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oai:www.ideals.illinois.edu:2142/50589 2023-07-11 col_2142_5131 com_2142_5130

Tyson, Jeremy T. Wu, Jang-Mei Tyson, Jeremy T. Dunfield, Nathan M. Hinkkanen, Aimo Athreya, Jayadev S. Lukyanenko, Anton 2014-09-16T17:24:10Z 2014-09-16T17:24:10Z 2014-08 2014-09-16 2014-08 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. Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-05-19T14:03:52Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Lukyanenko_Anton.pdf: 4087419 bytes, checksum: 9db4dfcb4832469c5a46ad54a4c81103 (MD5) Made available in DSpace on 2014-09-16T17:24:10Z (GMT). No. of bitstreams: 2 Anton_Lukyanenko.pdf: 4087419 bytes, checksum: 9db4dfcb4832469c5a46ad54a4c81103 (MD5) license.txt: 4066 bytes, checksum: 7229650d7f2dec5b2b04edbb085f66b3 (MD5) http://hdl.handle.net/2142/50589 en Copyright 2014 Anton Lukyanenko Heisenberg group complex hyperbolic space quasi-isometry quasi-conformal quasi-regular continued fraction co-Hopf Lattice Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces text Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD