  The existence of massive ($10^{11}$ solar masses) elliptical galaxies by
redshift z~4 (when the Universe was 1.5 billion years old) necessitates the
presence of galaxies with star-formation rates exceeding 100 solar masses per
year at z>6 (corresponding to an age of the Universe of less than 1 billion
years). Surveys have discovered hundreds of galaxies at these early cosmic
epochs, but their star-formation rates are more than an order of magnitude
lower. The only known galaxies with very high star-formation rates at z>6 are,
with only one exception, the host galaxies of quasars, but these galaxies also
host accreting supermassive (more than $10^9$ solar masses) black holes, which
probably affect the properties of the galaxies. Here we report observations of
an emission line of singly ionized carbon ([CII] at a wavelength of 158
micrometres) in four galaxies at z>6 that are companions of quasars, with
velocity offsets of less than 600 kilometers per second and linear offsets of
less than 600 kiloparsecs. The discovery of these four galaxies was
serendipitous; they are close to their companion quasars and appear bright in
the far-infrared. On the basis of the [CII] measurements, we estimate
star-formation rates in the companions of more than 100 solar masses per year.
These sources are similar to the host galaxies of the quasars in [CII]
brightness, linewidth and implied dynamical masses, but do not show evidence
for accreting supermassive black holes. Similar systems have previously been
found at lower redshift. We find such close companions in four out of
twenty-five z>6 quasars surveyed, a fraction that needs to be accounted for in
simulations. If they are representative of the bright end of the [CII]
luminosity function, then they can account for the population of massive
elliptical galaxies at z~4 in terms of cosmic space density.
