Contributed talks

Schedule

Monday: Naomi Andrew, Alexandra Edletzberger, Sam Fisher, Sam Hughes, Amina Ladjali

Tuesday: Merlin Incerti-Medici, Lawk Mineh, Ismael Morales, Sangrok Oh, Eduardo Oregon Reyes

Wednesday: Harry Petyt, Jose Andres Rodriguez Migueles, Donggyun Seo

Thursday: Kate Vokes, Motiejus Valiunas, Gabriele Viaggi, Eduardo Silva, Stefanie Zbinden

Titles and abstracts

Naomi Andrew (University of Southampton / Oxford)

Free-by-cyclic groups, their automorphisms, and actions on trees

Free-by-cyclic groups are defined by an automorphism of F_n. Their properties (for example hyperbolicity, or relative hyperbolicity) depend only on this defining automorphism, but not always transparently. I will introduce these groups and some of their properties, and how these connect to properties of the defining automorphism. I'll then discuss joint work with Armando Martino where we seek to understand their automorphisms, by finding useful actions on trees, as well as related questions involving centralisers of elements of Out(F_n).

Alexandra Edletzberger (University of Vienna)

Quasi-Isometries for RACGs

In the hunt for a solution to the Quasi-Isometry Problem of right-angled Coxeter groups (RACGs), we use a quasi-isometry invariant that is obtained by a certain splitting of the groups, the so-called Graph of Cylinders. I will introduce this splitting, which for a large family of RACGs can be read off the presentation graph. We will see how we can use it to "visually" distinguish certain RACGs up to quasi-isometry.

Sam Fisher (University of Oxford)

Algebraic fibring of IMM manifolds

A group algebraically fibres if it maps onto $\mathbb Z$ with finitely generated kernel, and a topological space algebraically fibres if its fundamental group does. Recently, Italiano--Martelli--Migliorini produced examples of 5-,6-,7-, and 8-dimensional hyperbolic manifolds that algebraically fibre and were able to show that the 5-dimensional manifold fibres over the circle. Moreover, they showed that the 7- and 8-dimensional manifolds algebraically fibre with finitely presented kernel. The goal of this short talk will be to explain how to use connections between $\ell^2$-Betti numbers and fibring to show that a finite cover of the 7-dimensional IMM manifold algebraically fibres with finitely presented kernel of type $\mathtt{FP}(\mathbb Q)$.

Sam Hughes (University of Oxford)

Regularity of quasigeodesics characterises hyperbolicity

In this talk we will describe a characterisation of hyperbolic groups in terms of quasigeodesics and regular languages. Based on joint work with Patrick S. Nairne and Davide Spriano.

Merlin Incerti-Medici (Karlsruher Institut für Technologie)

Hyperbolic projections and topological invariance of Morse boundaries

The visual boundary of a hyperbolic space is a crucial tool in the study of hyperbolic groups. One of its basic features is its invariance under quasi-isometry, a property that is lost when considering visual boundaries of spaces with weaker negative curvature properties, e.g. CAT(0) spaces. In order to circumvent this issue and to define boundaries of non-hyperbolic groups, a series of boundaries called Morse boundaries have been introduced. Morse boundaries admit topologies that make them quasi-isometry invariant, however, these topologies are quite different from visual topology and quasi-isometry invariance of the visual topology on Morse boundaries has remained open. In this talk, we sketch a strategy to prove quasi-isometry invariance of visual topology and present a large class of groups to which it can be applied.

Amina Ladjali (University of Southampton)

Quadratic growth of rank 2 coarse intervals in coarse median spaces

Coarse median spaces (and groups) were introduced by Bowditch in 2013 and can be thought of as `coarsened’ versions of CAT(0) cube complexes. They provide a unified approach to looking at different spaces, such as geodesic hyperbolic spaces and mapping class groups, allowing us to view all these spaces and groups under one umbrella. In this talk, we will introduce and define coarse median spaces and give a few examples. We will also briefly discuss some current work in showing polynomial growth of intervals in these spaces.

Lawk Mineh (University of Southampton)

Separability and quasiconvexity in (relatively) hyperbolic groups

We explore the link between the algebraic property of separability and the geometric property of quasiconvexity as it pertains to subgroups of hyperbolic and relatively hyperbolic groups, and discuss a recent joint work with Minasyan in this area.

Ismael Morales (University of Oxford)

On the residual nilpotence of cyclic splittings

The purpose of the talk is introducing new techniques aimed at proving that amalgamated products and HNN extensions of residually nilpotent groups are, again, residually nilpotent. The main motivation is the study of parafree groups and the main application consists on describing when the fundamental group of a graph of groups with cyclic edge groups is parafree. This is joint work with Andrei Jaikin-Zapirain

Sangrok Oh (Kyungpook National University)

Quasi-isometric invariants of special square complexes.

