# Geometry, Groups and

Dynamics/GEAR Seminar

The Geometry, Groups and Dynamics/GEAR Seminar, held at the Illinois hub of GEAR, features speakers of interest to network members. The talks are live streamed on the Illinois Department of Mathematics Youtube channel playlist. Seminars postings can also be found on the Illinois Mathematics Department Events Calendar.

Archive of past lectures.

##### Spring 2019

**12:00 pm, Thursday, January 24, 2019, 243 Altgeld Hall
Joshua Pankau (University of Iowa)
Stretch Factors Coming From Thurston's Construction**

Abstract: Associated to every pseudo-Anosov map is a real number called its stretch factor. Thurston proved that stretch factors are algebraic units, but it is unknown exactly which algebraic units are stretch factors. In this talk I will discuss a construction of pseudo-Anosov maps due to Thurston, and discuss my recent results where I classified (up to power) the stretch factors coming from this construction. We will primarily focus on a specific class of algebraic units known as Salem numbers. This talk is intended to be accessible to everyone.

** 12:00 pm, Thursday, January 17, 2019, 243 Altgeld Hall
Ilya Kapovich (Hunter College)
Index properties of random automorphisms of free groups**

Abstract: For automorphisms of the free group Fr, being "fully irreducible" is the main analog of the property of being a pseudo-Anosov element of the mapping class group. It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a "random" (in the sense of being generated by a long random walk) element ϕ of Out(Fr) is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles ϕ∈Out(Fr) have not been understood. We prove that for a "random" ϕ∈Out(Fr) (where r≥3), the attracting and repelling ℝ-trees of ϕ are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index <2r−2). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.

**Fall 2018**

**12:00 pm, Tuesday, November 13, 2018, 243 Altgeld Hall
Margaret Nichols (University of Chicago)
Taut sutured handlebodies as twisted homology products**

Abstract: A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’. One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such? We explore some classes of relatively simple sutured manifolds, and see one class is always a rational homology product, but that the next natural class contains examples which require twisting. We also find examples that require twisting by a representation which cannot be ‘too simple’.

**12:00 pm, 243 Altgeld Hall,Tuesday, October 2, 2018
Alex Zupan (University of Nebraska-Lincoln)
Generalized square knots and the 4-dimensional Poincare Conjecture**

Abstract: The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

**12:00 pm, 243 Altgeld Hall,Tuesday, September 25, 2018
Justin Lanier (Georgia Tech Math)
Normal generators for mapping class groups are abundant**

Abstract: For mapping class groups of surfaces, we provide a number of simple criteria that ensure that a mapping class is a normal generator, with normal closure equal to the whole group. We then apply these criteria to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator whenever genus is at least 3. We also show that every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. Showing that pseudo-Anosov normal generators exist at all answers a question of Darren Long from 1986. In addition to discussing these results on normal generators, we will describe several ways in which they can be leveraged to answer other questions about mapping class groups. This is joint work with Dan Margalit.

**12:00 pm, 243 Altgeld Hall,Thursday, September 20, 2018
Moira Chas (Stony Brook)
The generalization of the Goldman bracket to three manifold and its relation to Geometrization**

Abstract: In the eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. In the nineties, jointly with Dennis Sullivan, we generalized this Lie algebra structure to families of loops (defining the equivariant homology of the free loop space of a manifold). This Lie algebra, together with other operations in spaces of loops is now known as String Topology. The talk will start with a discussion of the Goldman Lie bracket in surfaces, and how it "captures" the geometric intersection number between curves. It will continue with the description of the string bracket, which generalizes of the Goldman bracket to oriented manifolds of dimension larger than two, and the space of families of loops where the string bracket is defined. The second part of the lecture describes how this structure in degrees zero and one plus the power operations in degree zero recognizes key features of the Geometrization, the above mentioned joint work. The lions share of effort concerns the torus decomposition of three manifolds which carry mixed geometry. This is joint work with Siddhartha Gadgil and Dennis Sullivan.

**12:00 pm, 243 Altgeld, Tuesday, September 11, 2018
Eric Samperton (UCSB)
From the dynamics of surface automorphisms to the computational complexity of 3-manifolds**

Abstract: Every 3-manifold admits a Heegaard splitting, and many 3-manifold invariants admit formulas using Heegaard splittings. These facts are one starting point for a common theme in the study of 3-manifolds: one can relate various topological or geometric properties of 3-manifolds to dynamical systems in 1 or 2 dimensions. We’ll explore this theme in the context of computational complexity. I’ll start with two examples (coloring invariants and the Jones polynomial) that translate dynamical properties of mapping class group actions into complexity-theoretic hardness properties of 3-manifold invariants. I’ll conclude with some brainstorming about future directions. I will introduce all of the necessary complexity theory as we go.