# Columbia Geometric Topology Seminar

## Spring 2015

The GT seminar meets on Fridays in Math. 520, at 1:15PM.
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

Date Speaker Title
January 23 Organizational meeting 1:15pm Room 520
January 30 Alessandro Sisto Deviation estimates for random walks and acylindrically hyperbolic groups
February 6 Alexei Fest at CUNY Joint meeting of NY Applied Algebra Colloquium, Model Theory, Algebraic Cryptography and NY Group Theory seminars.
February 13 Nick Salter Four-manifolds can be surface bundles over surfaces in many ways (except when they cant be)
February 20 Abibail Thompson Heegaard splittings and stabilizations
February 27 Thomas Koberda Convex cocompactness in right-angled Artin groups
March 6 Mark Feighn The geometry of Out(F_n)
March 13 Sebastian Hensel The handlebody Torelli group
March 20 No Seminar Spring Break
March 27 No seminar Various people are away
April 3 Kasra Rafi Teichmüller space is rigid.
April 10 No seminar No seminar
April 17 Lee Mosher Hyperbolic actions and second bounded cohomology for subgroups of Out(F_n) (joint with Michael Handel)
April 24 Dani Wise Counting cycles in graphs: A rank-1 version of the Hanna Neumann Conjecture
May 1 Ursula Hamenstaedt Typical properties of hyperbolic 3-manifolds
May 8 No Seminar Cornell Topology Festival starts 11:30am May 8

## Abstracts

Alessandro Sisto: January 30, 2015 “Deviation estimates for random walks and acylindrically hyperbolic groups”
Abstract: We will consider a class of groups that includes non-elementary (relatively) hyperbolic groups, mapping class groups, many cubulated groups and C'(1/6) small cancellation groups. Their common feature is to admit an acylindrical action on some Gromov-hyperbolic space and a collection of quasi-geodesics "compatible" with such action. As it turns out, random walks (generated by measures with exponential tail) on such groups tend to stay close to geodesics in the Cayley graph. More precisely, the probability that a given point on a random path is further away than L from a geodesic connecting the endpoints of the path decays exponentially fast in L. This kind of estimate has applications to the rate of escape of random walks (local Lipschitz continuity in the measure) and its variance (linear upper bound in the length). Joint work with Pierre Mathieu.

Alexei Fest: February 6, 2015 at CUNY Grad Center.
Program: 11:00-11:30 A. Mikhalev
11:40-12:10 V. Shpilrain
12:30-1:45 A. Miasnikov (at Model Theory)
1:45- 2:30 Lunch
2:30-3:10 M. Sohrabi
3:20-3:50 R. Gilman
4:00-4:40 R. Grigorchuk
4:50-5:30 P. Shupp
5:40-6:20 Yu Gurevich (to be confirmed)

Nick Salter: February 13, 2015 “Four-manifolds can be surface bundles over surfaces in many ways (except when they cant be)”
Abstract: An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the situation is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is finitely many? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on an asymptotic version of the how many? question. Time permitting, we will discuss some connections with the homology of the Torelli group, (non)-realization problems a la Nielsen and Morita, and symplectic topology.

Abigail Thompson: February 20, 2015 “Heegaard splittings and stabilizations”
Abstract: Let M be a closed, orientable 3-manifold. A Heegaard splitting of M is a splitting of the manifold into two handlebodies along a (closed, orientable) surface of genus g. Such a splitting is a convenient way to learn much about the structure of M. Every M has such a splitting; indeed every manifold has an infinite number of them. If we fix M, we can examine the set of all possible splittings of M. In particular, the Reidemeister-Singer theorem guarantees that any pair of splittings is stably equivalent. A natural question is, for two splittings of genus f and g, how many stabilizations are required to achieve equivalence? In work with J. Hass and W. Thurston, we showed that for some examples the least-genus equivalence is given by a surface of genus f+g. An outstanding conjecture is that f+g suffices. I'll discuss the examples from [H-T-T] and why the upper bound posited by the conjecture is reasonable, as well as some possible (but so far unsuccessful) approaches to proving it.

Thomas Koberda: February 27, 2015 “Convex cocompactness in right-angled Artin groups”
Abstract: Convex cocompact subgroups of mapping class groups were defined by Farb and Mosher, and they have attracted a lot of attention in recent years.  Despite this, many basic questions about these groups remain unresolved.  In this talk, we will discuss an analogue of convex cocompactness for right-angled Artin groups, and we will give geometric and algebraic characterizations of of such subgroups.  This represents joint work with J. Mangahas and S. Taylor.

Mark Feighn: March 6, 2015 “The geometry of Out(F_n)”
Abstract: The outer automorphism group Out(F_n) of the free group F_n of rank n acts naturally on a space CV_n of metric graphs called Culler-Vogtmanns Outer Space. Classically, the topology of CV_n has been used to explore the structure of Out(F_n). More recently, an (asymmetric) metric on CV_n has been used to the same end. I will discuss some of these more recent methods and advances.

Sebastian Hensel: March 13, 2015 “The handlebody Torelli group”
Abstract: The mapping class group of a surface has various topologically motivated subgroups. In this talk we combine two of them: the Torelli group (of those elements acting trivially on homology) and the handlebody group (of those elements extending to a given handlebody). We prove that it has an (infinite) generating set similar to the usual Torelli group (answering a question of Joan Birman) We also begin to develop a Johnson theory for the handlebody Torelli group, and highlight some of the many open questions about this group. This is joint work-in-progress with Andy Putman.

Kasra Rafi: April 3, 2015 “Teichmüller space is rigid.”
Abstract: We study the large scale geometry of Teichmüller space equipped with the Teichmüller metric. We show that, except for low complexity cases, any self quasi-isometry of Teichmüller space is a bounded distance away from an isometry of Teichmüller space. This is joint work with Alex Eskin and Howard Masur.

Lee Mosher: April 17, 2015 “Hyperbolic actions and second bounded cohomology for subgroups of Out(F_n) (joint with Michael Handel)”
Abstract: After surveying the co-evolution of the theories of hyperbolic actions and of second bounded cohomology of groups, we will report on recent progress on this topic for subgroups of Out(F_n).

Dani Wise: April 24, 2015 “Counting cycles in graphs: A rank-1 version of the Hanna Neumann Conjecture ”
Abstract: A "$W$-cycle" in a labelled digraph $\Gamma$ is a closed path whose label is the word $W$. I will describe a simple result about counting the number of $W$-cycles in a deterministically labelled connected digraph. Namely: the number of $W$-cycles in $\Gamma$ is bounded by $|E(\Gamma)| - |V(\Gamma)|+1$. I will outline the proof which uses left orderable groups. This is joint work with Joseph Helfer and has been proven independently by Lars Louder and Henry Wilton.

Ursula Hamenstaedt: May 1, 2015 “Typical properties of hyperbolic 3-manifolds ”
Abstract: By the solution of the virtual fibering conjecture, a closed hyperbolic 3-manifold has a finite cover which fibres over the circle. We explain a notion of randomness for such manifolds and describe some properties which hold true for typical manifolds with respect to this notion of randomness. We also discuss some open problems.

# Other relevant information.

## Previous semesters:

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.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting Walter Neumann.