A group or semigroup S defined by a set of generators
X and a set of defining relations R (i.e. defined by a
presentation) is said to have decidable word problem if there
exists an algorithm which, for any pair of words over the
alphabet, decides whether they represent the same element of S.
It is well known that, even if a group or semigroup is given by
a finite presentation, the word problem may be undecidable.
Given this, over the years quite a lot of work in this area has
been done on identifying and studying classes of finitely
presented groups and semigroups where the word problem is
decidable (e.g. hyperbolic groups, automatic semigroups and
groups). After giving some history and background on the word
problem in general, I will go on to speak about some recent
joint work with Igor Dolinka (Novi Sad) and Nik Ruskuc (St
Andrews) investigating the word problem for a particular class
of finitely presented semigroups, called free idempotent
generated semigroups. I will say something about why people are
interested in these semigroups, and explain how the groups that
they embed can be used to obtain results (both positive and
negative) about the word problem for semigroups in this class.