  We describe a fundamental domain for the punctured Riemann surface $V_{3,m}$
which parametrises (up to M\"obius conjugacy) the set of quadratic rational
maps with numbered critical points, such that the first critical point has
period three, and such that the second critical point is not mapped in $m$
iterates or less to the periodic orbit of the first. This gives, in turn, a
description, up to topological conjugacy, of all dynamics in all type III
hyperbolic components in $V_{3}$, and gives indications of a topological model
for $V_{3}$, together with the hyperbolic components contained in it.
