# Workshop on Immersed surfaces in 3-manifolds Mar 26-30, 2012

Organizers: Danny Calegari, Bill Goldman, Vlad Markovic, Jean-Marc Schlenker, Alan Reid.

### Invited speakers:

• Ian Agol (UC Berkeley)
• Nicolas Bergeron (Paris 6)
• Martin Bridgeman (Boston College)
• Jeff Brock (Brown)
• Danny Calegari (Cambridge)
• Will Cavendish (Princeton)
• François Guéritaud (Lille)
• Jeremy Kahn (Brown)
• Sang-hyun (Sam) Kim (KAIST)
• François Labourie (Orsay)
• Tao Li (Boston College)
• Brice Loustau (Orsay)
• Jason Manning (Buffalo)
• Jessica Purcell (Brigham Young)
• Claire Renard (ENS Cachan)
• Alden Walker (Caltech)

## Titles and Abstracts

### Ian Agol

Title: The virtual Haken conjecture.

Abstract: We prove that cubulated hyperbolic groups are virtually special. Work of Haglund and Wise on special cube complexes implies that they are therefore linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. The results depend on a recent result of Wise, the malnormal special quotient theorem; the cubulation of closed hyperbolic 3-manifolds by Bergeron-Wise using the existence of nearly geodesic surfaces by Kahn-Markovic; and a generalization of previous work with Groves and Manning to the case of torsion (which is joint with Groves and Manning).

### Nicolas Bergeron

Title: Cubulation of hyperbolic 3-manifolds.

Abstract: I will describe a simple criterion in terms of the boundary for the existence of a proper cocompact action of a hyperbolic group on a CAT(0) cube complex. Thanks to the proof, by Kahn and Markovic, of the surface subgroup conjecture, the criterion is satisfied by all compact hyperbolic 3-manifolds.

### Martin Bridgeman

Title: Pleated Surfaces, Convex Hulls and Domains.

Abstract:

### Jeff Brock

Title: Fat, exhaustive integer homology 3-spheres.

Abstract:

### Danny Calegari

Title: Stable commutator length - statistics, concentration and compression.

Abstract: We discuss the statistical distribution of stable commutator length in various classes of groups, and some applications. For certain classes of groups (e.g. central extensions of lattices in $$\mathrm{Sp}(2n, \mathbf{R})$$) stable commutator length is distributed like distance to the origin for a random walk in a finite dimensional Euclidean space. For other classes of groups (e.g. hyperbolic groups, braid groups) there is a concentration of values, clustered around some fixed scale $$Cn/\log(n)$$ where the constant $$C$$ should conjecturally be derived in a simple manner from the (growth) entropy. This concentration should be thought of as a random analogue of the phenomenon of Mostow rigidity for hyperbolic manifolds. Finally, the growth rate of stable commutator length is an obstruction to the existence of (nonelementary) homomorphisms to hyperbolic groups, or actions on certain hyperbolic spaces. Some of this is joint work with Koji Fujiwara, Joseph Maher and Alden Walker.

### Will Cavendish

Title: Towers of Covering Spaces of 3-manifolds and Mapping Solenoids.

Abstract: This talk concerns the following question: given a map $$f:X\toY$$ between compact CW complexes and a collection $$C$$ of finite-sheeted covering spaces of $$Y$$, which covering spaces of $$X$$ appear as pull-backs of covers in the collection $$C$$? We will begin by introducing a construction called the mapping solenoid of $$f$$, and to show how the cohomology groups of this object can be used to give qualitative answers to this question. To demonstrate some uses of this construction we will study $$\pi_1$$-injective immersions $$f$$ from surfaces into 3-manifolds. In this setting, we will show that certain cohomology classes on the mapping solenoid of $$f$$ can be viewed as obstructions to solving lifting problems in the tower $$C$$. We will go on to show that if $$C$$ is a tower of non-Haken 3-manifolds, then the cohomology of the mapping solenoid can often be computed exactly, and gives rise to $$\pi_1(M)$$-modules with curious algebraic properties.

### François Guéritaud

Title: On Lorentzian spacetimes with constant curvature.

Abstract: I will discuss a characterization of Lorentzian spacetimes with constant negative curvature (i.e. complete AdS manifolds) in terms of contracting equivariant self-maps of the hyperbolic plane. I will also describe an infinitesimal version (corresponding to complete flat Lorentz manifolds), and describe how geodesic laminations arise in the boundary of the deformation space of such objects. Joint work with F. Kassel.

