  We study a class of inverse monoids of the form M = Inv< X | w=1 >, where the
single relator w has a combinatorial property that we call sparse. For a sparse
word w, we prove that the word problem for M is decidable. We also show that
the set of words in (X \cup X^{-1})^* that represent the identity in M is a
deterministic context free language, and that the set of geodesics in the
Schutzenberger graph of the identity of M is a regular language.
