# Geometry and Topology Seminar 2014

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.

• Tuesday January 28, 2014 at 14:00, Wachman 617
GeoTop Seminar

Timothy Susse, CUNY, SCL in Torus Knot Complements

Given a group $G$ and an element $g$ of its commutator subgroup, its stable commutator length is the growth rate of the smallest number of commutators whose product is $g^n$. This quantity is closely related to the topology of surfaces with boundary mapping to a topological space with fundamental group $G$.

When $G$ is the fundamental group of a torus knot complement, or more generally an amalgamated free product of free abelian groups, we will construct a finite sided polyhedron which parameterizes surfaces with a specified boundary. We will then show that scl is rational in these groups, giving a topological solution to a conjecture of Calegari in this special case.

• Tuesday February 4, 2014 at 14:00, Wachman 617
GeoTop Seminar

Chris Hays, University of Pennsylvania, Constructing symplectic 4-manifolds

Symplectic 4-manifolds play an important role in the theory of smooth 4- manifolds for two reasons. First, they typically have a non-trivial Seiberg-Witten invariant. Second, there are methods that allow one to create new symplectic 4-manifolds from known ones. These properties allow one to construct infinitely many smooth 4-manifolds with the same underlying homeomorphism type.

In the talk, I will outline a new program for creating symplectic 4-manifolds. This method relies on creating both interesting concave and convex fillings of contact 3-manifolds, and attaching these fillings together. I will discuss manifolds that can be created in this manner, and the ease with which one can determine that these symplectic manifolds are 'non-standard'.

• Tuesday February 11, 2014 at 14:00, Wachman 617
GeoTop Seminar

Tarik Aougab, Yale University, Minimally intersecting filling pairs

Let $S_{g}$ denote the closed orientable surface of genus $g$. We show the existence of exponentially many mapping class group orbits of pairs of simple closed curves on $S_{g}$ which fill the surface, and intersect minimally amongst all filling pairs. We will demonstrate the main idea of the construction, and we'll discuss applications to the complex of curves. This is joint work with S. Huang (applications are joint with S. Taylor and R. Webb).

• Tuesday February 18, 2014 at 14:00, Wachman 617
GeoTop Seminar

Tudor Dimofte, Institute for Advanced Study, A Spectral Perspective on Neumann-Zagier

Thurston's gluing equations for ideal hyperbolic triangulations have certain symplectic properties, initially discovered by Neumann and Zagier, that underlie the formulation of many classical and quantum 3-manifold invariants. It has long been suspected that these symplectic properties have an intrinsic topological interpretation. I will explain one such interpretation, which trivializes the symplectic properties, based on branched covers of 3-manifolds and their boundaries. (Joint work with R. van der Veen.)

• Tuesday February 18, 2014 at 16:30, Wachman 617
GeoTop Seminar

-Note different time-

Martin Bridgeman, Boston College, The Pressure metric for convex Anosov representations

Using thermodynamic formalism, we introduce a notion of intersection for convex Anosov representations. We also produce an Out-invariant Riemannian metric on the smooth points of the deformation space of convex, irreducible representations of a word hyperbolic group $G$ into $SL(m, R)$ whose Zariski closure contains a generic element. In particular, we produce a mapping class group invariant Riemannian metric on Hitchin components which restricts to the Weilâ€“Petersson metric on the Fuchsian locus. This is joint work with R. Canary, F. Labourie and A. Sambarino.

• Tuesday February 25, 2014 at 14:00, Wachman 617
GeoTop Seminar

Dave Futer, Temple University, Volumes of hyperbolic 3-manifolds I

This will be an introductory talk about estimating the volume of hyperbolic 3-manifolds. By the Mostow rigidity theorem (which I will explain), a 3-dimensional manifold admits at most one complete hyperbolic metric. Hence, the volume of this metric is an important topological invariant.

After sketching the background, I will describe a program for obtaining explicit estimates on the volume of a hyperbolic 3-manifold directly from combinatorial data. To date, this program works for the broad class of 3-manifolds that fiber over the circle. All new results mentioned in these talks are joint work with J. Purcell and S. Schleimer.

• Tuesday March 4, 2014 at 16:00, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Kathryn Mann, University of Chicago, Surface groups, representation spaces, and rigidity

Let $S_g$ denote the closed, genus g surface. In this talk, we'll discuss the space of all circle bundles over $S_g$, namely $Hom(\pi_1(S_g), Homeo^+(S^1))$. The Milnor-Wood inequality gives a lower bound on the number of components of this space ($4g-3$), but until very recently it was not known whether this bound was sharp. In fact, we still don't know whether the space has infinitely many components!

I'll report on recent work and new tools to understand $Hom(\pi_1(S_g), Homeo^+(S^1))$. In particular, I use dynamical methods to give a new lower bound on the number of its components, and show that certain geometric representations are surprisingly rigid.

• Tuesday March 4, 2014 at 17:30, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Adam Levine, Princeton University, Non-orientable surfaces in homology cobordisms

We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a non-orientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle.

• Tuesday March 25, 2014 at 11:00, Wachman 447
GeoTop Seminar

Special Geometry Day:

Anton Lukyanenko, University of Illinois, Uniformly quasi-regular mappings on sub-Riemannian manifolds

A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. $f(z)=z^2$. The notion generalizes both covering and quasi-conformal mappings and is well-studied for Riemannian manifolds. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:

1. Every lens space admits a uniformly QR (UQR) self-mapping.
2. Every UQR mapping leaves invariant a measurable conformal structure.
The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.

