Abstracts
29-04-24 - Nicolas Vaskou (University of Bristol)
Title: Artin groups and quasi-isometric rigidity
Abstract: Artin groups form a large family of groups that includes free groups, free abelian groups, braids groups, and much more. In many ways, they can be seen as "cousins" of Coxeter groups. Despite having a relatively easy presentation, most conjectures about Artin groups are yet to be solved. In this talk, I will introduce Artin groups and I will give several reasons why one might want to care about them. I will introduce some of their general properties (algebraic, topological, geometric, etc.), and I will explain how one can exploit their geometry to recover various results about them. Lastly, to exhibit an application of this I will talk about an ongoing work with Huang and Osajda, in which we show that a large family of Artin groups has strong quasi-isometric rigidity properties.
22-04-24 - Sarah Martin (University of Bristol)
Title: Vertex-Transitive Graphs of Polynomial Growth
Abstract: A graph G=(V,E) has polynomial growth if there exist c, n > 0 such that for all r > 0 the ball of radius r in G has size bounded above by cr^n. If G is a Cayley graph of a group Γ, then it is known that G has polynomial growth if and only if Γ is virtually nilpotent, by a famous theorem of Gromov. In this talk I will outline some properties of (infinite) groups of polynomial growth and discuss whether these can be generalised to vertex-transitive graphs. If time allows, I will also discuss what information we can extract from these results for finite groups/graphs (since these trivially have polynomial growth).
15-04-24 - Kaitlin Ragosta (Brandeis University)
Title: Smallest non-cyclic quotients of braid groups
Abstract: The braid group on n strands has a quotient isomorphic to the symmetric group on n elements, induced by a natural projection map. Margalit conjectured that this symmetric group is the smallest non-cyclic quotient of a braid group, and his conjecture was proved in recent work of Kolay. In this talk, I will give an overview of Kolay’s proof and briefly discuss some work in progress (joint with Sam Hughes, Thomas Ng, Nancy Scherich, and Yvon Verberne) which generalizes these techniques from braid groups to other finite-type Artin groups.
11-03-24 - Ana Isakovic (University of Cambridge)
Title: Incoherence of trivalent triangle groups
Abstract: A triangle group can be seen as a symmetry group of a triangular tiling of a model space. Similarly, generalized triangle groups can be seen as symmetry groups of spaces with more branching. In this talk, we will focus on the case of hyperbolic trivalent triangle groups and, keeping this example in mind, define and discuss coherence and virtual fibring of groups.
Title: How many times can two curves on a surface intersect?
Abstract: Given a surface, one might want to understand how "complex" the surface is, in terms of curves.
More specifically, we may ask how many times two curves on this surface can intersect.
Of course, longer curves might intersect more times.
KVol is a quantity measuring how many times curves can intersect, modulo their length.
We will give an overview of some cases for which this quantity has been calculated,
with particular focus on Veech surfaces, a class of flat surfaces with a rich group of symmetries.
Title: The Real Locus of the Shape Variety
Abstract: A triangulation of a three-manifold is a collection of tetrahedra and face gluings.
We can look at how these gluings interact with the edges of the tetrahedra and the triangulation to acquire a series of equations, the intersection of which we call the shape variety.
The real locus of the shape variety coincides with when all these tetrahedra are flat.
We will use the figure-eight knot complement as an example throughout, drawing pictures of its triangulation.
We will define what a veering triangulation is and give an outline of the argument as to why veering triangulations do not lie on the real locus of the shape variety.
Title: The regularity number of a finite group
Abstract: Let $G$ be a transitive permutation group on a finite set $\Omega$ with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser in $G$ is trivial. The minimal size of a base is called the base size of $G$, denoted b(G), and this classical invariant has been widely studied for many years, finding many applications.
Observe that $b(G)$ coincides with the smallest positive integer $k$ such that $G$ has a regular orbit on the Cartesian product $(G/H)^k$. As a natural generalisation, we can consider the minimal number $r$ such that $G$ has a regular orbit on $(G/H_1) \times \cdots \times (G/H_r)$ for any core-free subgroups $H_1, \ldots, H_r$ of $G$. We call this invariant the regularity number of $G$.
In this talk, we will explain how a combination of probabilistic, combinatorial and computational methods can be used to study the regularity number of almost simple groups, as well as several other related problems. In particular, we will report on some recent progress towards natural generalisations of well studied base size conjectures of Cameron-Kantor and Vdovin. This is joint work with Tim Burness.
12-02-24 - Merlin Haith Rowlatt (University of Bristol)
Title: Growth and Harmonic Functions on Vertex-Transitive Graphs
Abstract: In 2016, Tointon proved that given an infinite group G with finite generating set S, the space of harmonic functions on the corresponding Cayley graph is finite dimensional if and only if G contains a finite-index subgroup isomorphic to Z.
Since every Cayley graph is vertex-transitive, and G containing a finite index subgroup isomorphic to Z is equivalent to G having linear growth, it makes sense to consider generalising this to vertex-transitive graphs. I will discuss some recent work I have made towards proving such a generalisation.
