  A quasi-automatic semigroup is a finitely generated semigroup with a rational
set of representatives such that the graph of right multiplication by any
generator is a rational relation. A asynchronously automatic semigroup is a
quasi-automatic semigroup for which these rational relations are also
recognisable by two-tape automata. We show that when such a semigroup happens
to be a group, the converse actually holds, meaning quasi-automatic groups are
asynchronously automatic.
