Redbud Topology Conference
## Titles and Abstracts

I will review a "dictionary" between mapping class groups and Out(F_n), with an emphasis on the concepts important for Teichmuller theory, such as Teichmuller space, distance and geodesics, measured laminations, Nielsen-Thurston classification, curve complex, space of ending laminations.

I will discuss boundary amenability and how to prove it for basic groups for most of the hour. The main interest in boundary amenability is that it implies the Novikov conjecture in manifold theory. I will then outline the main ideas in the proof of boundary amenability of Out(F_n). This is joint work with Vincent Guirardel and Camille Horbez.

A hyperbolic n-manifold is said to bound geometrically if it is isometric to the boundary of a hyperbolic (n+1)-manifold with totally geodesic boundary. We might expect that most hyperbolic manifolds will not bound geometrically, and if they do then their volumes should be quite big. In this talk I will discuss how the number of arithmetic hyperbolic manifolds which bound geometrically grows with volume. This is joint work with Sasha Kolpakov.

Artin groups are generalizations of braid groups and they arises naturally from the studying of Coxeter groups and hyperplane arrangements. Despite the simple presentation of Artin groups, it is easy to ask a basic question which is not known for all Artin groups. One conjectural approach to understand Artin groups is to put non-positively curvature like structure on such groups. I will talk about several ways to metrize certain classes of Artin groups, some are classical and some are more recent. Mostly I will focus on 2-dimensional Artin groups, and will switch to higher dimensional case when time allows. Part of the talk is based on joint work with D. Osajda.

We call a finitely generated group strongly rigid if any self quasi-isometry of the group is uniformly close to an automorphism of the group. We show many 2-dimensional Artin groups are strongly rigid. This is based on the understanding of certain tilings of planes arising from these Artin groups. This is joint work with D. Osajda.

The cubical dimension of a group G is the infimum n such that G admits a proper action on an n-dimensional CAT(0) cube complex. I will discuss a construction of finitely presented small cancellation groups with arbitrarily large cubical dimension (while their geometric dimension and CAT(0) dimension are always two). The construction also yields examples (joint work with D.Wise) of finitely presented groups of cohomological dimension two that act properly on a locally finite CAT(0) cube complex, but whose cubical dimension is infinite. I will also discuss the connection of the construction to the question of uniform exponential growth of cubulated groups.

Church and Farb proved that the homology of the pure braid group satisfies a property that they called representation stability. With Kevin Kordek, we study a subgroup of the pure braid group called the level 4 braid group. By a result with Tara Brendle, this group is equal to the subgroup generated by all squares of pure braids. Kordek and I compute the first Betti number of the level 4 braid group and show that the first homology satisfies representation stability. I will explain our result and also explain various applications to the hyperelliptic Torelli group, the level 4 mapping class group, and the characteristic variety of the pure braid group.

I will give a broad survey concerning the cohomology of the mapping class group, especially its stable cohomology.

I will prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as the ordinary moduli space: a polynomial ring in the Miller-Morita-Mumford classes.

Croke and Kleiner showed that visual boundary of CAT(0) groups such as right-angled Artin groups(RAAG) is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function , we consider a subset of the visual boundary called sublinear boundary and show that it is a QI-invariant. This is to say, the sublinear-boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin group, we show that the Poisson boundary is naturally identified with the (log t)-boundary. This talk is based on projects with Kasra Rafi and Giulio Tiozzo.

In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties. The topology of such a bundle is completely encoded by its monodromy representation (a homomorphism to a mapping class group), and it is a fundamental problem to understand precisely how the topology of the bundle is reflected in algebraic properties of the monodromy. The main result of this talk is that the Atiyah–Kodaira bundles have arithmetic monodromy groups. A corollary of this result is that Atiyah–Kodaira bundles fiber in exactly two ways. This is joint work with Nick Salter.

I will discuss several instances of hyperbolicity phenomena in outer automorphisms of free groups and free group extensions. In particular, I will illustrate by examples how the dynamics and the geometry of the Out(F_N) action reflects on the algebraic structure of Out(F_N) itself. Moreover, I will talk about a new subgroup classification theorem for Out(F_N) which is joint work with Matt Clay.