Fall Redbud Topology Conference

The conference takes place in Nielsen Hall room 251 on the University of Oklahoma campus. Abstracts are below.

Abstracts:

Matt Day, University of Arkansas

Title: Automorphism group orbits of sets of conjugacy classes in right-angled Artin groups

Abstract: A finitely presented group whose only relations are that some pairs of generators commute is a right-angled Artin group (RAAG). Automorphism groups of RAAGs include automorphism groups of free groups, integer general linear groups, and many other groups. In this talk, I will present an algorithm to check whether two sets of conjugacy classes in a RAAG are in the same orbit under the action of its automorphism group, and I will discuss a recent theorem of mine that proves the correctness of this algorithm. This work also implies that the stabilizer of a set of conjugacy classes in a RAAG is always a finitely presented subgroup of the automorphism group. This work generalizes the peak-reduction method from free groups and also involves row reduction of matrices in an interesting way.

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Funda Gultepe, University of Oklahoma

Title: Essential tori and Dehn twists in Outer Space

Abstract: Fully irreducible automorphisms of the outer automorphism group Out(F_n) of the free group of rank n are particularly important in understanding the geometry of Out(F_n) and their construction is important as well. Fully irreducibles are considered to be the closest analogs of pseudo Anosov elements of the mapping class group. Following a procedure similar to Thurston's construction of Pseudo Anosov elements for a surface with one puncture, we will construct fully irreducible elements by taking a 3-manifold as a model space for Out(F_n) and twisting essential tori in this 3-manifold.

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Dale Rolfsen, University of British Columbia

Title: Ordering knot groups

Abstract: The group of a knot (in 3-space) is the fundamental group of the knot's complement. A group is left-orderable if its elements can be given a strict total ordering, < , which is left-invariant: g < h implies fg < fh. I will explain why all knot groups are left-orderable, and some have two-sided invariant orderings, while some others do not. I will also try to explain why this is interesting, including some connections with Heegaard-Floer homology.

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Jeremy Van Horn-Morris, University of Arkansas

Title: Arbitrarily long positive factorizations in surface mapping class groups

Abstract: Lefschetz intruduced the notion of a pencil as a topological/combinatorial object in order to study algebraic varieties. Through work of Donaldson and Gompf, it has become a central tool in the study of more general symplectic 4-manifolds. Its 3-dimensional analogue is the open book decomposition which due to Giroux has become a central tool in the study of contact 3-manifolds. In each case, the question of factorizations of a surface mapping class as a product of positive Dehn twists is at the core of the relationship. We will present the first examples of mapping classes which at admit arbitrarily long positive factorizations, answering questions of Smith, and Ozbagci and Stipsicz. This has the unpleasant corollary that there is no bound on the complexity of a symplectic 4-manifold coming from the genus of its supporting Lefschetz pencil and similarly, that there is no bound on the complexity of a symplectic filling of a contact manifold coming from the genus of its supporting open book. This is joint work with I. Baykur.

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Updated October 26th, 2012.