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converges uniformly with respect to u ∈ B(x, µ). The derivative is
Df (x)u = lim δµ f (x)’1 f (xδµ u)

and the virtual tangent is
V T f (x) : V Tx G1 ’ V Tf (x) G2 , V T f (x)LN1 ,x = LG(x) LN2 (x)y Lf (x) 2
y f Df

Theorem 4.37 Let φ : G ’ G be a bi-Lipschitz map with respect to the Carnot-
Carath´odory distance on G induced by the left invariant distribution D. Then for
almost all x ∈ G the map φ is commutative derivable at x and the virtual tangent to φ
is an isomorphism of conical groups.

We don™t give the proof of this theorem. The reader might ¬nd interesting to adapt
the proof of the mentioned Margulis & Mostow theorem in our particular case.

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