Pham and Teissier showed in the late 60's that any two plane curve germs with the same outer Lipschitz geometry have equivalent embeddings into $\mathbb{C}^2$. We consider to what extent the same holds in higher dimensions, giving examples of normal surface singularities which have the same topology and outer Lipschitz geometry but whose embeddings into $\mathbb{C}^3$ are topologically inequivalent. Joint work with Anne Pichon.

Keywords: bilipschitz - Lipschitz geometry - normal surface singularity - Zariski equisingularity - Lipschitz equisingularity[-]