Scientific Program

Tentative Schedule:

Minicourse Abstracts:

Harry Petyt: Curtains and wallspaces

With the successes of CAT(0) cube complexes in geometric group theory, it is reasonable to try to export some of the tools used in studying them to other settings. One way of doing this is by trying to use the combinatorics of 'walls'. The goal of this minicourse is to outline some of the main ideas involved in recent works with Abdul Zalloum and Davide Spriano that centre around using walls to explore nonpositive curvature.

Eduardo Reyes: Deformation spaces of isometric group actions

Geometric group theory studies groups through their isometric actions. It turns out that many groups with non-positively curved behavior admit infinitely many (qualitatively distinct) isometric actions. On the other hand, many classical geometric structures can be encoded as isometric actions, including Teichmüller spaces, Outer spaces, Hitchin components, and random walks, among others. The goal of this minicourse is to give an introduction to deformation spaces of isometric actions. We plan to define some of these spaces, provide tools for their study, and survey some applications and recent results. We will focus on two aspects of these spaces: Manhattan curves/geodesics, and spaces of cubulations for cubulated hyperbolic groups.

Abstracts:

Penelope Azuelos: A guide to constructing free transitive actions on median spaces

Median spaces are an increasingly important class of spaces which includes both CAT(0) cube complexes and real trees. Finitely generated groups which admit free transitive (or proper cocompact) actions on discrete median spaces (i.e. the 0-skeletons of CAT(0) cube complexes) are an active area of study. However not much is known about their continuous counterpart: groups which act freely and transitively on connected median spaces. I will present several methods of constructing such actions, focusing on actions on real trees and their products, and discuss some of the surprising behaviours that show up. Even when considering real trees, the class of groups acting on such spaces is vastly more diverse than in the discrete setting: while any simplicial tree admits at most one free vertex transitive action, we will see that there are $2^{2^{\aleph_0}}$ pairwise non-isomorphic groups which admit a free transitive action on the universal real tree with continuous valence.

Inhyeok Choi: Metric WPD for Homeo(S)

Recently, Bowden-Hensel-Webb introduced the notion of fine curve graph as an analogue of the classical curve graph. They used this to construct nontrivial quasi-morphisms (in fact, infinitely many independent ones) on $\mathrm{Homeo}_0(S)$. Their method crucially uses independent pseudo-Anosov conjugacy classes, whose existence follows from the WPD-ness of pseudo-Anosov mapping classes on the curve graph. In this talk, I will explain an ongoing work regarding an analogue of WPD-ness for point-pushing pseudo-Anosov maps on the fine curve graph. If time allows, I will explain why this is related to constructing new quasi-morphisms on Homeo(S).

Koji Fujiwara: Growth rates in families of groups of negative curvature

Let e(G,S) be the exponential growth rate of a finite generated group G with a finite generating set S. Let $\xi(G)$ be the set $\{e(G,S)\}$ for all finite generating sets S of G. If G is a non-elementary hyperbolic group, then $\xi(G)$ is well-ordered. I will also discuss the set of growth rates in a family of groups of negative curvature.

Radhika Gupta: Conformal dimension of Bowditch boundary of certain Coxeter groups

Quasi-isometry (QI) classification of finitely generated groups is an important problem in geometric group theory. When two Gromov hyperbolic groups are quasi-isometric, then they have homeomorphic visual boundaries. But the converse is not true. One tool to distinguish two hyperbolic groups with the same visual boundary is to show that the conformal dimension, which is an analytic QI invariant, of the two boundaries are different. In this talk, we consider the family of Coxeter groups with defining graph a complete graph on at least 5 vertices and edge labels at least three. Such groups are hyperbolic relative to free abelian subgroups (CAT(0) with isolated flats). We give the first computation of bounds for the conformal dimension of the Bowditch boundary of a non-hyperbolic group. As a corollary, we show that there are infinitely many QI classes of groups in this family of Coxeter groups (which all have the same visual boundary by the work of Haulmark-Hruska-Sathaye). In the process, we also exhibit a geometrically finite action on a CAT(-1) space for the non-hyperbolic Coxeter groups in our family. This is joint work with Elizabeth Field, Robbie Lyman and Emily Stark.

Mark Hagen: Hierarchical hyperbolicity among free-by-Z groups

Let G be the mapping torus of an automorphism of a finite-rank free group F. Brinkmann characterised hyperbolicity of G in terms of the dynamics of the automorphism, and more recent results by Dahmani-Li and Ghosh show that G admits a (possibly trivial) relatively hyperbolic structure where the peripheral subgroups are mapping tori of polynomial-growth automorphisms. The latter can be studied using their virtual splittings: Macura produced a cyclic virtual hierarchy terminating in mapping tori of linear-growth automorphisms, and results of Andrew-Martino and Dahmani-Touikan provide nice splittings of the latter along $\mathbf{Z}^2$ subgroups. This sets the stage for asking about the coarse median, cubical, or HHG structures that are possible on G. In the hyperbolic case, G is cocompactly cubulated, by Hagen-Wise, but an observation of Gersten shows that this cannot hold in general. Very recently, Munro-Petyt have provided powerful general obstructions to a group even having a coarse median, and shown that these apply for many free-by-cyclic groups G. Building on their work, all of the aforementioned structural results about G, and combination theorems for hierarchical hyperbolicity, we show that G is coarse median if and only if it is actually quasi-isometric to a cube complex, and that this happens if and only if G is virtually hierarchically hyperbolic. We also characterise such G algebraically as well as in terms of relative train track representatives of the automorphism. This talk is on forthcoming joint work with Eliot Bongiovanni, Pritam Ghosh, and Funda Gültepe.

Yulan Qing: Sublinearly directions under first passage percolation

In this talk we will discuss the behavior of sublinearly Morse directions under first passage percolation. We first recall the construction of the sublinearly Morse boundary, which is a metrizable and QI-invariant topological space that enjoys some genericity properties. First passage percolation is a model of random perturbation of a given geometry. Assuming only strict positivity and finite expectation of the random lengths, we prove that if an infinite graph has bounded degree, then almost surely, all sublinearly Morse directions are preserved by first passage percolation on the graph. This is a work in progress joint with Sagnik Jana.

Kasra Rafi: Automatic continuity for big mapping class groups

We provide a complete classification of when the homeomorphism group of a stable surface, Σ, has the automatic continuity property: Any homomorphism from Homeo(Σ) to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of Σ has the automatic continuity property.

Michah Sageev: Topological median algebra structures on euclidean spaces

We will discuss an initial exploration into a characterization of topological median algebra structures on euclidean spaces in terms what we term “web structures”. This characterization works for ER homology manifolds as well. This is joint work with Bestvina and Bromberg.

Alessandro Sisto: Construction and uses of asymptotically CAT(0) metrics

After defining asymptotically CAT(0) metrics, I will discuss how to construct them on mapping class groups and more generally on colourable hierarchically hyperbolic groups. I will also briefly discuss a notion of boundary for asymptotically CAT(0) spaces and some applications. Based on joint work with Matt Durham and Yair Minsky.

Giulio Tiozzo: The Poisson-Furstenberg boundary of discrete subgroups of semisimple Lie groups without moment conditions

The Poisson(-Furstenberg) boundary is a measure-theoretic object attached to a group equipped with a probability measure, and is closely related to the notion of harmonic function on the group. In many cases, the group is also endowed with a topological boundary arising from its geometric structure, and a recurring research theme is to identify the Poisson boundary with the topological boundary. In this talk, we prove that the Poisson boundary of a random walk with finite entropy on a discrete subgroup of a semisimple Lie group can be identified with its Furstenberg boundary, without assuming any moment condition on the measure. Note that no pivoting theory will be needed. Joint with K. Chawla, B. Forghani, and J. Frisch.

Renxing Wan: Proper actions on finite products of hyperbolic spaces

A group $G$ is said to have property (PH') if there exist finitely many hyperbolic spaces $X_1,\cdots,X_n$ on which $G$ acts coboundedly such that the diagonal action of $G$ on the product $\prod_{i=1}^nX_i$ equipped with $\ell^1$-metric is proper. A group $G$ has property (PH) if it virtually has property (PH'). This notion is a generalization of both word-hyperbolicity and property (QT) introduced by Bestvina-Bromberg-Fujiwara. In this talk, I will introduce some recent progress on groups with property (PH). In particular, by considering a central extension of groups $1\to Z\to E\to G\to 1$, we prove that $E$ has property (PH') (resp. (PH) or (QT)) if and only if $G$ has property (PH') (resp. (PH) or (QT)) and the Euler class of the extension is bounded. As corollaries, we obtain more interesting examples of groups with property (QT) including the central extension of residually finite hyperbolic groups, the mapping class group of any finite-type surface and the outer automorphism group of torsion-free one-ended hyperbolic groups. This talk is based on the joint work with Bingxue Tao.

Wenyuan Yang: Hausdorff Dimension of non-conical and Myrberg limit sets

In this talk, we study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set for Fuchsian groups, Kleinian groups and groups actings on trees. We also show that the Hausdorff dimension of the Myrberg limit set is the same as the critical exponent confirming a conjecture of Falk-Matsuzaki. This is based on a joint work with Mahan Mj.