ar X iv : m at h / 05 05 11 4 v 2 [ m at h . G T ] 2 8 M ay 2 00 5 Shadows of mapping class groups : capturing convex co - compactness

A Kleinian group Γ is a discrete subgroup of PSL2(C). When non-elementary, such a group possesses a unique non-empty minimal closed invariant subset ΛΓ of the Riemann sphere, called the limit set. A Kleinian group acts properly discontinuously on the complement ∆Γ of ΛΓ and so this set is called the domain of discontinuity. Such a group is said to be convex co-compact if it acts co-compactly on the convex hull HΓ in H3 of its limit set ΛΓ. This is equivalent to the condition that an orbit of Γ is quasi-convex in H3—or that the orbit defines a quasi-isometric embedding Γ → H3. Equivalent to each of these is the property that every limit point of Γ is conical, and still another definition is that Γ has a compact Kleinian manifold—meaning that Γ acts co-compactly on H∪∆Γ. We refer the reader to [10] and the references therein for the history of these notions and the proof of their equivalence (see also [53]). Let S be a closed oriented hyperbolic surface, Mod(S) its group of orientation preserving self–homeomorphisms up to isotopy, and T(S) the Teichmüller space of S equipped with Teichmüller’s metric. The mapping class group Mod(S) acts on Teichmüller space T(S) by isometries, and W. Thurston discovered a Mod(S)–equivariant compactification of T(S) by an ideal sphere, the sphere of projective measured laminations PML(S). J. McCarthy and A. Papadopoulos have shown that a subgroup G of Mod(S) has a well defined limit set ΛG, although it need not be unique or minimal, and that there is a certain enlargement ZΛG of ΛG on whose complement G acts properly discontinuously [42]. So such a group has a domain of discontinuity ∆G = PML(S)−ZΛG. In general, the limit set of a subgroup of Mod(S) has no convex hull to speak of, as there are pairs in PML(S) that are joined by no geodesic in T(S). Nevertheless, if every pair of points in ΛG are the negative and positive directions of a geodesic in T(S), one can define the weak hull HG of ΛG to be the union of all such geodesics. This is precisely what B. Farb and L. Mosher do in [19], where they develop a notion of convex co-compact mapping class groups. They prove the following ∗The second author was supported by an N.S.F. postdoctoral fellowship.