  The Hubble constant can be constrained using the time delays between multiple
images of gravitationally lensed sources. In some notable cases, typical
lensing analyses assuming isothermal galaxy density profiles produce low values
for the Hubble constant, inconsistent with the result of the HST Key Project
(72 +- 8 km/s/Mpc). Possible systematics in the values of the Hubble constant
derived from galaxy lensing systems can result from a number of factors, e.g.
neglect of environmental effects, assumption of isothermality, or contamination
by line-of-sight structures. One additional potentially important factor is the
triaxial structure of the lensing galaxy halo; most lens models account for
halo shape simply by perturbing the projected spherical lensing potential, an
approximation that is often necessary but that is inadequate at the levels of
triaxiality predicted in the CDM paradigm. To quantify the potential error
introduced by this assumption in estimates of the Hubble parameter, we strongly
lens a distant galaxy through a sample of triaxial softened isothermal halos
and use an MCMC method to constrain the lensing halo profile and the Hubble
parameter from the resulting multiple image systems. We explore the major
degeneracies between the Hubble parameter and several parameters of the lensing
model, finding that without a way to accurately break these degeneracies
accurate estimates of the Hubble parameter are not possible. Crucially, we find
that triaxiality does not significantly bias estimates of the Hubble constant,
and offer an analytic explanation for this behaviour in the case of isothermal
profiles. Neglected triaxial halo shape cannot contribute to the low Hubble
constant values derived in a number of galaxy lens systems.