A quasi-isometry between the universal covers of special square complexes preserves flats up to finite Hausdorff distance. From this fact we can define an intersection complex of the cover, which turns out to be a quasi-isometry invariant. In this talk, I will introduce an isometry between intersection complexes induced from the quasi-isometry and talk about its applications to 2-dimensional right-angled Artin groups and graph 2-braid groups.

Eduardo Oregon Reyes (University of California, Berkeley)

The space of metric structures on hyperbolic groups

I will talk about a generalization of Teichmüller/Outer space for an arbitrary hyperbolic group, where we consider the space of all left-invariant, hyperbolic metrics on the group that are quasi-isometric to the word metric, up to rough similarity. Endowed with the appropriate metric, this space is contractible, unbounded, and has a natural Out(G)-invariant geodesic bicombing. Based on joint work with Stephen Cantrell.

Harry Petyt (University of Bristol)

Rank-one isometries in CAT(0) spaces

CAT(0) spaces have a family of isometries than can naturally be thought of as hyperbolic-like, namely rank-one isometries. One theme in recent geometric group theory has been to try to use hyperbolic spaces to understand the "hyperbolic directions" of a group. In this talk, we shall discuss how, given a CAT(0) group G, one can construct a hyperbolic space X such that every rank-one element of G acts on X in a negatively curved way. Joint with Davide Spriano and Abdul Zalloum.

Jose Andres Rodriguez Migueles (LMU München)

On volumes of link complements and filling pairs of multicurves

Every oriented closed geodesic on a hyperbolic surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. Foulon and Hasselblatt showed that given a collection of closed geodesics that fill the surface, the associated link complement is a finite-volume hyperbolic 3-manifold. The objective of this talk is to estimate the volume of the link complement in terms of properties of the closed geodesics. For example, when the collection of geodesics is a filling pair of multicurves, I will show that the volume of the link complement is coarsely comparable to expressions involving distances in the pants graph.

Donggyun Seo (Seoul National University)

Intersection number and RAAG subgroups of MCGs

Many researchers have been studying how right-angled Artin groups (RAAGs) can embed into mapping class groups (MCGs). In this talk, we will see the reasons why they are finding RAAGs in MCGs and what I am working on. This is joint work with KyeongRo Kim.

Eduardo Silva (École Normale Supérieure Paris)

Dead ends on wreath products and lamplighter groups

Let G be a group and S a finite generating set. We say that (G,S) has unbounded depth if for any positive integer n, there is an element g which is at distance at least n from any element of word length |g|_S+1. In this talk we will cover some basic results about depth properties of groups, and then focus on wreath products A wr B. In this family of groups, the word metric has a combinatorial interpretation in terms of the solutions to the Traveling Salesperson Problem inside the Cayley graph of B, and thus we can apply combinatorial tools to study when G has unbounded depth, generalizing a classical result of Cleary and Taback.

Motiejus Valiunas (University of Wrocław)

Biautomaticity and non-positive curvature

In this talk I will give a brief introduction to the class of biautomatic groups and its relation to various classes of non-positively curved groups. Namely, I will advertise some methods one can use to show that certain CAT(0) groups are not subgroups of biautomatic groups, and that certain hierarchically hyperbolic groups are not biautomatic. Partially joint work with Sam Hughes.

Gabriele Viaggi (University of Heidelberg)

Properly convex domains divided by non-hyperbolic groups.

Properly convex domains in projective spaces are a rich source of geometry, dynamics, and group theory: Such objects are endowed with a natural Hilbert metric invariant under projective transformations that preserve the domain. Starting with the groundbreaking work of Benoist, a lot of effort has been made to understand the correspondence between the algebra and geometry of discrete groups of symmetries and the structure and regularity of the properly convex domain.

Of particular interest is the case where the domain is divisible, that is, there is a discrete group of isometries acting cocompactly on it. In this situation, by a theorem of Benoist, a divisible domain is strictly convex if and only if the group dividing it is hyperbolic. By contrast, in this talk, I will show that there are examples of properly convex domains divided by Zariski dense non-hyperbolic groups in any dimension D. These groups will be relatively hyperbolic with respect to a family of subgroups of the form Z times a lattice in SO(1,D-2).

This is joint work with Pierre-Louis Blayac.

Kate Vokes (University of Luxembourg)

Geometry of graphs of multicurves

We will present joint work with Jacob Russell classifying a wide family of graphs associated to surfaces according to whether the graph is Gromov hyperbolic, relatively hyperbolic or not relatively hyperbolic.

Stefanie Zbinden (Heriot-Watt University)

Morse boundary

The Morse boundary is a generalization of the Gromov boundary for non-hyperbolic spaces. I will introduce the Morse boundary and then present some extension results that allow the characterization of the Morse boundary arising from certain graph of groups.