Speaker: Bill Floyd, Virginia Tech
Title: Finite subdivision rules and rational maps
Abstract: A finite subdivision rule gives an essentially combinatorial method for recursively subdividing planar complexes. The theory was developed (as part of an approach to Cannon’s conjecture) as a tool for studying the recursive structure at infinity of Gromov-hyperbolic groups, but it is becoming increasingly useful for studying postcritically finite rational maps. I’ll give an overview (with lots of graphic images) of some of the connections between finite subdivision rules and rational maps.
We have Thomas Koberda (Yale), Eriko Hironaka (Florida State) and Andy Putman (Rice) speaking on things vaguely related to the mapping class group:
Speaker Thomas Koberda
Title: The complex of curves for a right-angled Artin group
Abstract: I will discuss an analogue of the curve complex for right-angled Artin groups and describe some of its properties. I will then show how it guides parallel results between the theory of mapping class groups and the theory of right-angled Artin groups. Joint with Sang-hyun Kim.
Speaker: Eriko Hironaka
Title: Small dilatation pseudo-Anosov mapping classes
Abstract: A pseudo-Anosov mapping classes on a compact finite-type oriented surface S has the property that the growth rate of lengths of an essential simple closed curve under iterations of the mapping class is exponential, and the growth rate is independent of the choice of curve and the of the choice of metric. This growth rate is called that dilatation of the mapping class. In this talk, we discuss the problem of describing small dilatation pseudo-Anosov mapping classes, i.e., those such that the dilatation raised to the topological Euler characteristic of the surface is bounded. We describe small dilatation mapping classes in terms of deformations within fibered faces, and give some explicit examples. We finish the talk with a conjecture concerning the “shape” of small dilatation mapping classes.
Speaker: Andy Putman
Title: Stability in the homology of congruence subgroups
Abstract: I’ll discuss some recent results which uncover new patterns in the homology groups of congruence subgroups of SLn(ℤ) and related groups.
Igor Rivin and Dave Futer are both speaking at the Hyperbolic Geometry conference at FDR’s house.
Speaker: Dieter Kotschick, U. of Munich
Title: Fundamental groups of Kahler manifolds and combinatorial group theory.
The study of fundamental groups of Kähler manifolds is a fascinating enterprise at the crossroads of various branches of geometry and topology, with strong relations to algebra and analysis as well. This lecture will survey some recent developments in this area, focussing on applications of the so-called Albanese map (which will be introduced during the lecture). Several results pertaining to groups of interest in low-dimensional topology and in combinatorial and geometric group theory will be discussed.
Brian will speak September 4th and 11th (at 3:30pm)
Title: An introduction to subdivision rules and Cannon’s conjecture
Abstract: Hyperbolic 3-space has a useful sphere at infinity, and any group acting geometrically on it has a sphere at infinity as well. It is not known if the converse is true; this is Cannon’s conjecture about Gromov hyperbolic groups with a 2-sphere at infinity. Subdivision rules were developed in an attempt to solve this conjecture. We will discuss the background of Cannon’s conjecture, subdivision rules, and what it means for a subdivision rule to be conformal.
The PATCH seminar, a joint venture with Bryn Mawr College, Haverford College, and the University of Pennsylvania, has its last meeting of Spring 2012 on Friday, April 20, at Haverford. The two speakers are Liam Watson of UCLA and David Gay of the University of Georgia.
The talks will take place at 3:00 and 4:30, in room KINS H108 at Haverford College.
Watson is speaking first, at 3:00pm.
Title: L-spaces and Left-orderability
Abstract: A group is left-orderable if it admits a strict total order of its elements that is invariant under multiplication on the left. As an immediate consequence (exercise!), left-orderable groups are torsion free. For example, a finite cyclic group cannot be left-ordered; hence the fundamental group of a lens space is not left-orderable. L-spaces provide a generalizations of lens spaces in the context of Heegaard Floer homology. These manifolds have simplest possible Heegaard Floer homology, though they need not have cyclic fundamental group. This talk will describe some evidence supporting the conjecture that L-spaces are equivalent to 3-manifolds with non-left-orderable fundamental group.
Gay is speaking second, at 4:30pm.
Title: Using Morse 2-Functions to trisect 4-manifolds
Abstract: Morse 2-functions are generic smooth maps to 2-manifolds, just as ordinary Morse functions are generic smooth maps to 1-manifolds. The goal of this talk is to get the audience to feel comfortable thinking about Morse 2-functions and to make the case that they are worth thinking about. As a vehicle for this agenda I’ll show how to construct a Morse 2-function on an arbitrary closed oriented 4-manifold X which yields a very natural decomposition of X into three diffeomorphic pieces. This seems to be the correct 4-dimensional analog of a Heegaard splitting; each piece is a 4-dimensional 1-handlebody, and the pieces are glued together along naturally arising 3-dimensional handlebodies. I will also throw in some other tidbits and hints about our larger Morse 2-function vision. This is a report on joint work with Rob Kirby.
This Tuesday at 5:00, the seminar hosts Jeff Brock of Brown University.
Title: Fat, exhausted integer homology spheres
Abstract: Since Perelman’s groundbreaking proof of the geometrization conjecture for three-manifolds, the possibility of exploring tighter correspondences between geometric and algebraic invariants of three-manifolds has emerged. In this talk, we address the question of how homology interacts with hyperbolic geometry in 3-dimensions, providing examples of hyperbolic integer homology spheres that have large injectivity radius on most of their volume. (Indeed such examples can be produced that arise as -Dehn filling on knots in the three-sphere). Such examples fit into a conjectural framework of Bergeron, Venkatesh and others providing a counterweight to phenomena arising in the setting of arithmetic Kleinian groups. This is joint work with Nathan Dunfield.