  We associate a 2-complex to the following data: a presentation of a semigroup
$S$ and a transitive action of $S$ on a set $V$ by partial transformations. The
automorphism group of the action acts properly discontinuously on this
2-complex. A sufficient condition is given for the 2-complex to be simply
connected. As a consequence we obtain simple topological proofs of results on
presentations of Sch\"utzenberger groups. We also give a geometric proof that a
finitely generated regular semigroup with finitely many idempotents has
polynomial growth if and only if all its maximal subgroups are virtually
nilpotent.
