Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces

We discuss the Heisenberg group \Heisⁿ and its mappings from three perspectives. As a nilpotent Lie group, \Heisⁿ can be viewed as a generalization of the real numbers, leading to new notions of base-$b$ expansions and continued fractions. As a metric space, \Heisⁿ serves as an infinitesimal model (metric tangent space) of some sub-Riemannian manifolds and allows one to study derivatives of mappings between such spaces. As a subgroup of the isometry group of complex hyperbolic space \Hypⁿ⁺¹_\C, \Heisⁿ becomes a large-scale model of a rank-one symmetric space and provides rigidity results in \Hypⁿ⁺¹_\C. After discussing homotheties and conformal mappings of \Heisⁿ, we show the convergence of base-$b$ and continued fraction expansions of points in \Heisⁿ, and discuss their dynamical properties. We then generalize to sub-Riemannian manifolds and their quasi-conformal and quasi-regular mappings. We show that sub-Riemannian lens spaces admit uniformly quasi-regular (UQR) self-mappings, and use Margulis--Mostow derivatives to construct for each UQR self-mapping of an equiregular sub-Riemannian manifold an invariant measurable conformal structure. Turning next to hyperbolic spaces, we recall the relationship between quasi-isometries of Gromov hyperbolic spaces and quasi-symmetries of their boundaries. We show that every quasi-symmetry of \Heisⁿ lifts to a bi-Lipschitz mapping of \Hypⁿ⁺¹_\C, providing a rigidity result for quasi-isometries of \Hypⁿ⁺¹_\C. We conclude by showing that if $\Gamma$ is a lattice in the isometry group of a non-compact rank one symmetric space (except \Hyp¹_\C = \Hyp²_\R), then every quasi-isometric embedding of $\Gamma$ into itself is, in fact, a quasi-isometry.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Lukyanenko, Anton

Contributors dc:contributor

• Tyson, Jeremy T.
• Wu, Jang-Mei
• Dunfield, Nathan M.
• Hinkkanen, Aimo
• Athreya, Jayadev S.

• Heisenberg group
• complex hyperbolic space
• quasi-isometry
• quasi-conformal
• quasi-regular
• continued fraction
• co-Hopf
• Lattice

Statement dc:rights

• Copyright 2014 Anton Lukyanenko

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/50589

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/50589