Romney, Matthew; Vellis, Vyron Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces. (English) Zbl 1385.53019 Math. Res. Lett. 24, No. 4, 1177-1203 (2017). Summary: 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 J.-M. Wu [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 2, 633–644 (2015; Zbl 1329.53048)]. Cited in 6 Documents MSC: 53C17 Sub-Riemannian geometry 30L05 Geometric embeddings of metric spaces 30L10 Quasiconformal mappings in metric spaces Keywords:generalized Grushin plane; quasiplane; bi-Lipschitz embedding Citations:Zbl 1329.53048 PDFBibTeX XMLCite \textit{M. Romney} and \textit{V. Vellis}, Math. Res. Lett. 24, No. 4, 1177--1203 (2017; Zbl 1385.53019) Full Text: DOI arXiv