Titles and Abstracts

Here you can find the titles and abstracts for the talks at the graduate workshop and the main conference. All talks will take place in Room 201 of the Physical Sciences Center (PHSC), located at 601 Elm Avenue.

To go back to the main conference page, click here.

Tara Brendle

Title: Abelian quotients of the Torelli group

Abstract: The mapping class group MCG(S) of a surface S is the group of symmetries of S, that is, the group of self-homeomorphisms of S up to isotopy. MCG(S) acts naturally on the first homology group of S, giving rise to a representation MCG(S) → Sp(2g,Z). The Torelli group of the surface S is the kernel of this representation, and is often described as the “nonlinear” or “mysterious” part of the mapping class group. A basic question about any group is: what is its abelianization, and what does that tell us about the group? In 1978, Birman-Craggs discovered the first abelian quotients of the Torelli group, giving a family of maps to Z/2 via the Rokhlin invariant of 4-manifolds. A few years later, Dennis Johnson discovered a new abelian quotient of the Torelli group onto a free abelian group. He further showed that this map, now known as the Johnson homomorphism, together with the Birman-Craggs maps, are sufficient to calculate the abelianization of the Torelli group. In this talk, we will explain both “pieces” of the abelianization: we will describe the Birman-Craggs maps and also give two ways to define the Johnson homomorphism, one in terms of certain 3-manifolds and one that is more algebraic in flavor. We will also briefly survey more recent work of Masatoshi Sato and of Tudur Lewis on the abelianization of a closely related group, the level 2 congruence subgroup of MCG(S), and describe how this sheds new light on the work of Johnson and Birman-Craggs.

Title: Semi-direct product structures in mapping class groups of 3-manifolds

Abstract: There is a natural map from the mapping class group of any manifold to the (outer) automorphism group of its fundamental group. In general, this map is neither injective nor surjective. In this talk, we will show that an associated short exact sequence splits, which gives rise to a semi-direct product structure on the mapping class group when the 3-manifold contains copies of S2 × S1. (This is joint work with Nathan Broaddus and Andrew Putman.)

Nathan Broaddus

Title: The Steinberg Module of the braid group

Abstract: The braid group on n strands is a Bieri-Eckmann duality group due to the fact that it is torsion free and has a finite index subgroup (the pure braid group) which is an iterated extension of free groups by free groups. I will talk about current joint work with Lindsey-Kay Lauderdale (Southern Illinois University), Emille Lawrence (University of San Francisco), Anisah Nu’Man (Spelman College) and Robin Wilson (Loyola Marymount University) in which we seek to improve upon the known presentations of the dualizing module of the braid group as a braid group module.

Lei Chen

Title: Mapping class groups of circle bundles over a surface

Abstract: In this talk, we study the algebraic structure of mapping class group Mod(M) of 3- manifolds M that fiber as a circle bundle over a surface S1 → M → Sg. We prove an exact sequence 1 → H1(Sg) → Mod(M) → Mod(Sg) → 1, relate this to the Birman exact sequence, and determine when this sequence splits. This is joint work with Bena Tshishiku.

Rita Jiménez Rolland

Title: Cohomology of mapping class groups: known and unknown

Abstract: A well-studied group that captures the symmetries of a topological surface is its mapping class group: the group of isotopy classes of the self-homeomorphisms of the surface. In this talk we will recall its definition and describe some of its properties for surfaces of finite type. Our goal is to present an overview of known results, and some open questions, about homology and cohomology of mapping class groups of orientable surfaces of finite type. If type permits, we will talk about our contributions to this topic.

Title: On normalizers and commensurators of abelian subgroups of mapping class groups

Abstract: Let Mod(S) be the mapping class group of a connected surface S of finite type with negative Euler characteristic. In joint work with León Álvarez and Sánchez Saldaña, we show that the commensurator of any infinite abelian subgroup of Mod(S) can be realized as the normalizer of a subgroup in the same commensuration class. As a consequence, we give an upper bound for the virtually abelian dimension of Mod(S). In this talk we will introduce the necessary definitions and explain how these results are obtained.

Annette Karrer

Title: From Stallings’ Theorem to connected components of Morse boundaries of graph of groups

Abstract: Every finitely generated group G has an associated topological space, called a Morse boundary. It was introduced by a combination of Cordes and Charney–Sultan and captures the hyperbolic-like behavior of G at infinity.

    In this talk, I will first explain Stallings’ theorem – a fundamental theorem in geometric group theory. Afterward, I will explain an analogous statement for Gromov boundaries of Gromov- hyperbolic groups. As Morse boundaries generalize Gromov boundaries, this raises the question whether it is possible to formulate an analog for Morse boundaries. Motivated by this question, we will study connected components of Morse boundaries of graph of groups. We will focus on the case where the edge groups are undistorted and do not contribute to the Morse boundary of the ambient group. Results presented are joint with Elia Fioravanti.

Rylee Lyman

Title: Deformation Spaces of Free Splittings and Coarse Lipschitz Retractions

Abstract: A free splitting of a group is a nontrivial action of that group on a tree with trivial edge stabilizers. Building on work of Forester and Culler–Vogtmann, Guirardel and Levitt define a deformation space of free splittings to which that tree action belongs. In ongoing joint work with Mosher, we provide a flexible construction of coarse Lipschitz retractions between (pointed) deformation spaces of free splittings and certain solvable Lie groups. As a corollary, we recover and extend work of Bridson–Vogtmann and Handel–Mosher on the Dehn function of outer automorphism groups of free groups.

Alan Reid

Title: Distinguishing 3-manifold groups by finite quotients

Abstract: By Perelman’s resolution of Thurston’s Geometrization Conjecture, the fundamental groups of compact 3-manifolds are residually finite and thereby have a rich supply of finite quotients. This talk will be a gentle discussion centered around the questions: To what extent are 3-manifold groups determined by their finite quotients and what properties of 3-manifolds are recognized by finite quotients of their fundamental groups.

Title: Profinite rigidity, direct products and finite presentability

Abstract: A finitely generated residually finite group G is called profinitely rigid, if for any other finitely generated residually finite group H, whenever the profinite completions of H and G are isomorphic, then H is isomorphic to G. In this talk we will discuss some recent work that constructs finitely presented groups that are profinitely rigid amongst finitely presented groups but not amongst finitely generated one.

Daniel Studenmund

Title: Hidden symmetries of groups via topology of solenoidal spaces

Abstract: “Hidden symmetries” of a group G are packaged up in its abstract commensurator, the group of isomorphisms between finite-index subgroups of G, modulo equivalence. When G is of type F, the abstract commensurator can be identified with the group of self-homotopy equivalences of the inverse limit of all finite covers of a K(G,1) complex, a so-called solenoidal space. We will explain these constructions and provide context in classical work of McCord, the use of shape theory, and related dynamical constructions of Sullivan and others. In the second part of the talk, we will apply these ideas to show that any countable union of finite groups can be realized as a group of hidden symmetries of the free group F2. This work is joint with Edgar A. Bering IV.

Ric Wade

Title: Aut-invariant quasimorphisms on groups

Abstract: For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. This has some pleasing applications to Aut-invariant norms on groups. Joint work with Francesco Fournier-Facio.