Title: The hyperelliptic Torelli group: a survey (I & II)

Abstract: The hyperelliptic mapping class group and the Torelli group have long been known to be important in many areas of group theory and low-dimensional topology. These two subgroups of the mapping class group are seemingly orthogonal in nature: the Torelli group consists of precisely those elements acting by the identity (Id) on homology, whereas a hyperelliptic involution of a surface is characterized by action by -Id on homology, and hyperelliptic mapping classes commute with this involution. However, the hyperelliptic Torelli group (the intersection of these two groups) turns out to be of great interest in its own right, with deep connections to algebraic geometry and number theory as well.

In these talks we will survey basic facts about the hyperelliptic Torelli group, including combinatorial and cohomological properties, as well as connections with classical objects such as the Burau representation of the braid group and Brunnian braids. Topics we will cover include:

• the resolution of a conjecture of Hain, giving a nice generating set for the group, and implications for the topology of moduli space;
• algorithmic factorizations with applications to an old question of Dennis Johnson;
• connections to palindromic automorphisms of free groups; and
• a symplectic representation of the braid group, and corresponding connections between Brunnian braids and congruence subgroups.

Title: Automorphisms of the free factor complex

Abstract: I shall describe recent work proving that the natural map from Aut(F) to the group of simplicial automorphisms of the free-factor complex is an isomorphism. The corresponding theorem for Out(F) is also proved. If time allows, I'll discuss applications. This is joint work with Mladen Bestvina.

Title: Projections, quasi-trees and the mapping class group

Abstract: Geodesics have in hyperbolic space have the property that the nearest point projection from a ball, disjoint from the geodesic, to the geodesic will have uniformly bounded diameter. In general, geodesics with this property are "strongly contracting". Given a collection of strongly contracting geodesics in a metric space we will build a quasi-tree from the geodesics where the projections in the quasi-tree agree with the projections in the original metric. We will use this construction to embed the mapping class group in a product of trees. This is joint work with M. Bestvina and K. Fujiwara.

Title: Train track splittings of free-by-cyclic groups

Abstract: The semidirect product of a free group with the integers can often be expressed as such in infinitely many ways with different base free groups and defining monodromy automorphisms. This talk will explore this family of splittings and describe ways in which the monodromy automorphisms are related to each other. Some properties we'll consider are full irreducibility and the existence of irreducible train track representatives. Joint work with Ilya Kapovich and Chris Leininger.

Title: The boundary of the free splitting complex

Abstract: I will talk about ongoing work with Mladen Bestvina and Patrick Reynolds using fold paths in Culler-Vogtmann’s Outer space as a tool to explore/describe the boundary of the free splitting complex.

Title: Boundaries revisited

Abstract: We give a description of the boundary of the free splitting complex which is very similar to the well known description of the boundary of the curve graph. We use this to describe subgroups of Out(Fn) which quasi-isometrically embed into the free splitting complex

Title: Homology of covers of finite graphs

Abstract: In the 1930s Chevalley and Weil gave a formula to describe the homology of a finite regular cover of a Riemann surface as a representation of the deck group. A completely analogous formula exists for covers of finite graphs. In this talk we will describe the beginnings of a dictionary between topological and representation-theoretic properties of the homology of a finite cover of a graph. This point of view allows on the one hand to answer natural, basic questions on finite covers of graphs (including actions of Out(F_n)) and on the other hand suggests a rich class of unsolved problems. This is joint work with Benson Farb.

Title: The Tits alternative for the automorphism group of a free product

Abstract: A group \(G\) is said to satisfy the Tits alternative if every subgroup of \(G\) either contains a nonabelian free subgroup, or is virtually solvable. I will present a version of this alternative for the automorphism group of a free product of groups. A classical theorem of Grushko states that every finitely generated group \(G\) splits as a free product of the form \( G_1* \ldots *G_k*F_N \), where \(F_N\) is a finitely generated free group, and all groups \(G_i\) are nontrivial, non isomorphic to Z, and freely indecomposable. In this situation, I prove that if all groups \(G_i\) and Out\((G_i)\) satisfy the Tits alternative, then so does Out\((G)\). I will give some applications, and present a proof of this theorem, in parallel to a new proof of the Tits alternative for mapping class groups of compact surfaces. The proof relies on the study of the actions of some subgroups of Out\((G)\) on a version of the outer space, and on a hyperbolic simplicial graph.

Title: The primitivity index function for a free group, and untangling closed geodesics on hyperbolic surfaces.

Abstract: Motivated by the results of Scott and Patel about lifting closed geodesics to simple closed geodesics in finite covers of hyperbolic surfaces, we consider several related index functions for free groups.
Given a nontrivial element \(g\in F_N=F(A)\), we define the "primitivity index" \(d(g)\) to be the smallest index of a subgroup of \(F_N\) containing \(g\) as a primitive element, define the "simplicity index" \(d_s(g)\) to be the smallest index of a subgroup of \(F\) containing \(g\) as an element belonging to a proper free factor of this subgroup, and define the "filling index" \(d_f(g)\) to be the smallest index of a subgroup \(H\) of \(F\) such that \( g\in H\) and that $g$ is "non-fillng" in \(H\), that is, such that $g$ is elliptic for some very small splitting of \(H\). It is not hard to see that we always have \(d_f(g)\le d_s(g)\le d(g)\le ||g||_A\), but lower bounds are much harder to obtain.
We show that there exists \(c=c(N)>0\) such that for the element \(w_n\in F_N\) obtained by a simple non backtracking random walk of length \(n\) with respect to \(A\), with probability tending to \(1\) as \(n\t\infty\) one has \(d_s(w_n)\ge c \log^{1/3} n\) and \(d_f(w_n)\ge c \log^{1/5} n\).
We also discuss applications of these results to the original setting of Scott and Patel of untangling closed geodesics on hyperbolic surfaces.
This talk is based on joint work with Neha Gupta.

