# Isoperimetric Inequalities in Carnot Groups

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 Title: Isoperimetric Inequalities in Carnot Groups Author(s): Wang, Liming Advisor(s): Wenger, Stefan Department / Program: Mathematics, Statistics, and Computer Science Graduate Major: Mathematics Degree Granting Institution: University of Illinois at Chicago Degree: MS, Master of Science Genre: Masters Subject(s): Isoperimetric inequality Carnot group Geometric group theory Abstract: The isoperimetric problem is a very classical problem whose history dates back to more than two thousand years ago. Roughly speaking, the isoperimetric problem is to determine the largest possible area enclosed by a closed curve which has a specified length. In this thesis, we give proofs of a few theorems on isoperimetric inequalities in Carnot groups. Specifically, for a free nilpotent Carnot group $G$ of step 2, we show the filling function $FA_0(r)$ of the central product $G\cp G$ has a quadratic isoperimetric inequality. Moreover, for Carnot group $G$ of step 2 which satisfies quadratic isoperimetric inequality, we show the filling function of its quotient group satisfies $\fa \preceq r^2\log r$. As a result of two previous theorems, we prove the following result: For a Carnot group $G$ of step $2$, the filling function of the central product $G\times_z G$ satisfies $FA_0 (r) \preceq r^2\log (r)$. Issue Date: 2012-12-07 Genre: thesis URI: http://hdl.handle.net/10027/8858 Rights Information: Copyright 2011 Liming Wang Date Available in INDIGO: 2012-12-07 Date Deposited: 2011-08

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