Luminous compact blue galaxies (LCBGs) are a diverse class of galaxies characterized by high luminosities, blue colors, and high surface brightnesses. Residing at the high luminosity, high mass end of the blue sequence, LCBGs sit at the critical juncture of galaxies that are evolving from the blue to the red sequence. Yet we do not understand what drives the evolution of LCBGs, nor how they will evolve. Based on single-dish HI observations, we know that they have a diverse range of properties. LCBGs are HI-rich with M(HI)=10^{9-10.5} M(sun), have moderate M(dyn)=10^{10-12} M(sun), and 80% have gas depletion timescales less than 3 Gyr. These properties are consistent with LCBGs evolving into low-mass spirals or high mass dwarf ellipticals or dwarf irregulars. However, LCBGs do not follow the Tully-Fisher relation, nor can most evolve onto it, implying that many LCBGs are not smoothly rotating, virialized systems. GMRT and VLA HI maps confirm this conclusion revealing signatures of recent interactions and dynamically hot components in some local LCBGs, consistent with the formation of a thick disk or spheroid. Such signatures and the high incidence of close companions around LCBGs suggest that star formation in local LCBGs is likely triggered by interactions. The dynamical masses and apparent spheroid formation in LCBGs combined with previous results from optical spectroscopy are consistent with virial heating being the primary mechanism for quenching star formation in these galaxies.
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