Lyman-alpha spectroscopy of extreme [OIII] emitting galaxies at
  $z\simeq2-3$: Implications for Ly$\alpha$ visibility and LyC leakage at $z>6$
  Spectroscopic observations of massive $z>7$ galaxies selected to have
extremely large [OIII]+H$\beta$ equivalent width (EW $\sim1500$ \r{A}) have
recently revealed large Ly$\alpha$ detection rates, in contrast to the weak
emission seen in the general population. Why these systems are uniquely visible
in Ly$\alpha$ at redshifts where the IGM is likely significantly neutral is not
clear. With the goal of better understanding these results, we have begun a
campaign with MMT and Magellan to measure Ly$\alpha$ in galaxies with similar
[OIII]+H$\beta$ EWs at $z\simeq2-3$. At these redshifts, the IGM is highly
ionized, allowing us to clearly disentangle how the Ly$\alpha$ properties
depend on the [OIII]+H$\beta$ EW. Here we present Ly$\alpha$ EWs of $49$
galaxies at $z=2.2-3.7$ with intense [OIII]+H$\beta$ line emission (EW
$=300-3000$ \r{A}). Our results demonstrate that strong Ly$\alpha$ emission (EW
$>20$ \r{A}) becomes more common in galaxies with larger [OIII]+H$\beta$ EW,
reflecting a combination of increasingly efficient ionizing photon production
and enhanced transmission of Ly$\alpha$. Among the galaxies with the most
extreme [OIII]+H$\beta$ emission (EW $\sim1500$ \r{A}), we find that strong
Ly$\alpha$ emission is not ubiquitous, with only $50$ per cent of our
population showing Ly$\alpha$ EW $>20$ \r{A}. Our data suggest that the range
of Ly$\alpha$ strengths is related to the observed ellipticity, with those
systems that appear edge-on or elongated having weaker Ly$\alpha$ emission. We
use these results to interpret the anomalous Ly$\alpha$ properties seen in
$z>7$ galaxies with extreme [OIII]+H$\beta$ emission and discuss implications
for the escape of ionizing radiation from these extreme line emitting galaxies.
