Javier Aramayona

Injections of mapping class groups

We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of oncepunctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.

Homomorphisms between mapping class groups

Suppose that X and Y are surfaces of nite topologi- cal type, where X has genus g 6 and Y has genus at most 2g 1; in addition, suppose that Y is not closed if it has genus 2g 1. Our main result asserts that every non-trivial homomorphism Map(X)! Map(Y ) is induced by an embedding, i.e. a combina- tion of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if X has no boundary then every non-trivial endomorphism Map(X)! Map(X) is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if X and Y have nite analytic type then every non-constant holomorphic mapM(X)! M(Y ) between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless X and Y have the same genus and Y has at most as many marked points as X.

FINITE RIGID SETS IN CURVE COMPLEXES

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex 𝔛 of the curve complex such that every locally injective simplicial map is the restriction of an element of , unique up to the (finite) pointwise stabilizer of 𝔛 in . Furthermore, if S is not a twice-punctured torus, then we can replace in this statement with the extended mapping class group Mod±(S).

An obstruction to the strong relative hyperbolicity of a group

Abstract We give a simple combinatorial criterion for a group that, when satisfied, implies that the group cannot be strongly relatively hyperbolic. The criterion applies to several classes of groups, such as surface mapping class groups, Torelli groups, and automorphism and outer automorphism groups of free groups.

The proper geometric dimension of the mapping class group

We show that the mapping class group of a closed surface admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension.

Constructing convex planes in the pants complex

Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, referred to as the pants graph, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the pants graph of any surface of complexity at least 3.

Convexity of strata in diagonal pants graphs of surfaces

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

Exhausting curve complexes by finite rigid sets

Let

Hyperbolic Structures on Surfaces and Geodesic Currents

S

Free subgroups of surface mapping class groups

be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex