We present new H I imaging and spectroscopy of the 14 UV-selected star-forming galaxies in the Lyman Alpha Reference Sample (LARS), aimed for a detailed study of the processes governing the production, propagation, and escape of Ly$alpha$ photons. New H I spectroscopy, obtained with the 100m Green Bank Telescope (GBT), robustly detects the H I spectral line in 11 of the 14 observed LARS galaxies (although the profiles of two of the galaxies are likely confused by other sources within the GBT beam); the three highest redshift galaxies are not detected at our current sensitivity limits. The GBT profiles are used to derive fundamental H I line properties of the LARS galaxies. We also present new pilot H I spectral line imaging of 5 of the LARS galaxies obtained with the Karl G. Jansky Very Large Array (VLA). This imaging localizes the H I gas and provides a measurement of the total H I mass in each galaxy. In one system, LARS 03 (UGC 8335 or Arp 238), VLA observations reveal an enormous tidal structure that extends over 160 kpc from the main interacting systems and that contains $>$10$^9$ M$_{odot}$ of H I. We compare various H I properties with global Ly$alpha$ quantities derived from HST measurements. The measurements of the Ly$alpha$ escape fraction are coupled with the new direct measurements of H I mass and significantly disturbed H I velocities. Our robustly detected sample reveals that both total H I mass and linewidth are tentatively correlated with key Ly$alpha$ tracers. Further, on global scales, these data support a complex coupling between Ly$alpha$ propagation and the H I properties of the surrounding medium.
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