  The present work splits in two parts: first, we perform a straightforward
generalization of results from [Re], proving autoquasitriangularity of quantum
groups $ U_q(\frak{g}) $ and their unrestricted specializations at roots of 1,
in particular the function algebra $ F[H] $ of the Poisson group $ H $ dual of
$ G $; second, as a main contribution, we prove the convergence of the
(specialized) $ R $--matrix action to a birational automorphism of a
2$\ell$--fold ramified covering of the specialization of $ U_q(\frak{g}) $ at a
primitive $ \ell $--th root of 1, and of a 2-fold ramified covering of $ H $,
thus giving a geometric content to the notion of triangularity (or
autoquasitriangularity) for quantum groups.
