  In this paper we build a link between the Teichmuller theory of hyperbolic
Riemann surfaces and isomonodromic deformations of linear systems whose
monodromy group is the Fuchsian group associated to the given hyperbolic
Riemann surface by the Poincare' uniformization. In the case of a one-sheeted
hyperboloid with n orbifold points we show that the Poisson algebra D_n of
geodesic length functions is the semiclassical limit of the twisted q-Yangian
for the orthogonal Lie algebra defined by Molev, Ragoucy and Sorba. We give a
representation of the braid group action on this algebra in terms of an adjoint
matrix action. We characterize two types of finite-dimensional Poissonian
reductions and give an explicit expression for the generating function of their
central elements. Finally, we interpret the algebra D_n as the Poisson algebra
of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple
point.
