  In this article we provide simple and provable bounds on the size and shape
of the quasiconformal deformation space of the groups $\IZ_p*\IZ_q$, the free
product of cyclic groups of order $p$ and $q$, in $\PSL(2,\IC)$ for $3\leq p,q
\leq \infty$. Though simple, these bounds are sharp, meeting the highly fractal
boundary of the deformation space in four cusp groups. Such bounds have great
utility in computer assisted searches for extremal Kleinian groups so as to
identify universal constraints (volume, length spectra, etc) on the geometry
and topology of hyperbolic $3$-orbifolds. As an application, we prove a
strengthened version of a conjecture by Morier-Genoud, Ovsienko, and Veselov on
the faithfulness of the specialised Burau representation.
