It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell.

We prove that negative alternating chainmail links are L-space links. Many of these can be shown to have asymmetric link complements. We also show that augmented chainmail links admit many L-space surgeries, showing that any 3-manifold is surgery on a generalized L-space link. This answers a question of Yajing Liu. We can show that some of these L-spaces have non-orderable fundamental group.

For a curve complex C of a finite type surface S we construct a tower of hyperbolic Mod(S)-complexes

We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures of a (relatively) acylindrical manifold. Using a toy model for the flow, we show that the flow has a global attracting fixed point at the structure M_geod the unique structure with totally geodesic convex core boundary. This is joint work with Kenneth Bromberg and Franco Vargas Pallete.

We consider properly embedded oriented surfaces of genus 1 in the 4-ball with boundary a knot K in the 3-sphere up to the action of the group of isotopies of the 4-ball. The study of (the equivalence classes of) those surfaces has a long history. We restrict to the subclass of Seifert surfaces: those surfaces that can be isotoped into the 3-sphere.

Motivated by a recent example of Hayden-Kim-Miller-Park-Sundberg of a pair of Seifert surfaces that can be distinguished by a classical algebraic topology invariant (namely, by the intersection form on the second homology of the double branched cover), we essentially characterize in what cases this invariant can be used to distinguish Seifert surfaces of genus 1 in the 4-ball.

The key algebraic player will be Gauss's group G_D: primitive integral binary quadratic forms with fixed non-zero discriminant D considered up to the canonical SL₂(Z)-action. For square-free D, G_D is studied in algebraic number theory under a different name: the ideal class group of the ring of integers of Q[√D]. While determining the ideal class group for a fixed D is hard in general, it turns out that the task of distinguishing Seifert surfaces amounts to understanding a much simpler-to-calculate subgroup.

Disclaimers: Based on joint work in progress with M. Akka, A. Miller, and A. Wieser. No knowledge of knot theory or number theory will be assumed. No ideals were harmed in preparation of this talk.

We construct an example of a hyperbolic 5-manifold fibering over the circle. As a consequence, we show that being hyperbolic is not a hereditary property of subgroups, even when one limits themselves in considering only subgroups with the strongest finiteness assumptions: there is a type F subgroup of a hyperbolic group that is not hyperbolic. The construction relies on Bestvina-Brady Morse theory, and it is joint work with B. Martelli and M. Migliorini.

Anosov representations can be considered a generalization of convex-cocompact representations to groups of higher-rank. In this talk we are considering connected components of Anosov representations from the fundamental group of a closed hyperbolic manifold N, and which contains Fuchsian representations, and their associated domains of discontinuity. We will prove that the quotient of these domains of discontinuity are always smooth fiber bundles over N. Determining the topology of the fiber is hard in general, but we are able to describe it for representations in Sp(4,C), and for the domain of discontinuity in the space of complex Lagrangians in C⁴ by using the classification of smooth 4-manifolds. This is joint work with Daniele Alessandrini, Nicolas Tholozan and Anna Wienhard.

Sigma invariants are geometric invariants associated to a finitely generated group that can be used to determine the homological and homotopical finiteness properties of coabelian subgroups. The Sigma invariants for right angled Artin groups have been computed by Meier, Meinert and VanWyck. In this talk we will talk about some results that allow one to compute a bit portion of the invariants for many Artin groups and sometimes to determine them completely. This is a joint work with Marcos Escartín.

Since their introduction by Thurston, pleated surfaces have been extensively deployed to study the geometry of hyperbolic 3-manifolds. In this talk, we present a generalization of this notion to study representations of the fundamental group G of a closed orientable surface of genus larger than 1 inside PSL(d,C). This allows us to construct holomorphic coordinates on suitable open subsets of the corresponding representation variety, generalizing Bonahon's shear-bend coordinates for classical pleated surfaces, and Bonahon-Dreyer's coordinates on the Hitchin component. This is joint work with Sara Maloni, Giuseppe Martone, and Tengren Zhang.

