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Boston College

HMI

NSF

# Surface Groups in Low Dimensional Topology and Geometric Group Theory

## Schedule of Talks

All talks will be held in the Hamilton Building, TCD.

 Time and Location Speaker Talk Thursday Joly Lecture Theatre 9.00 - 10.00 Benson Farb Surface groups in discrete groups 10:15 - 11:15 Chris Leininger Surface groups in Mod(S) with "accidental parabolics" 11:15 - 11:45 DISCUSSION 11:45 - 1:30 LUNCH 1:30 - 2:30 Genevieve Walsh Virtual homology slopes and the group determinant 2:45 - 3:45 Richard Kent Being purely pseudo-Anosov 4:00 - 5:00 Benson Farb Representations of surface groups in mapping class groups Friday MacNeill Lecture Theatre 9:00 - 10:00 Daniel Groves Dehn filling in relatively hyperbolic groups 10:15 - 11:15 Darren Long Subgroup separability and virtual retractions 11:15 - 11:45 DISCUSSION 11:45 - 1:30 LUNCH 1:30 - 2:30 Josh Barnard Distortion of surface groups in free-by-cyclic groups 2:45 - 3:45 Caroline Series Lines of minima and Teichmuller geodesics 4:00 - 5:00 Michah Sageev Surface subgroups of right-angled Artin groups Saturday Maxwell Lecture Theatre 9:00 - 10:00 Bert Wiest Surface groups in right-angled Artin groups and braid groups 10:15 - 11:15 Denis Osin Cut points in asymptotic cones of finitely presented groups 11:15 - Noon Summary Discussion. Problems. Directions. END OF WORKSHOP

## Abstracts

 Speaker Title and Abstract Benson Farb: 1st talk Surface groups in discrete groups In this talk I will give an introduction and overview of the main theme of this conference: the ubiquity of surface groups in geometry and topology. I'll explain some of the fundamental constructions and connections to a wide variety of topics. Chris Leininger: Surface groups in Mod(S) with "accidental parabolics" I'll discuss a construction (joint work with Alan Reid) of surface subgroups of the mapping class group, Mod(S) which are  almost purely pseudo-Anosov:  every non-trivial element with one exception (up to powers and conjugacy) is pseudo-Anosov. I'll also discuss a generalization of that construction, along with a description of the geometry of such groups Genevieve Walsh: Virtual homology slopes and the group determinant Given a finite group $G$, the group determinant is the representation determinant of the right regular representation. We use the group determinant to study the homology of a finite regular cover of a 3-manifold with covering group $G$.  In particular we are interested in understanding virtual homology slopes.  A virtual homology slope $\alpha$ for a cover $\tilde Y$ of $Y$ is a slope such that, in particular, filling along pre-images of $\alpha$ in $\tilde Y$ results in a manifold with positive first Betti number. We use the existence of such slopes to show that every hyperbolic 1-cusped 3-manifold admits infinitely many virtually Haken Dehn-fillings. This is joint work with Daryl Cooper. Richard Kent: Being purely pseudo-Anosov I'll discuss some criteria for a subgroup of the mapping class group to be purely pseudo-Anosov (meaning that all non-identity elements are pseudo-Anosov), and survey the examples of such subgroups that are known.  This is joint work with Chris Leininger. Benson Farb: 2nd talk Representations of surface groups in mapping class groups In this talk I will discuss the problem of classifying, up to conjugacy, representations of a fixed (closed) surface group in a fixed mapping class group.  I will explain how this problem relates to problems in topology and complex geometry, and will show (via joint work with J. Crisp) how to construct a huge number of faithful, irreducible representations.  I will then describe various approaches to the classification problem, from algorithmic aspects to volume invariants to Teichmuller-geometric properties Daniel Groves: Dehn filling in relatively hyperbolic groups (Joint work with Jason Manning) We prove an algebraic analogue of the Gromov-Thurston 2pi-Theorem. In this talk, I will discuss the formulation of the theorem, and also tools used in the proof.  In particular, we define a cusped space' which is analogous to hyperbolic space for the fundamental groups of finite-volume hyperbolic manifolds.  I will then discuss preferred paths' which are a key technical innovation in our work, and some other applications of this technology.  Finally, I will say something about the proof of the Dehn filling theorem. Time permitting, I will discuss the (im)possibility of new surface subgroups arising under Dehn filling. Darren Long: Subgroup separability and virtual retractions We show that one can usefully take a somewhat different point of view on subgroup separability and this naturally leads to a new sort of property for groups. Applications to the virtual Betti number problem will be discussed. Josh Barnard: Distortion of surface groups in free-by-cyclic groups We give examples of polynomially distorted closed surface groups in CAT(0) free-by-cyclic groups, and of exponentially distorted closed surface groups in hyperbolic free-by-cyclic groups. Caroline Series: Lines of minima and Teichmuller geodesics.  A pair \mu, \nu of measured laminations which fill up a hyperbolisable surface S determines two paths through Teichmuller space: a Teichmuller geodesic; and a line of minima (along which convex combinations of hyperbolic lengths of \mu, \nu are minimised). One can ask whether and when these two paths enter the thin parts of Teichmuller space. Rafi (for Teichmuller geodesics) and Choi-Rafi-Series (for lines of minima) have answered this by estimating the hyperbolic length of a short simple closed curve \gamma along these paths in terms of the combinatorics of \mu,\nu relative to \gamma.  We discuss these results and how they can be used to study the relative behaviour of the two paths. In particular, we recover Lenzhen's recent example of a Teichmuller geodesic which does not converge to a point in the Thurston boundary PML, and show that the same statement applies to the corresponding line of minima. Michah Sageev: Surface subgroups of right-angled Artin groups  We will discuss some results which deal with the question: which right-angled Artin groups have surface subgroups. This is joint work with John Crisp and Mark Sapir. Bert Wiest: Surface groups in right-angled Artin groups and braid groups This is joint work with John Crisp. We prove that the fundamental group of a surface $S$ can be embedded in a RAAG if and only if $S$ is not the projective plane, Klein bottle, or connected sum of three projective planes. This embedding, when it exists, can even be made quasi-isometric. We also prove that all the RAAGs in question can be embedded, again quasi-isometrically, in a pure braid group. Indeed, every RAAG can be embedded quasi-isometrically in some surface braid group. Denis Osin: Cut points in asymptotic cones of finitely presented groups In this talk I will discuss the question of how to generalize the definition of relatively hyperbolic groups in order to encompass such examples as mapping class groups, right angled Artin groups, etc. One natural way is to require the existence of cut points in asymptotic cones. I will survey known results in this direction.

## Format

The three day directed workshop  will consist of background talks in the morning, followed by a question-and-answer session, and then afternoon lectures which will describe current research.

The first two days of the workshop, there will be two morning talks and three afternoon talks. On the last day, there will be two morning talks followed by an afternoon  problem session in which participants will be encouraged to describe parallels that they observed between the problems and techniques used in the various fields. A list of observed parallels, open problems,  and likely future research directions will be compiled for general dissemination.

The HMI will publish  the workshop lectures and results as a series of notes. The notes will be disseminated in paper form to other institutes as well as be posted on the HMI website.

 Trinity College Dublin, College Green, Dublin 2. Tel: +353-1-608-1000. |