Title: Searching strong `spin'-orbit coupled one-dimensional hole gas in strong magnetic fields

Abstract: We show that a strong `spin'-orbit coupled one-dimensional (1D) hole gas is achievable via applying a strong magnetic field to the originally two-fold degenerate hole gas confined in a cylindrical Ge nanowire. Both strong longitudinal and strong transverse magnetic fields are feasible to achieve this goal. The induced low-energy subband dispersion of the hole gas can be written as $E=\hbar^{2}k^{2}_{z}/(2m^{*}_{h})+\alpha\sigma^{z}k_{z}+g^{*}_{h}\mu_{B}B\sigma^{x}/2$, a form that is exact the same as that of the electron gas in the conduction band. Also, the induced hole effective $m^{*}_{h}$ ($0.07\sim0.08~m_{e}$) and the `spin'-orbit coupling $\alpha$ ($0.4\sim0.65$ eV~\AA) have a small magnetic field dependence, while the effective $g$-factor $g^{*}_{h}$ of the hole `spin' only has a small magnetic field dependence in the large fields region.