05-02-24 - Lewis Molyneux (Royal Holloway)
Title: An introduction to BNSR Invariants
Abstract: Finiteness properties are a frequent topic of interest among discrete groups. BNSR invariants provide an extension to the study of finiteness properties that, along with allowing us to distinguish between groups with similar finiteness properties, provides deep insight into the finiteness properties of subgroups. In this talk, I shall provide an introduction to the concepts surrounding BNSR invariants, provide an example of calculation, and explain their application in understanding finiteness properties of subgroups. I shall also introduce my own research linking the BNSR invariant of a group with that of its finite index subgroup.
29-01-24 - Sophie Wright (University of Bristol)
Title: Mapping graphs into hyperbolic surfaces and counting subgroups of Fuchsian groups
Abstract: A realisation is a continuous map from a finite graph into a hyperbolic surface with edges mapped to geodesic arcs. The sum of the lengths of these arcs gives us a natural notion of the length of a realisation. We are able to count the number of critical realisations of length less than some L and understand the asymptotic behaviour as L tends to infinity.
A realisation induces a homomorphism from the fundamental group of the graph to the fundamental group of the surface. I am interested in counting subgroups of Fuchsian groups by studying these induced homomorphisms. The aim of this talk is to introduce the basics of realisations and how to count them, beginning with a quick refresher of hyperbolic surfaces and Fuchsian groups.
22-01-24 - Tom Adams (University of Cambridge)
Title: From the Bruhat-Tits Tree to the Representation Theory of GL2(F)
Abstract: I will define a non-Archimedean field F and the Bruhat-Tits tree of GL2(F). I will then explain how, in the language of Drinfeld, this tree can be viewed as a homogeneous non-Archimedean analogue of the complex upper half plane and why we might be interested in studying it.
Title: Spectral gaps for scl and negative curvature of groups
Abstract: Stable commutator length is a measure of homological complexity of group elements. The goal of this talk is to explain some of its connections with negative curvature. We will discuss spectral gaps, and present a geometric proof of the celebrated result of Duncan and Howie that gives a sharp spectral gap in free groups. Along the way, we will encounter angle structures on 2-dimensional cell complexes and a combinatorial Gauß–Bonnet formula.
04-12-23 - Yago Moreno Alonso (University of Bristol)
Title: The Farfalle Mystery
Abstract: In a public lecture in 2017, Mickaël Launay described a remarkable property of the Fibonacci sequence: when the Fibonacci numbers are put on the circle modulo N, and segments joining consecutive elements of the sequence are drawn, a “farfalle” shape appears (a popular pasta shape in France). Launay proposed a prize of 3.14euros for they whom explain “le mystère de la farfalle”. In this talk, we describe how this problem can be understood using simple ideas from ergodic theory, and we will discuss one (time permitting, two) explanations of this phenomenon, while keeping things low-tech. This is joint work with Serge Cantat and François Maucourant.
Title: Surgery and its Effect on Hyperbolic 3-Manifolds
Abstract: Surgery is a method to modify a 3-manifold by removing a solid torus and gluing it back in a different way. This innocuous-looking procedure is shockingly powerful, namely any two 3-manifolds are connected by a sequence of surgeries. Hence, studying the behavior of 3-manifolds under surgery is a key theme in low-dimensional topology with many classical results and open conjectures. This talk will give an introduction to hyperbolic 3-manifolds, surgery, and the relation between these. The focal point of the talk will be the classical result that gluing a solid torus to the boundary of a hyperbolic 3-manifold almost always gives a manifold that is still hyperbolic. No prerequisite knowledge of 3-manifolds will be needed. We will also mention recent results relating surgery to the contact topology of a hyperbolic 3-manifold.
Title: Recognising spaces for stable subgroups
Abstract: In hyperbolic groups, the quasiconvex subgroups can be thought of as hyperbolic subgroups which inherit their hyperbolicity from the group itself. There are several ways of generalising quasiconvexity outside of hyperbolic groups, but the one that arguably best captures the idea of inherited hyperbolicity is the notion of stable subgroups. We will look at when such subgroups are "recognised" by an action of the group on a hyperbolic space, that is when they are quasi-isometrically embedded via the orbit map, particularly in the case that this action is acylindrical. The aim is for this to be less of a research talk, and more of an introduction to a question that I'm interested in.
Title: The Hanna Neumann conjecture beyond free groups
The classical Howson property of a free group F states that, for all finitely generated subgroups U and V of F, the intersection of U and V is finitely generated. It was then conjectured by Hanna Neumann that a stronger quantified property should hold, namely that the rank of the intersection of U and V should, in fact, be bounded above by the product of the ranks of U and V. In this talk we discuss a bit of the history of this problem and how L^2-Betti numbers came into the picture.
