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We investigate the relationships between the Lipschitz outer geometry and theembedded topological type of a hypersurface germ in $(\mathbb C^n,0)$. It iswell known that the Lipschitz outer geometry of a complex plane curve germdetermines and is determined by its embedded topological type. We prove thatthis does not remain true in higher dimensions. Namely, we give two normalhypersurface germs $(X_1,0)$ and $(X_2,0)$ in $(\mathbb C^3,0)$ having the sameouter Lipschitz geometry and different embedded topological types. Our pairconsist of two superisolated singularities whose tangent cones form anAlexander-Zariski pair having only cusp-singularities. Our result is based on adescription of the Lipschitz outer geometry of a superisolated singularity. Wealso prove that the Lipschitz inner geometry of a superisolated singularity iscompletely determined by its (non embedded) topological type, or equivalentlyby the combinatorial type of its tangent cone.

Lipschitz geometry does not determine embedded topological type