### Jeremy Kahn

Title: Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds.

Abstract: We prove that every closed hyperbolic 3-manifold has an essential immersed surfaces, and we present some results on the number of these surfaces of a given genus. In particular, we will sketch a proof that in certain cases, a typical closed immersed surface of large genus will not be evenly distributed in the given 3-manifold.

### Sang-hyun (Sam) Kim

Title: Doubles of free groups and hyperbolic surface subgroups.

Abstract: Let G be the double of a rank-two free group. We prove that if G is one-ended, then G has a hyperbolic surface subgroup. The proof is done by a partial resolution of a stronger graph theoretical conjecture, and is based upon a previous joint work with Henry Wilton. We also discuss a generalization to a graph of free groups with cyclic edge groups. This is a joint work with Sang-il Oum.

### François Labourie

Title: Random complex projective structures.

Abstract: In this talk, I will present a way to describe as a dynamical system the space of all complex projective structure on surfaces.

Quotienting by the action of a closed 3-manifold group yields a compact space sharing many properties of the usual geodesic flow. This space which is foliated by Riemann surfaces have the usual chaotic properties: genericity of dense leaves, density of the space of closed leaves, stability, many invariant ergodic measures of full support (associated to complex cross ratios). These results are not new but they present a framework to ask questions suscited by Kahn-Markovic theory.

### Tao Li

Title: Rank and genus of 3-manifolds.

Abstract: We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.

### Brice Loustau

Title: Minimal surfaces in almost-Fuchsian manifolds and renormalized volume.

Abstract: Minimal surfaces in quasifuchsian 3-manifolds provide a parametrization of the set of almost-Fuchsian structures by holomorphic quadratic differentials. This parametrization is different from the classical Schwarzian parametrization. We use the notion of renormalized volume to compare the symplectic structures induced on the set of almost-Fuchsian structures.

### Jason Manning

Title: Height, multiplicity, and Dehn filling.

Abstract: We show that given a virtually special quasiconvex subgroup $$H$$ of a hyperbolic group $$G$$, and some $$g \notin H$$, there is a hyperbolic quotient of $$G$$ in which $$H$$ projects to a finite subgroup, disjoint from the image of the element $$g$$. This statement is used in Ian Agol's recent proof of the Virtual Haken Conjecture. The method of proof is an induction on the height of the quasiconvex subgroup, using relatively hyperbolic Dehn filling to reduce height. Along the way, we give a new characterization of height as the multiplicity of a certain map, and a new proof (which does not depend on torsion-freeness) that Dehn filling reduces height of quasi-convex subgroups. This is joint work with Ian Agol and Daniel Groves.

### Jessica Purcell

Title: State surfaces in knot complements: fibers, guts, and the Jones polynomial.

Abstract: In the last decade, there have been several breakthroughs in hyperbolic geometry and 3-manifold topology. One goal in the research community is to use these new results to shed light on related problems, for example problems in knot and link theory. In this talk, we will describe some of our work in this direction. We determine geometric information on incompressible spanning surfaces in a large class of knot and link diagrams. Study of the geometric properties of these surfaces allows us to detect fibers and bound volumes, and relate them to certain coefficients of the Jones polynomial. This is joint with David Futer and Efstratia Kalfagianni.

### Claire Renard

Title: Finite covers of a hyperbolic 3-manifold and virtual fibrations.

Abstract: One can ask if there are some 'nice' conditions for a finite cover of a hyperbolic 3-manifold $$M$$ to fiber over the circle, or at least to contain a virtual fiber. In this talk, we provide an inequality on the Heegaard genus $$g$$ of a finite cover $$M'$$ of a hyperbolic 3-manifold $$M$$ and the degree $$d$$ of the cover such that if this inequality is satisfied, $$M'$$ contains a fiber which is an embedded surface of genus at most $$g$$. This inequality involves an explicit constant $$k$$ depending only on the volume and the injectivity radius of $$M$$. This machinery also applies to the setting of a circular decomposition associated to a non trivial homology class, to obtain sufficient conditions for a non trivial homology class of $$M$$ to correspond to a fibration over the circle. Similar methods lead also to a sufficient condition for an incompressible embedded surface in $$M$$ to be a fiber.

### Alden Walker

Title: Ziggurats and rotation numbers.

Abstract: (Joint with Danny Calegari) We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e. the conjecture of Jankins--Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.