Title: Convex cocompactness in right-angled Artin groups

Abstract: I will talk about joint work with Thomas Koberda and Sam Taylor on a version for right-angled Artin groups of convex cocompactness for subgroups of mapping class groups. The latter were defined by Farb and Mosher in partial analogy to convex cocompact hyperbolic manifold groups, and there are interesting connections between the three settings.

Title: Combinatorial models for mapping class groups

Abstract: A celebrated theorem of Nikolai Ivanov states that the automorphism group of the complex of curves is isomorphic to the mapping class group. This theorem has applications in the theories of Teichmuller space and mapping class groups. After many similar theorems were proven, Ivanov made a metaconjecture that any "sufficiently rich" combinatorial complex associated to a surface should have automorphism group isomorphic to the mapping class group. In joint work with Tara Brendle, we study complexes whose vertices correspond to connected subsurfaces. We give necessary and sufficient conditions for such complexes to have automorphism group the mapping class group and give group-theoretic applications to the mapping class group.

Title: Hyperbolic actions and second bounded cohomology for subgroups of Out\( (F_n) \)

Abstract: Motivated by a general desire to understand the large scale geometry of Out\( (F_n) \), and a specific desire to understand second bounded cohomology of subgroups of Out\( (F_n) \), we shall study various actions of subgroups of Out\((F_n)\) on hyperbolic complexes, and the dynamics of individual elements of those actions. As an application, we shall prove that finitely generated subgroups \( H < \mbox{Out}(F_n)\) satisfy an alternative: either \( H \) is virtually abelian; or the second bounded cohomology of \( H \) with real coefficients is of uncountable dimension. This work is joint with Michael Handel.

Title: The high-dimensional cohomology of the mapping class group

Abstract: I will discuss the high-dimensional cohomology of both the mapping class group and its linear congruence subgroups. Parts of this are joint work with Tom Church and Benson Farb, and other parts are joint work in progress with Neil Fullarton.

Title: Teichmüller space is rigid

Abstract: We study the large scale geometry of Teichmüller space equipped with the Teichmüller metric. We show that, except for low complexity cases, any self quasi-isometry of Teichmüller space is a bounded distance away from an isometry of Teichmüller space. This is joint work with Alex Eskin and Howard Masur.

Title: End invariants of splitting sequences

Abstract: Thurston introduced train tracks and geodesic laminations as tools to study surface diffeomorphisms and Kleinian groups. We'll start the talk with a relaxed introduction to these. Then, in analogy with the end invariants of Kleinian groups and Teichmüller geodesics, we will define the end invariants of an infinite splitting sequence of train tracks. These end invariants determine the set of laminations that are carried by all tracks in the infinite splitting sequence. If there is time, we'll use these ideas to sketch a new proof of Klarreich's theorem, determining the boundary of the curve complex.

Title: Growth Tight Actions

Abstract: Let G be a group equipped with a finite generating set S. G is called growth tight if its exponential growth rate relative to S is strictly greater than that of every quotient G/N with N infinite. This notion was first introduced by Grigorchuk and de la Harpe. Examples of groups that are growth tight include free groups relative to bases and, more generally, hyperbolic groups relative to any generating set. In this talk, I will provide some sufficient conditions for growth tightness which encompass all previous known examples.

Title: Hyperbolic extensions of free groups

Abstract: Every subgroup \(G\) of the outer automorphism group of a finite-rank free group \(F\) naturally determines a free group extension \(1\to F \to E_G \to G\to 1\). In this talk, I will discuss joint work with Spencer Dowdall which gives geometric conditions on the subgroup \(G\) that imply its corresponding extension \(E_G\) is hyperbolic. These conditions are in terms of the action of \(G\) on the free factor complex of \(F\) and allow one to easily build new examples of hyperbolic free group extensions. Besides explaining some geometric properties of these hyperbolic extensions, I will also discuss joint work with Giulio Tiozzo which implies that such extensions are (in an appropriate sense) generic.

Title: Cycles in moduli spaces of graphs

Abstract: The rational homology of Out\((F_n)\) and Aut\((F_n)\) is known to vanish in high and low dimensions, but Euler characteristic computations indicate that there is actually a huge amount of homology somewhere. Using work of Kontsevich, Morita found an infinite series of cycles, the first three of which have been shown to give non-trivial homology classes. We use representation theory and "assembly maps" to produce many more cycles, including the only two other known non-trivial homology classes. We also give a new geometric interpretation of some of these cycles, including Morita's original series, as maps of closed manifolds into the moduli space of graphs.

Title: Splittings of free groups via systems of surfaces

Abstract: There is a pleasing correspondence between splittings of a free group over finitely generated subgroups and systems of surfaces in a doubled handlebody. One can use this to describe a family of hyperbolic complexes on which Out\((F_n)\) acts. This is joint work with Camille Horbez.