The set of idempotents of an arbitrary semigroup has the structure of a so called biordered set (or regular biordered set in the case of von Neumann regular semigroups). These structures were studied in detail in work of Nambooripad (1979) and Easdown (1985). There is a notion of a free idempotent generated semigroup on a biordered set, and it was conjectured by McElwee in 2002 that the maximal subgroups of such a semigroup are all free (in fact, this had been conjectured since the early 1980's). The first counterexample to this conjecture was given by Brittenham, Margolis and Meakin (2009), where it was shown that the free abelian group of rank 2 is a maximal subgroup of the free idempotent generated semigroup arising from a certain 72-element semigroup.

In this talk I will present some recent work (joint with Nik Ruskuc) which shows that every group is a maximal subgroup of some free idempotent generated semigroup.