##
IHP program

Geometry and analysis of surface group representations

January-March 2012

#
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)

## Schedule

## 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.