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Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces. (English) Zbl 1385.53019

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)].

MSC:

53C17 Sub-Riemannian geometry
30L05 Geometric embeddings of metric spaces
30L10 Quasiconformal mappings in metric spaces

Citations:

Zbl 1329.53048
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