We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup G of MCG defining an extension 1 → π 1 (S) → Γ G → G → 1, we prove that if Γ G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky subgroup of MCG, the converse is true as well; a semidirect product of π 1 (S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G = Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup. AMS Classification numbers Primary: 20F67, 20F65Secondary: 57M07, 57S25

Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable.We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of nonrelatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil-Peterson metric, in contrast with Brock-Farb's hyperbolicity result in low complexity.

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We study the large scale geometry of mapping class groups MCG.S/, using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG.S / (outside a few sporadic cases) is a bounded distance away from a leftmultiplication, and as a consequence obtain quasi-isometric rigidity for MCG.S /, namely that groups quasi-isometric to MCG.S / are equivalent to it up to extraction of finite-index subgroups and quotients with finite kernel. (The latter theorem was proved by Hamenstädt using different methods).As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG.S/; a characterization of the image of the curve complex projections map from MCG.S/ to Q Y ÂS C.Y /; and a construction of †-hulls in MCG.S /, an analogue of convex hulls.

Given a free group F n of finite rank n ≥ 2, a free splitting over F n is a minimal, simplicial action of the group F n on a simplicial tree T such that the stabilizer of each edge of T is the trivial subgroup of F n . A free splitting is denoted F n T , or just T when the group and its action are understood. Although the tree T is allowed to have vertices of valence 2, there is a unique natural cell structure on T the vertices of which are the points of valence ≥ 3. We say that T is a k-edge free splitting if k is the number of natural edge orbits, a number which can take on any value from 1 to 3n − 3. The equivalence relation amongst free splittings is conjugacy, where two free splittings of F n are conjugate if there exists an F n -equivariant homeomorphism between them. See the beginning of Section 1 for the details of these definitions.The free splitting complex of F n , denoted FS(F n ), is a simplicial complex of dimension 3n − 4 having a simplex T of dimension k for each conjugacy class of k + 1-edge free splittings F n T . Given another free splitting F n S, the simplex S is a face of T if and only if there is a collapse map T → S, which collapses to a point each edge in some F -invariant set of edges of T . We write T ≻ S for the relation "T collapses to S", and S ≺ T for the inverse relation "S expands to T ". There is a natural left action of the outer automorphism group Out(F n ) on FS(F n ), where φ ∈ Out(F n ) acts on the conjugacy class of a free splitting F n T by precomposing the action by an automorphism of F n representing φ. The free splitting complex was introduced by Hatcher in [Hat95] in its role as the sphere complex of a connected sum of n copies of the 3-manifold S 2 × S 1 . A careful construction of an isomorphism between the 1-skeletons of FS(F n ) and Hatcher's sphere complex can be found in [AS11], and that proof extends with little trouble to the entire complexes. In Section 1.3 we shall give a rigorous construction of the free splitting complex given purely in tree language.The complex FS(F n ) is regarded as one of several Out(F n ) analogues of the curve complex of a surface -another competing analogue is the free factor complex of F n introduced by Hatcher and Vogtmann in [HV98]. The analogies are imperfect in each case: Hatcher and Vogtmann showed that the free factor complex, like the curve complex, has the homotopy type of a wedge of spheres of constant dimension [HV98]; by contrast, Hatcher showed that FS(F n ) is contractible [Hat95]. On the other hand we showed in 0 → · · · → T 0 J . Consider also a zig-zag path T 0, which may be regarded as a W diagram. We do not assume that this W diagram is a geodesic, nor even that it is normalized, but we do assume that T 2 J = β J ∪ ρ J . Consider finally a stack of four combing rectangles combined into one commutative diagram as shown in Figure 10, where the given fold sequence occurs as the T 0 row along the bottom of the diagram, and the W zig-zag