  We study how an inversion-breaking quantum critical point affects the ground
state of a one-dimensional electronic liquid with repulsive interaction and
spin-orbit coupling. We find that regardless of the interaction strength, the
critical fluctuations always lead to a gap in the electronic spin-sector. The
origin of the gap is a two-particle backscattering process, which becomes
relevant due to renormalization of the Luttinger parameter near the critical
point. The resulting spin-gapped state is a one-dimensional version of spin
triplet superconductivity. Interestingly, in the case of a ferromagnetic
critical point the Luttinger parameter is renormalized in the opposite manner,
such that the system remains non-superconducting.
