We study the Zeeman spin-splitting in hole quantum wires oriented along the $[011]$ and $[01bar{1}]$ crystallographic axes of a high mobility undoped (100)-oriented AlGaAs/GaAs heterostructure. Our data shows that the spin-splitting can be switched `on (finite $g^{*}$) or `off (zero $g^{*}$) by rotating the field from a parallel to a perpendicular orientation with respect to the wire, and the properties of the wire are identical for the two orientations with respect to the crystallographic axes. We also find that the $g$-factor in the parallel orientation decreases as the wire is narrowed. This is in contrast to electron quantum wires, where the $g$-factor is enhanced by exchange effects as the wire is narrowed. This is evidence for a $k$-dependent Zeeman splitting that arises from the spin-3/2 nature of holes.
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