertation Type: | Single |
| Date of Defense: | 18 June 2025 |
| Subjects: | 500 Science > 510 Mathematics |
| Institute / Center: | 08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |
| Depositing User: | Hammer Igor |
| Date Deposited: | 25 Aug 2025 16:38 |
| Last Modified: | 25 Aug 2025 16:38 |
| URI: | https://boristheses.unibe.ch/id/eprint/6619 |
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