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