Danny Calegari

Shrinkwrapping and the taming of hyperbolic 3-manifolds

We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.

Laminations and groups of homeomorphisms of the circle

Abstract.If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that π1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so π1(M) is isomorphic to a subgroup of Homeo(S1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S1. As a corollary, the Weeks manifold does not admit a tight essential lamination with solid torus guts, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston’s universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem.

The Geometry of R{covered foliations

We study R{covered foliations of 3{manifolds from the point of view of their transverse geometry. For an R{covered foliation in an atoroidal 3{manifold M , we show that f M can be partially compactied by a canonical cylinder S 1 R on which 1(M ) acts by elements of Homeo(S 1 )Homeo(R), where the S 1 factor is canonically identied with the circle at innity of each leaf of e F . We construct a pair of very full genuine laminations transverse to each other and to F , which bind every leaf of F . This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for F , analogous to Thurston’s structure theorem for surface bundles over a circle with pseudo-Anosov monodromy. A corollary of the existence of this structure is that the underlying manifold M is homotopy rigid in the sense that a self-homeomorphism homotopic to the identity is isotopic to the identity. Furthermore, the product structures at innity are rigid under deformations of the foliation F through R{covered foliations, in the sense that the representations of 1(M )i nHomeo((S 1 )t) are all conjugate for a family parameterized by t. Another corollary is that the ambient manifold has word-hyperbolic fundamental group. Finally we speculate on connections between these results and a program to prove the geometrization conjecture for tautly foliated 3{manifolds. AMS Classication numbers Primary: 57M50, 57R30 Secondary: 53C12

Stable commutator length in word-hyperbolic groups

In this paper we obtain uniform positive lower bounds on the stable commutator length of elements in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on an associated Bass-Serre tree). If G is a word-hyperbolic group that is δ-hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on δ and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length of the element is at least equal to C. By Bavard’s theorem, these lower bounds on stable commutator length imply the existence of quasimorphisms with uniform control on the defects; however, we show how to construct such quasimorphisms directly. We also prove various separation theorems on families of elements in such groups, constructing homogeneous quasimorphisms (again with uniform estimates) which are positive on some prescribed element while vanishing on some family of independent elements whose translation lengths are uniformly bounded. Finally, we prove that the first accumulation point for stable commutator length in a torsion-free word-hyperbolic group is contained between 1/12 and 1/2. This gives a universal sense of what it means for a conjugacy class in a hyperbolic group to have a small stable commutator length, and can be thought of as a kind of “homological Margulis lemma”.

Statistics and compression of scl

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting non-degenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length

Stable commutator length is rational in free groups

n

Distortion in transformation groups

is of order

Combable functions, quasimorphisms, and the central limit theorem

n/ \log n

Faces of the scl norm ball

. This establishes quantitative refinements of qualitative results of Bestvina and Fujiwara and others on the infinite dimensionality of two-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length

Promoting essential laminations

n