  We study invariance of multiplicity of complex analytic germs and degree of
complex affine sets under outer bi-Lipschitz transformations (outer
bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz
homeomorphims at infinity in the second case). We prove that invariance of
multiplicity in the local case is equivalent to invariance of degree in the
global case. We prove invariance for curves and surfaces. In the way we prove
invariance of the tangent cone and relative multiplicities at infinity under
outer bi-Lipschitz homeomorphims at infinity, and that the abstract topology of
a homogeneous surface germ determines its multiplicity.
