  We study mappings on sub-Riemannian manifolds which are quasi-regular with
respect to the Carnot-Caratheodory distances and discuss several related
notions. On H-type Carnot groups, quasiregular mappings have been introduced
earlier using an analytic definition, but so far, a good working definition in
the same spirit is not available in the setting of general sub-Riemannian
manifolds. In the present paper we adopt therefore a metric rather than
analytic viewpoint.
  As a first main result, we prove that the sub-Riemannian lens space admits
nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular
mappings with a uniform bound on the distortion of all the iterates. In doing
so, we also obtain new examples of UQR maps on the standard sub-Riemannian
spheres. The proof is based on a method for building conformal traps on
sub-Riemannian spheres using quasiconformal flows, and an adaptation of this
approach to quotients of spheres.
  One may then study the quasiregular semigroup generated by a UQR mapping. In
the second part of the paper we follow Tukia to prove the existence of a
measurable conformal structure which is invariant under such a semigroup. Here,
the conformal structure is specified only on the horizontal distribution, and
the pullback is defined using the Margulis--Mostow derivative (which
generalizes the classical and Pansu derivatives).
