  We investigate the groups of units of one-relator and special inverse
monoids. These are inverse monoids which are defined by presentations where all
the defining relations are of the form $r=1$. We develop new approaches for
finding presentations for the group of units of a special inverse monoid, and
apply these methods to give conditions under which the group admits a
presentation with the same number of defining relations as the monoid. In
particular our results give sufficient conditions for the group of units of a
one-relator inverse monoid to be a one-relator group. When these conditions are
satisfied these results give inverse semigroup theoretic analogues of classical
results of Adjan for one-relator monoids, and Makanin for special monoids. In
contrast, we show that in general these classical results to not hold for
one-relator and special inverse monoids. In particular, we show that there
exists a one-relator special inverse monoid whose group of units is not a
one-relator group (with respect to any generating set), and we show that there
exists a finitely presented special inverse monoid whose group of units is not
finitely presented.
