{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/50589"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/50589","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces","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.","abstract_html":"We discuss the Heisenberg group <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span> and its mappings from three perspectives. As a nilpotent Lie group, <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span> 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, <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span> 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 <span class=\"etd-inline-math\">\\Hyp<sup>n+1</sup><sub>\\</sub>C</span>, <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span> becomes a large-scale model of a rank-one symmetric space and provides rigidity results in <span class=\"etd-inline-math\">\\Hyp<sup>n+1</sup><sub>\\</sub>C</span>. After discussing homotheties and conformal mappings of <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span>, we show the convergence of base-$b$ and continued fraction expansions of points in <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span>, 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 <span class=\"etd-inline-math\">\\Heis<sup>n</sup></span> lifts to a bi-Lipschitz mapping of <span class=\"etd-inline-math\">\\Hyp<sup>n+1</sup><sub>\\</sub>C</span>, providing a rigidity result for quasi-isometries of <span class=\"etd-inline-math\">\\Hyp<sup>n+1</sup><sub>\\</sub>C</span>. We conclude by showing that if $\\Gamma$ is a lattice in the isometry group of a non-compact rank one symmetric space (except <span class=\"etd-inline-math\">\\Hyp<sup>1</sup><sub>\\</sub>C = \\Hyp<sup>2</sup><sub>\\</sub>R</span>), then every quasi-isometric embedding of $\\Gamma$ into itself is, in fact, a quasi-isometry.","abstract_has_math":true,"creators":["Lukyanenko, Anton"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Tyson, Jeremy T.","Wu, Jang-Mei","Dunfield, Nathan M.","Hinkkanen, Aimo","Athreya, Jayadev S."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-09-16T17:24:10Z","date_published":"2014-09-16T17:24:10Z","updated_at":"2026-07-22T22:25:40Z","subjects":["Heisenberg group","complex hyperbolic space","quasi-isometry","quasi-conformal","quasi-regular","continued fraction","co-Hopf","Lattice"],"languages":["en"],"rights":["Copyright 2014 Anton Lukyanenko"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/50589","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Tyson, Jeremy T.","Wu, Jang-Mei","Dunfield, Nathan M.","Hinkkanen, Aimo","Athreya, Jayadev S."]},{"key":"dc:creator","label":"Author","values":["Lukyanenko, Anton"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-09-16T17:24:10Z","2014-08","2014-09-16"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Heisenberg group","complex hyperbolic space","quasi-isometry","quasi-conformal","quasi-regular","continued fraction","co-Hopf","Lattice"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Anton Lukyanenko"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/50589"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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)"]},{"key":"dc:title","label":"Title","values":["Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces"]}]}],"canonical_facts":{"dc:contributor":["Tyson, Jeremy T.","Wu, Jang-Mei","Dunfield, Nathan M.","Hinkkanen, Aimo","Athreya, Jayadev S."],"dc:creator":["Lukyanenko, Anton"],"dc:date":["2014-09-16T17:24:10Z","2014-08","2014-09-16"],"dc:description":["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)"],"dc:identifier":["http://hdl.handle.net/2142/50589"],"dc:language":["en"],"dc:rights":["Copyright 2014 Anton Lukyanenko"],"dc:subject":["Heisenberg group","complex hyperbolic space","quasi-isometry","quasi-conformal","quasi-regular","continued fraction","co-Hopf","Lattice"],"dc:title":["Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:40Z"}