The space of measured laminations on a hyperbolic surface is a generalisation of the set of weighted multi curves. The action of the mapping class group on this space is an important tool in the study of the geometry of the surface. For orientable surfaces, orbit closures are now well-understood and were classified by Lindenstrauss and Mirzakhani. In particular, it is one of the pillars of Mirzakhani's curve counting theorems. For non-orientable surfaces, the behaviour of this action is very different and the classification fails. In this talk I will review some of these differences. I will talk about some of these results and classify mapping class group orbit closures of (projective) measured laminations for non-orientable surfaces. This is joint work with Erlandsson, Gendulphe and Souto.

Historically, closed exotic 4-manifolds are built using cut and paste constructions, and their gauge theoretic invariants are computed using gluing formulae. In this talk, I'll define some new smooth 4-manifold invariants which we can compute by hand using a (Dehn) surgery formula. Armed with this I'll build explicit closed exotic 4-manifolds out of elementary handle cobordisms. We'll see some new phenomena, like closed exotic definite 4-manifolds (with fundamental group Z/2), closed 4-manifolds with homologically essential square 0-spheres and nonvanishing invariants, and instances when knot surgery on an Alexander polynomial 1 knot can change the smooth structure. We'll try not to work too hard. This is joint work with Adam Levine and Tye Lidman.

The holonomy of a projective surface is a representation of its fundamental group in PSL(3,R). I plan to describe a neighborhood of the deformation space in PSL(4,R), which is singular, and apply it to deformations of projective 3-manifolds with boundary.

The theory of convex cocompact subgroups of the mapping class group contains two intertwining threads. One is the rich analogy between these subgroups of the mapping class group and the convex cocompact Kleinian group that inspired them. The other is work of Farb and Mosher plus Hamenstädt that shows convex cocompactness is precisely the property that characterizes when an extension of a surface group is Gromov hyperbolic. Both of these threads have natural generalizations that are unresolved. Among Kleinian groups, convex cocompactness is a special case of geometric finiteness, yet no robust notion of geometric finiteness has emerged for the mapping class group. In geometric group theory, there are a variety of generalizations of Gromov hyperbolicity, but there is no characterization of these geometries for surface group extensions. Mosher suggested these two threads should continue to intertwine with geometric finiteness in the mapping class group (however it is eventually defined) being equivalent to some generalization of Gromov hyperbolicity of the corresponding surface group extension. Inspired by their work on Veech groups, Dowdall, Durham, Leininger, and Sisto conjectured that this generalized hyperbolicity could be the hierarchical hyperbolicity of Behrstock, Hagen, and Sisto. We provide evidence for this conjecture by showing that several classes of subgroups that should be considered geometrically finite (stabilizers of multicurves, twist subgroups, cyclic subgroups) correspond to surface group extensions that are hierarchically hyperbolic.

Building on the ideas of Blanchet, Habegger, Masbaum and Vogel, Khovanov recently used their universal construction of topological theories in dimension two to construct interesting topological theories that are not TQFTs. In this talk we describe how the universal construction coupled with the interpretation of the partition category via 2-dimensional cobordisms by Comes, leads to multi-parameter monoidal generalizations of the partition and Deligne categories. We show that these generalizations correspond to rational functions in one variable which in turn determine finite rank universal theories in dimension two over a field by encoding values of the invariant on connected closed surfaces over all genera. This is joint work with Mikhail Khovanov.

Given a subgroup of a mapping class group, there is a corresponding extension group, and fundamental groups of surface bundles (with injective monodromy) are exactly these extension groups. Farb and Mosher introduced the notion of convex-cocompact subgroup of a mapping class group, and a subgroup is convex-cocompact if and only if the corresponding extension group is hyperbolic. Moving beyond this, one may wonder what happens for subgroups that are "close" to convex-cocompact. I will discuss a couple of cases, and in particular focus on the case of Veech groups. In that case the corresponding extension groups are hierarchically hyperbolic; I will explain what this means, what consequences this has, and speculate about notions of geometric finiteness in mapping class groups. Based on joint work with Spencer Dowdall, Matt Durham, and Chris Leininger.

For any right-angled Artin group A_G there is a contractible "outer space" O_G on which the outer automorphism group Out(A_G) acts properly. I will show how the combinatorics of the defining graph G can be used to find a fixed point for the action of a finite subgroup of Out(A_G). This is joint work with Corey Bregman and Ruth Charney.