Mathematical Research Letters

Volume 24 (2017)

Number 4

Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Pages: 1177 – 1203

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n4.a11

Authors

Matthew Romney (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Vyron Vellis (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Abstract

We show that, for all $\alpha \geq 0$, the generalized Grushin plane $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to a 2-dimensional quasiplane in the Euclidean space $\mathbb{R}^{\lfloor \alpha \rfloor +2}$, where $\lfloor \alpha \rfloor$ is the integer part of $\alpha$. The target dimension is sharp. This generalizes a recent result of Wu [J.-M.Wu, “Bilipschitz embedding of Grushin plane in $\mathbb{R}^3$”, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (2015), no. 2, 633–644].

Keywords

generalized Grushin plane, quasiplane, bi-Lipschitz embedding

2010 Mathematics Subject Classification

Primary 53C17. Secondary 30L05, 30L10.

Received 23 March 2015

Accepted 12 February 2016

Published 9 November 2017