# Geometry and Topology Seminar

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.

• April 27, 2016 at 14:30, Wachman 617
The simple loop conjecture for 3-manifolds modeled on Sol

Drew Zemke, Cornell University

The $p$-local commensurability graph ($p$-local graph) of a group has vertices consisting of all finite-index subgroups, where an edge is drawn between two subgroups if their commensurability index is a power of $p$. Sitting at the interface between intersection graphs, containment graphs, and commensurability, these $p$-local graphs give insights to Lubotzky-Segal's subgroup growth functions. In this talk, we connect topological properties of $p$-local graphs to nilpotence, solvability, and largeness (containing a free subgroup of finite index) of the target group. This talk covers joint work with Daniel Studenmund and Chen Shi.

• April 20, 2016 at 14:30, Wachman 617
Unsmoothable group actions on one-manifolds

Thomas Koberda, University of Virginia

I will discuss virtual mapping class group actions on compact one-manifolds. The main result will be that there exists no faithful $C^2$ action of a finite index subgroup of the mapping class group on the circle, which generalizes results of Farb-Franks and establishes a higher rank phenomenon for the mapping class group, mirroring a result of Ghys and Burger-Monod.

• April 14, 2016 at 17:30, PATCH seminar, at Penn, DRL room 4C8
Loop products, index growth, and closed geodesics

Nancy Hingston, The College of New Jersey

• April 14, 2016 at 16:30, PATCH seminar, at Penn, DRL room 4C8
Controlling Ray Bundles with Reflectors

Andrew Hicks, Drexel University

• April 6, 2016 at 14:30, Wachman 617
Veering Dehn surgery

Saul Schleimer, University of Warwick

It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced veering triangulations'' for studying punctured surface bundles over the circle. Veering triangulations are very rigid; one current conjecture is that any fixed three-manifold admits only finitely many veering triangulations.

After giving an overview of these ideas, we will introduce veering Dehn surgery''. We use this to give the first infinite families of veering triangulations with various interesting properties. This is joint work with Henry Segerman.

• March 25, 2016 at 16:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

John Etnyre, Georgia Tech

• March 25, 2016 at 14:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

Genevieve Walsh, Tufts University

• March 18, 2016 at 13:30, Wachman 617
Counting curves on hyperbolic surfaces

-Note different day and time-

Viveka Erlandsson, Aalto University

In this talk I will discuss the growth of the number of closed geodesic of bounded length, and the length grows. More precisely, let $c$ be a closed curve on a hyperbolic surface $S=S(g,n)$ and let $N_c(L)$ denote the number of curves in the mapping class orbit of $c$ with length bounded by $L$. Mirzakhani showed that when $c$ is simple, this number is asymptotic to $L^{6g-6+2n}$. Here we consider the case when $c$ is an arbitrary closed curve, i.e. not necessarily simple. This is joint work with Juan Souto.

• March 9, 2016 at 14:30, Wachman 617
$k$-geodesics and lifting curves simply

Tarik Aougab, Brown University

Let $\gamma$ be a closed curve on a surface $S$ with negative Euler characteristic, and suppose gamma has at most $k$ self-intersections. We construct a hyperbolic metric with respect to which $\gamma$ has length (on the order of) $\sqrt{k}$, and whose injectivity radius is bounded below by $1/\sqrt{k}$; these results are optimal. As an application, we give sharp upper bounds on the minimum degree of a cover for which gamma lifts to a simple closed curve. This is joint work with Jonah Gaster, Priyam Patel, and Jenya Sapir.

• February 26, 2016 at 16:30, Wachman 617

Josh Greene, Boston College [PATCH seminar, joint with Bryn Mawr, Haverford, and Penn]

I will describe a characterization of alternating links in terms intrinsic to the link exterior and use it to derive some properties of these links, including algorithmic detection and new proofs of some of Tait's conjectures.

• February 26, 2016 at 15:00, Wachman 617
Knot contact homology and string topology

Lenny Ng, Duke University [PATCH seminar, joint with Bryn Mawr, Haverford, and Penn]

Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach, combined with the modern theory of Legendrian contact homology (which I'll also introduce), to construct a rather powerful invariant of knots called "knot contact homology".

• February 24, 2016 at 14:30, Wachman 617
Compactifying spaces of Riemannian manifolds, with applications

Ian Biringer, Boston College

We will describe how to compactify sets of Riemannian manifolds with constrained geometry (e.g. locally symmetric spaces), where the added limit points are transverse measures on some universal foliated space. As an application, we study the ratio of the $k$-th Betti number of a manifold to its volume, and give a strong convergence result for higher rank locally symmetric spaces.

• February 10, 2016 at 14:30, Wachman 617
The topology of local commensurability graphs

Khalid Bou-Rabee, City College of New York

The p-local commensurability graph (p-local graph) of a group has vertices consisting of all finite-index subgroups, where an edge is drawn between two subgroups if their commensurability index is a power of p. Sitting at the interface between intersection graphs, containment graphs, and commensurability, these p-local graphs give insights to Lubotzky-Segal's subgroup growth functions. In this talk, we connect topological properties of p-local graphs to nilpotence, solvability, and largeness (containing a free subgroup of finite index) of the target group. This talk covers joint work with Daniel Studenmund and Chen Shi.

• February 3, 2016 at 14:30, Wachman 617
Spacious knots

Richard Kent, University of Wisconsin

Brock and Dunfield showed that there are integral homology spheres whose thick parts are very thick and take up most of the volume. Precisely, they show that, given $R$ big and $r$ small, there is an integral homology 3-sphere whose $R$-thick part has volume $(1-r) vol(M)$. Purcell and I find knots in the 3-sphere with this property, answering a question of Brock and Dunfield.

• January 27, 2016 at 14:30, Wachman 617
Topological constructions of manifolds with geometric structures

Matthew Stover, Temple University

Classical uniformization implies that the existence of a complete hyperbolic metric on a Riemann surface depends only on its topological type. In dimension 3, Thurston's geometrization program also gives a necessary and sufficient topological condition. I will discuss topological methods for proving existence of a metric of constant holomorphic sectional curvature -1 on the complement of curves in a smooth complex projective surface. I will mainly focus on an interesting example due to Hirzebruch, and hopefully turn to some applications of these topological constructions, e.g., to questions about betti number growth. This is mostly joint with Luca Di Cerbo.

• January 20, 2016 at 14:30, Wachman 617
Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

Dave Futer, Temple University

I will discuss a proof that a cusped hyperbolic 3-manifold M contains an abundant collection of immersed, quasifuchsian surfaces. These surfaces are abundant in the sense that their boundaries separate any pair of points on the sphere at infinity. As a corollary, we recover Wise's theorem that the fundamental group of M is cubulated. This is joint work with Daryl Cooper.