  The electron Land\'e g factor ($g^{*}$) is investigated both experimentally
and theoretically in a series of GaBi$_{x}$As$_{1-x}$/GaAs strained epitaxial
layers, for bismuth compositions up to $x = 3.8$%. We measure $g^{*}$ via
time-resolved photoluminescence spectroscopy, which we use to analyze the spin
quantum beats in the polarization of the photoluminescence in the presence of
an externally applied magnetic field. The experimental measurements are
compared directly to atomistic tight-binding calculations on large supercells,
which allows us to explicitly account for alloy disorder effects. We
demonstrate that the magnitude of $g^{*}$ increases strongly with increasing Bi
composition $x$ and, based on the agreement between the theoretical
calculations and experimental measurements, elucidate the underlying causes of
the observed variation of $g^{*}$. By performing measurements in which the
orientation of the applied magnetic field is changed, we further demonstrate
that $g^{*}$ is strongly anisotropic. We quantify the observed variation of
$g^{*}$ with $x$, and its anisotropy, in terms of a combination of epitaxial
strain and Bi-induced hybridization of valence states due to alloy disorder,
which strongly perturbs the electronic structure.
