  The category of all idempotent generated semigroups with a prescribed
structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has
an initial object called the free idempotent generated semigroup over
$\mathcal{E}$, defined by a presentation over alphabet $E$, and denoted by
$\mathsf{IG}(\mathcal{E})$. Recently, much effort has been put into
investigating the structure of semigroups of the form
$\mathsf{IG}(\mathcal{E})$, especially regarding their maximal subgroups. In
this paper we take these investigations in a new direction by considering the
word problem for $\mathsf{IG}(\mathcal{E})$. We prove two principal results,
one positive and one negative. We show that, for a finite biordered set
$\mathcal{E}$, it is decidable whether a given word $w \in E^*$ represents a
regular element; if in addition one assumes that all maximal subgroups of
$\mathsf{IG}(\mathcal{E})$ have decidable word problems, then the word problem
in $\mathsf{IG}(\mathcal{E})$ restricted to regular words is decidable. On the
other hand, we exhibit a biorder $\mathcal{E}$ arising from a finite idempotent
semigroup $S$, such that the word problem for $\mathsf{IG}(\mathcal{E})$ is
undecidable, even though all the maximal subgroups have decidable word
problems. This is achieved by relating the word problem of
$\mathsf{IG}(\mathcal{E})$ to the subgroup membership problem in finitely
presented groups.
