  We present a sample of 22 massive galaxies with stellar masses $>10^{10}
M_{\odot}$ at $3<z<4$ with deep H and K-band high resolution spectra
(R=3500-3000) from Keck/MOSFIRE and VLT/KMOS near-infrared spectrographs. We
find a large fraction have strong [OIII]5007 and H$\beta$ emission lines with
large line widths ($\sigma$ 100 -- 450 km/s). We measure the sizes of our
galaxies from Hubble Space Telescope images and consider the potential
kinematic scaling relations of our sample; and rule out an explanation for
these broad lines in terms of galaxy-wide kinematics. Based on consideration of
the [OIII]5007 $/$ H$\beta$ flux ratios, their location in the Mass--Excitation
diagram, and the derived bolometric luminosities, we conclude that Active
Galactic Nuclei (AGN) and their Narrow Line Regions most likely give rise to
this emission. At redshifts $3<z<4$, we find significantly high AGN fractions
in massive galaxies, ranging from 60--70\% for the mass range
$10<\log(M_{\star}/M_{\odot})<11$, with a lower limit 30\% for all galaxies
within that redshift range when we apply our most stringent AGN criteria. We
also find a considerably lower AGN fraction in massive quiescent galaxies,
ranging from 20-30\%. These fractions of AGN point to the period between
$3<z<4$ being a time of heightened activity for the development of supermassive
black holes in the massive end of the galaxy population and provide evidence
for their role in the emergence of the first massive quenched galaxies at this
epoch.
