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

Lukyanenko, Anton

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https://hdl.handle.net/2142/50589 Copy

Description

Title

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

Author(s)

Lukyanenko, Anton

Issue Date

2014-09-16

Director of Research (if dissertation) or Advisor (if thesis)

Tyson, Jeremy T.

Doctoral Committee Chair(s)

Wu, Jang-Mei

Committee Member(s)

• Tyson, Jeremy T.
• Dunfield, Nathan M.
• Hinkkanen, Aimo
• Athreya, Jayadev S.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

We discuss the Heisenberg group $\Heis^n$ and its mappings from three perspectives. As a nilpotent Lie group, $\Heis^n$ 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, $\Heis^n$ 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 $\Hyp^{n+1}_\C$, $\Heis^n$ becomes a large-scale model of a rank-one symmetric space and provides rigidity results in $\Hyp^{n+1}_\C$. After discussing homotheties and conformal mappings of $\Heis^n$, we show the convergence of base-$b$ and continued fraction expansions of points in $\Heis^n$, 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 $\Heis^n$ lifts to a bi-Lipschitz mapping of $\Hyp^{n+1}_\C$, providing a rigidity result for quasi-isometries of $\Hyp^{n+1}_\C$. We conclude by showing that if $\Gamma$ is a lattice in the isometry group of a non-compact rank one symmetric space (except $\Hyp^1_\C = \Hyp^2_\R$), then every quasi-isometric embedding of $\Gamma$ into itself is, in fact, a quasi-isometry.

Graduation Semester

2014-08

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

Copyright and License Information

Copyright 2014 Anton Lukyanenko

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