Juan Souto

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.

A finiteness theorem for hyperbolic 3-manifolds

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to arithmetic manifolds is also given.

A characterization of round spheres in terms of blocking light

A closed Riemannian manifold is said to have cross blocking if whenever distinct points p and q are at distance less than the diameter, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each point p, all the light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres.

Minimality of the well-rounded retract

We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.

Geometric limits of knot complements

We prove that any complete hyperbolic 3-manifold with finitely generated fundamental group, with a single topological end, and which embeds into is the geometric limit of a sequence of hyperbolic knot complements in . In particular, we derive the existence of hyperbolic knot complements that contain balls of arbitrarily large radius. We also show that a complete hyperbolic 3-manifold with two convex cocompact ends cannot be a geometric limit of knot complements in .

A Cantor set with hyperbolic complement

We construct a Cantor set in S whose complement admits a complete hyperbolic metric.

Dimension invariants of outer automorphism groups

The geometric dimension for proper actions

Geometric filling curves on surfaces

\underline{\mathrm{gd}}(G)

Three techniques for obtaining algebraic circle packings

of a group