# Columbia Geometric Topology Seminar

## Fall 2016 & Spring 2017

The GT seminar meets on Fridays in Math 520, at 2 PM (overflow seminars Thursday 2:40pm Room 507).
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

Date Speaker Title
Friday Sept 9 Organizational meeting 2pm Room 520
September 16 Hung C Tran (UGA) Geometric embedding properties of Bestvina-Brady subgroups
September 22 Thursday 2:40pm BoGwang Jeon The Unlikely Intersection Theory and the Cosmetic Surgery Conjecture
September 30  No GT seminar
October 7 Henry Segerman, Oklahoma State Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves
October 14 Mehdi Yazdi (Princeton) On Thurston's Euler class one conjecture
October 21 David Hume, MSRI and Oxford The wide world of relative hyperbolicity
October 28 Linus Kramer, Universitaet Muenster Automatic continuity of abstract homomorphisms between topological groups
November 4 No GT Seminar Symplectic Seminar
November 11 Tim Susse, U Nebraska Geometry of the word problem in 3-manifold groups
November 17 Thursday 2:40pm Brendan Owens, Glasgow University Searching for slice alternating knots
November 18 Daniel Groves, UIC Hyperbolic groups acting improperly
November 25 No Seminar Happy Thanksgiving
December 1 Thursday 2:40pm Cornelia Drutu, Oxford University Strong versions of Kazhdan's Property (T) and random groups
December 2 David Futer, Temple U Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds
December 8 Thursday 2:40pm Corey Bregman, Rice Kaehler Groups and Surface Bundles over Tori
December 9 Hongbin Sun, UCB NonLERFness of arithmetic hyperbolic manifold groups
December 15 Thursday 2:40pm Nick Salter, Chicago On the monodromy group of the family of smooth plane curves

## Abstracts

Hung C Tran, UGA
September 16
Title: Geometric embedding properties of Bestvina-Brady subgroups
We compute the subgroup distortion of Bestvina-Brady subgroups. We use the result of this computation to show that for each integer $n\geq 3$, there is a free subgroup of rank $n$ of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This corollary answers the question of Carr about the minimum rank $n$ such that some right-angled Artin group has a free subgroup of rank $n$ whose inclusion is not a quasi-isometric embedding. It is also well-known that a right-angled Artin group $A_\Gamma$ is the fundamental group of a graph manifold whenever the defining graph $\Gamma$ is a tree. We show that the Bestvina-Brady subgroup $H_\Gamma$ in this case is a horizontal surface subgroup.

BoGwang Jeon, Columbia
THURSDAY September 22, 2:40pm
Title: The Unlikely Intersection Theory and the Cosmetic Surgery Conjecture
The main result of this talk is the following theorem: Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is not quadratic, and M(p/q) be its p/q-Dehn filled manifold. If p/q is not equal to p'/q' for sufficiently large |p|+|q| and |p'|+|q'|, there is no orientation preserving isometry between M(p/q) and M(p'/q'). This resolves the conjecture of C. Gordon, which is so called the Cosmetic Surgery Conjecture, for hyperbolic 3-manifolds belonging to the aforementioned class except for possibly finitely many exceptions for each manifold. We also consider its generalization to more cusped manifolds. The key ingredient of the proof is the unlikely intersection theory developed by E. Bombieri, D. Masser, and U. Zannier.

Henry Segerman, Oklahoma State U
October 7
Title: Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves.
Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the "2-3 Pachner graph" of such triangulations is connected. This is useful for defining invariants of a three-manifold based on the triangulation. However, there are "would-be" invariants that can only be defined on triangulations with certain properties, for example 1-efficiency or having only essential edges. Unfortunately, there are no similar connectivity results for the subgraphs of the Pachner graph with such properties. In this talk, I will describe a new connectivity result for a weaker property than either 1-efficiency or essential edges: that of a triangulation having no degree one edges.

Mehdi Yazdi, Princeton
October 14
Title: On Thurston's Euler class one conjecture
In 1976, Thurston proved that taut foliations on closed hyperbolic 3–manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. I construct counterexamples to this conjecture and suggest an alternative conjecture.

David Hume, MSRI and Oxford
October 21
Title: The wide world of relative hyperbolicity
I will describe part of a joint project with Matthew Cordes in which, for a given finitely generated group $H$ satisfying some very mild assumptions, we build infinitely many quasi-isometry types of 1-ended groups $G_i$ which are hyperbolic relative to $H$. With care, one can even ensure that all the $G_i$ satisfy Kazhdan's property T. The invariant we use is a refinement of the stable dimension which we introduced in a previous paper.

Linus Kramer, Univ Muenster
October 28
Title: Automatic continuity of abstract homomorphisms between topological groups
It is a classical result that an ’abstract’ group isomorphism between real semisimple Lie groups is automatically continuous. In this talk we consider more generally abstract homomorphisms of certain topological groups onto certain types of Lie groups and compact groups. The approach is axiomatic and yields new continuity results both for Polish and for locally compact groups.

November 11
Title: Geometry of the word problem in 3-manifold groups
Since the introduction of automatic groups, there have been attempts by many mathematicians to find a common solution to the word problem in any 3-manifold groups that uses only a finite state automaton. However, not all 3-manifold groups are automatic. Several generalizations of automaticity have been introduced, including the notion of an autostackable group by Brittenham, Hermiller and Holt. We will define autostackability, as well as autostackability respecting a subgroup and prove that the fundamental group of any closed 3-manifold is autostackable.

Brendan Owens, Glasgow University
November 17
Title: Searching for slice alternating knots
Many geometric properties of alternating knots have been shown to be computable or discernible from their alternating diagrams. This talk is based on the possibility that sliceness of an alternating knot may be algorithmically detectable from an alternating diagram. I will describe a computer search which has found approximately 29,000 new slice alternating knots, and discuss its limitations and some related questions. This is joint work with Frank Swenton.

Daniel Groves, University of Illinois at Chicago
November 18
Title: Hyperbolic groups acting improperly
Suppose that a hyperbolic group G acts cocompactly on a CAT(0) cube complex, and that vertex stabilizers are quasiconvex and virtually special. Then G is virtually special. In case vertex stabilizers are finite, this is a result of Agol (which implies the Virtual Haken Conjecture and the Virtual Fibering Conjecture), and in case the cube complex is a tree it is a result of Wise (a key ingredient in Agol's proof and result with significant consequences in its own right). I will discuss the background and ingredients of the proof of this result and also some possible applications. This is joint work with Jason Manning (Cornell).

Cornelia Drutu, Oxford University
December 1, 2:40pm
Title: Strong versions of Kazhdan's Property (T) and random groups
Various strengthened versions of property (T) have been formulated in recent years. Among them, those involving actions on Lp spaces are particularly interesting, because they manage to achieve a separation between rank one and higher rank lattices, because of their presumed connection to the conformal dimension of the boundary of hyperbolic groups, and because of the increasing role that they play in operator algebras. In this talk I shall explain how random groups have all the strengthened Lp -versions of property (T), and how these connect to the conformal dimension of their boundary. This is joint work with J. Mackay.

David Futer, Temple University
December 2
Title: Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds
I will discuss a proof that every finite volume hyperbolic 3-manifold M contains an abundant collection of immersed, $\pi_1$-injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

Corey Bregman, Rice
December 8
Title: Kaehler Groups and Surface Bundles over Tori
Abstract: A question going back to Serre asks which finitely presented groups arise as the fundamental groups of compact Kaehler manifolds. In this talk we study extensions of abelian groups by hyperbolic surface groups. Topologically, these groups are realized as fundamental groups of surface bundles over tori. We show that if any such an extension is Kaehler, then it is virtually a product. This is joint work with Letao Zhang.

Hongbin Sun, UCB
December 9
Title: NonLERFness of arithmetic hyperbolic manifold groups
Abstract: We will show that, for "almost" all arithmetic hyperbolic manifolds with dimension >3, their fundamental groups are not LERF. The main ingredient in the proof is a study of certain graph of groups with hyperbolic 3-manifold groups being the vertex groups. We will also show that a compact irreducible 3-manifold with empty or tori boundary does not support a geometric structure if and only if its fundamental group is not LERF.

Nick Salter, Chicago
December 15
Title: On the monodromy group of the family of smooth plane curves
Abstract: Let P denote the space of smooth projective curves in the complex projective plane of degree d. Over P, there is the tautological family of smooth plane curves, and hence a monodromy representation, i.e. a subgroup of the mapping class group. In the 1980?s, algebraic geometers computed an ?approximation? to this monodromy group, namely the action on the homology of the fiber. I will discuss some recent work of mine concerning a characterization of the image of the full monodromy representation, including a complete description of the image for degree d=5, as well as a conjectural picture of the image for all degrees. This will involve a blend of ideas from algebraic geometry and the theory of the mapping class group, particularly the Torelli group.

# Other relevant information.

## Previous semesters:

Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.