  We study the evolution of stars that may be the progenitors of long-soft
gamma-ray burst (GRBs) -- rotating naked helium stars presumed to have lost
their envelopes to winds or companions. Our aim is to investigate the formation
and development of single and binary systems and from this population evaluate
the rates of interesting individual species. Using a rapid binary evolution
algorithm, that enables us to model the most complex binary systems and to
explore the effect of metallicity on GRB production, we draw the following
conclusions. First we find that, if we include an approximate treatment of
angular momentum transport by mass loss, the resulting spin rates for single
stars become too low to form a centrifugally supported disc that can drive a
GRB engine. Second massive stars in binaries result in enough angular momentum
-- due to spin-orbit tidal interactions -- to form a centrifugally supported
disc and are thus capable of supplying a sufficient number of progenitors. This
holds true even if only a small fraction of bursts are visible to a given
observer and the GRB rate is several hundred times larger than the observed
rate. Third low-metallicity stars aid the formation of a rapidly rotating,
massive helium cores at collapse and so their evolution is likely to be
affected by the local properties of the ISM. This effect could increase the GRB
formation rate by a factor of 5-7 at Z=Z_solar/200. Finally we quantify the
effects of mass loss, common-envelope evolution and black-hole formation and
show that more stringent constraints to many of these evolution parameters are
needed in order to draw quantitative conclusions from population synthesis
work.
