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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
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
- Permalink
- http://hdl.handle.net/2142/50589
- Copyright and License Information
- Copyright 2014 Anton Lukyanenko
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