  Building on the previous extensive study of Yang, Gould and the present
author, we provide a more precise insight into the group-theoretical
ramifications of the word problem for free idempotent generated semigroups over
finite biordered sets. We prove that such word problems are in fact equivalent
to the problem of computing intersections of cosets of certain subgroups of
direct products of maximal subgroups of the free idempotent generated semigroup
in question, thus providing decidability of those word problems under
group-theoretical assumptions related to the Howson property and the coset
intersection property. We also provide a basic sketch of the global
semigroup-theoretical structure of an arbitrary free idempotent generated
semigroup, including the characterisation of Green's relations and the key
parameters of non-regular $\mathscr{D}$-classes. In particular, we prove that
all Sch\"utzenberger groups of $\mathsf{IG}(\mathcal{E})$ for a finite
biordered set $\mathcal{E}$ must be among the divisors of the maximal subgroups
of $\mathsf{IG}(\mathcal{E})$.
