Contents Online

# Mathematical Research Letters

## Volume 24 (2017)

### Number 4

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

Pages: 1177 – 1203

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

#### Authors

#### 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.

The second author was supported by the Academy of Finland project 257482.

Paper received on 23 March 2015.