K g is not finitely generated

Let Σg be a closed orientable surface of genus g. The mapping class group Modg of Σg is defined to be the group of isotopy classes of orientationpreserving diffeomorphisms Σg → Σg. Recall that an essential simple closed curve γ in Σg is called a bounding curve, or separating curve, if it is nullhomologous in Σg or, equivalently, if γ separates Σg into two connected components. Let Kg denote the subgroup of Modg generated by the (infinite) collection of Dehn twists about bounding curves in Σg. Note that K1 is trivial. It has been a long-standing problem in the combinatorial topology of surfaces to determine whether or not the group Kg is finitely generated for g ≥ 2. For a discussion of this problem, see, e.g., [Jo1, Jo3, Bi, Mo1, Mo3, Ak]. McCullough-Miller [MM] proved that K2 is not finitely generated; Mess then proved thatK2 is in fact an infinite rank free group. Akita proved in [Ak] that for all g ≥ 2, the rational homology H∗(Kg;Q) is infinite-dimensional as a vector space over Q. Note that as Kg admits a free action on the Teichmuller space of Σg, which is contractible and finite-dimensional, Kg has finite cohomological dimension. For some time it was not known if Kg was equal to, or perhaps a finite index subgroup of, the Torelli group Ig, which is the subgroup of elements ∗This research was conducted during the period the first author served as a Clay Mathematics Institute Long-Term Prize Fellow. †Supported in part by NSF grants DMS-9704640 and DMS-0244542.