• Tuesday March 25, 2014 at 14:00, Wachman 617
GeoTop Seminar

Special Geometry Day:

Sara Maloni, Brown University, Combinatorial methods on actions on character varieties

In this talk we consider the $SL(2,C)$-character variety $X = Hom(\pi_1(S), SL(2,C) ) // SL(2,C)$ of the four-holed sphere $S$, and the natural action of the mapping class group $MCG(S)$ on it. In particular, we describe a domain of discontinuity for the action of $MCG(S)$ on the relative character varieties $X_{(a,b,c,d)}$, which is the set of representations for which the traces of the boundary curves are fixed. Time permitting, in the case of real characters, we show that this domain of discontinuity may be non-empty on the components where the relative Euler class is non-maximal.

• Tuesday March 25, 2014 at 17:00, Wachman 617
GeoTop Seminar

Special Geometry Day:

François Guéritaud, Université Lille 1, Spacetimes of constant curvature

I will survey recent results (joint with J. Danciger and F. Kassel) on 3-dimensional complete spacetimes of constant curvature $K$, also known as quotients of $PSL(2,R)$ ($K<0$) or of its Lie algebra ($K=0$). I will emphasize the transition phenomena as $K$ goes to 0 and, time permitting, discuss the so-called Crooked Plane Conjecture of Charette, Drumm and Goldman.

• Thursday March 27, 2014 at 14:00, Wachman 617
GeoTop Seminar

-Note different day-

Christian Millichap, Temple University, Geometric invariants of highly twisted hyperbolic pretzel knots

Given a hyperbolic knot $K$, the corresponding knot complement $M$ has a number of interesting geometric invariants. Here, we shall consider the systole of $M$, which is the shortest closed geodesic in $M$ and the volume of $M$. It is natural to ask how bad are these invariants at distinguishing hyperbolic $3$-manifolds and how do these invariants interact with one another. In this talk, we shall construct large families of hyperbolic pretzel knot complements with the same volume and the same systole. This construction will rely on mutating pretzel knots along four-punctured spheres, and then showing that such mutations often preserve the volume and the systole of a hyperbolic knot.

• Tuesday April 1, 2014 at 14:00, Wachman 617
GeoTop Seminar

Ara Basmajian, CUNY, Involution generating sets for isometries of hyperbolic n-space

The focus of this talk will be on the word length of the orientation preserving isometries of hyperbolic $n$-space (the Mobius group), denoted $G$, with respect to various generating sets of involutions. If the generating set consists of the conjugacy class of a single orientation preserving $k$-involution, we show that the word length of $G$ is comparable to $n$. Here a $k$-involution is an involution with a fixed point set of codimension $k$. We also discuss the percentage of involution conjugacy classes for which $G$ has length two as the dimension $n$ gets large. Most of this is joint work with Karan Puri.

• Tuesday April 8, 2014 at 14:00, Wachman 617
GeoTop Seminar

Andrew Zimmer, University of Michigan, Rigidity of complex convex divisible sets

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic dividing group. In this talk I will discuss a notion of convexity in complex projective space and show that every divisible complex convex set with $C^1$ boundary is projectively equivalent to the unit ball. The proof uses tools from dynamics, geometric group theory, and algebraic groups.

• Tuesday April 15, 2014 at 17:00, Wachman 617
GeoTop Seminar

-Note different time-

Feng Luo, Rutgers University, Choi's theorem on triangulated 3-manifolds and consequences

In her 2000 Ph.D thesis, Y. Choi proved a very nice theorem concerning Thurston's gluing equations on triangulated 3-manifolds. In this talk, we will give a new simple proof of it and discuss some consequences of Choi's theorem.

• Tuesday April 22, 2014 at 14:00, Wachman 617
GeoTop Seminar

Mark Hagen, University of Michigan, Cubulating hyperbolic free-by-cyclic groups

Let $G$ be a word-hyperbolic free-by-$\mathbb{Z}$ group. Then $G$ acts freely and cocompactly on a CAT(0) cube complex. I'll explain some of the consequences of this fact (notably, $\mathbb{Z}$-linearity) and discuss the main ingredients of the proof. This talk is on joint work with Dani Wise.

• Tuesday May 6, 2014 at 14:00, Wachman 617
GeoTop Seminar

Ruth Charney, Brandeis University, Hyperbolic-like geodesics

In spaces of non-positive curvature (CAT(0) spaces), some geodesics act like hyperbolic geodesics and others do not. In joint work with H. Sultan, we use hyperbolic-like geodesics to define a new boundary for a CAT(0) space. In this talk I will give various equivalent characterizations of "hyperbolic-like" geodesics and show how these can be used to understand explicit examples. In addition, I will discuss some recent work of M. Cordes generalizing some of these ideas to geodesic metric spaces with no curvature conditions. (This talk will expand on some ideas from the colloquium talk, but it will be self-contained.)

• Tuesday May 20, 2014 at 14:00, Wachman 617
GeoTop Seminar

Ben McReynolds, Purdue University, Homology of infinite volume manifolds

I will discuss a homological vanishing result for certain classes of real rank one, locally symmetric, infinite volume manifolds that models well-known homological vanishing results for closed manifolds. The talk will largely focus on the mechanism for the vanishing results which is blends analytic, dynamical, and geometric ideas. This is joint work with Chris Connell and Benson Farb.