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Title:  Metric geometry of the Grushin plane and generalizations 
Author(s):  Romney, Matthew 
Director of Research:  Tyson, Jeremy T. 
Doctoral Committee Chair(s):  Wu, JangMei G 
Doctoral Committee Member(s):  D'Angelo, John P.; Leininger, Christopher J. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Metric space
BiLipschitz embedding SubRiemannian geometry Quasiconformal mapping 
Abstract:  Given $\alpha>0$, the $\alpha$Grushin plane is $\mathbb{R}^2$ equipped with the subRiemannian metric generated by the vector fields $X = \partial_1$ and $Y = x_1^{\alpha} \partial_2$. It is a standard example in subRiemannian geometry, as a space which is Riemannian except on a small singular sethere the vertical axis, where the vector field $Y$ vanishes. The main purpose of this thesis is to study various problems related to the metric geometry of the $\alpha$Grushin plane and a generalization of it, termed {\it conformal Grushin spaces}. One such problem is the embeddablity of these spaces in some Euclidean space under a biLipschitz or quasisymmetric mapping. Building on work of Seo \cite{Seo:11} and Wu \cite{Wu:15}, we prove a sharp embedding theorem for the $\alpha$Grushin plane and a general embedding theorem for conformal Grushin spaces under appropriate hypotheses. We also study quasiconformal homeomorphisms of the $\alpha$Grushin plane. In the final section, we solve a separate problem regarding quasiconformal mappings in metric spaces. The main result states that if a metric space homeomorphic to $\mathbb{R}^2$ can be quasiconformally parametrized by a domain in $\mathbb{R}^2$, then one can find a mapping which improves the dilatation to within a universal constant. A nonsharp theorem of this type was recently proved by Rajala; our theorem gives the sharp bounds for this problem. 
Issue Date:  20171203 
Type:  Text 
URI:  http://hdl.handle.net/2142/99345 
Rights Information:  Copyright 2017 Matthew Romney 
Date Available in IDEALS:  20180313 
Date Deposited:  201712 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois