Mathematics

A variant of Gromov's Hölder equivalence problem for a class of jet spaces

Derek Jung (UIUC Math)

Abstract: A Carnot group is an $\mathbb{R}^n$ equipped with a product to make it into a Lie group with stratified Lie algebra. In a 1996 survey, Gromov considered the following question: How smoothly can one embed an $n$-dimensional surface into a Carnot group $G=(\mathbb{R}^n,\cdot)$? More specifically, if one equips $G$ with a natural path metric, what is the largest exponent $\alpha$ for which there exists a (locally) $\alpha$-Hölder homeomorphism $f:(\mathbb{R}^n,\text{euc})\to G$? In 2013, Balogh, Hajlasz, and Wildrick considered a variant of this problem for the Heisenberg group. In this talk, I will begin by introducing the model filiform groups, which is a collection of Carnot groups that can be realized as a class of jet spaces. I will then present my work in generalizing BHW's result to these model filiform groups. I will conclude by discussing my efforts in generalizing their result to all Carnot groups. Some prior knowledge of Lie group theory would be nice, but nothing else is necessary. Not a single epsilon will be used in this talk.