  We show that every word hyperbolic, surface-by-(noncyclic) free group Gamma
is as rigid as possible: the quasi-isometry group of Gamma equals the abstract
commensurator group Comm(Gamma), which in turn contains Gamma as a finite index
subgroup. As a corollary, two such groups are quasi-isometric if and only if
they are commensurable, and any finitely generated group quasi-isometric to
Gamma must be weakly commensurable with Gamma. We use quasi-isometries to
compute Comm(Gamma) explicitly, an example of how quasi-isometries can actually
detect finite index information. The proofs of these theorems involve ideas
from coarse topology, Teichmuller geometry, pseudo-Anosov dynamics, and
singular solv-geometry.