06-11-23 - Sam Mansfield (University of Bristol)
Title: A structural theorem for sets with few triangles
In 1946, Erdős conjectured that the minimum number of distinct distances determined by a set of N points is N(log N)^{-1/2}, the number achieved by an N^{1/2} by N^{1/2} square lattice. This was (almost) solved by Guth and Katz in 2015, but a harder variant -- that any point set with only this number of distinct distances must have a lattice structure -- is still open, and there are remarkably few results about the structure of such a set at all. Instead of considering distinct distances, we consider sets that determine few congruent triangles, showing that such sets either contain a polynomially-rich line or a positive proportion of the set lies on a circle. Our methods include classical tools from additive combinatorics combined with geometric structure within the affine group. This is based on joint work with Jonathan Passant.
30-10-23 - Tirumala Venkata Chakradhar (University of Bristol)
Title: Index bounds for free boundary minimal surfaces
Abstract: We consider the free boundary minimal hypersurfaces of Euclidean bodies and introduce the Jacobi operator. The (Morse) index of a minimal hypersurface is then given by the number of negative eigenvalues of this operator and is a measure of stability of the minimal surface under perturbations. After looking at some nice examples, we focus on lower bounds for the index. We briefly discuss the key ingredients of the proof of certain lower bounds in terms of the first absolute Betti number, using a relation between the eigenvalues of the stability operator and the Hodge Laplacian. The emphasis will be on describing the beautiful interplay of geometry, analysis, and topology. This is based on the work of Alessandro Savo and Pam Sargent.
Title: Currents on complex manifolds
Abstract: Currents are key objects of study in pluripotential theory and holomorphic dynamics. They also relate to areas like complex analysis, geometric measure theory, complex algebraic geometry and, surprisingly, tropical geometry (which gives a combinatorial flavour to them).
By definition, currents are dual to smooth forms. Hence, they carry some of the properties of forms, such as having an exterior derivative and (in some cases) a wedge product. On the other hand, we can regard submanifolds as currents, so that we can derivate and (sometimes) "wedge" submanifolds by using currents. These and some other important examples make us believe that currents can generalise (in a nice new way) theories like: Hodge theory, Intersection theory, ergodic theory, tropicalisation of varieties....
The goal of this talk is to introduce the basic definitions and the first important examples of currents, beginning with a quick recap of complex smooth forms. Time permitting, we will also discuss some interesting problems.
02-10-23 - Sofiya Yatsyna (Royal Holloway)
Title: Introduction to totally disconnected locally compact groups
Abstract: The study of locally compact groups can be divided into its connected and totally disconnected components. In this talk, we will focus on totally disconnected locally compact (tdlc) groups, which are a class of topological groups that contain a compact open subgroup. Examples of tdlc groups include the automorphism group of a connected locally-finite graph and Neretin’s group. We will introduce some important concepts in group cohomology and in particular, look at the representation theory and rational discrete cohomology for tdlc groups.
09-05-23 - David Guo (University of Bristol)
Title: Uniform D_α property in polycyclic groups
Abstract: Given a polycyclic group, Wolf proved that polynomial growth is equivalent to being virtually nilpotent. Ana Khukhro and Alain Valette introduced a notion called uniform D_α in their study on the box spaces of groups. Having uniform D_α is a weaker condition than having polynomial growth. It turns out that having uniform D_α is also equivalent to being virtually nilpotent. In this talk, we will introduce the definition and some facts about the uniform D_α property, as well as explore some related results. In the end, we will go through some of the ideas of the proof.
Title: Representations and product-free subsets in finite nonabelian groups
Abstract: Sum-free sets are a staple of additive combinatorics, however when looking at nonabelian groups, many of the tools used to study sum-free sets do not apply. In 2007, Gowers provided a lower bound for the size of the largest product-free subset of a group based on the size of the smallest non-trivial representation, which Babai, Nikolov and Pyber improved upon. I will give a brief introduction to product-free subsets and some constructions of some simple cases, and give a short introduction to representation theory and the nonabelian Fourier transform, before providing an alternative proof to the Babai-Nikolov-Pyber bound.
Title: The isomorphism problem for large-type Artin groups
Abstract: The study of random walks on groups has yielded interesting results about ‘generic’ elements and has deepened our understanding of certain groups (for example acylindrically hyperbolic groups). It is therefore natural to wonder what can be said about random walks on groups that are quasi-isometric to each other. Sadly, in general there is no meaningful notion of a random walk on a quasi-isometric group. In this talk, I will mention one way to resolve this by studying more general Markov chains. I will present some results about them and propose a new quasi-isometry invariant that can be obtained from these Markov chains. This is based on joint work with both Alessandro Sisto and Mark Hagen, Harry Petyt and Alessandro Sisto.
Title: The isomorphism problem for large-type Artin groups
Abstract: Artin groups are a large family of groups with interesting properties and various open problems. In this talk, we will talk about the isomorphism problem for Artin groups, and we will sketch a solution to this problem for Artin groups of large-type.
Title: Finiteness properties of kernels of maps to Z: Bestvina-Brady and beyond
Abstract: It is an idea dating back to the work of Stallings and Bieri from the 1960's and 70's, that one can obtain subgroups with interesting finiteness properties by studying kernels of homomorphisms onto the integers. Inspired by this, Bestvina-Brady developed a piecewise linear Morse theory and successfully applied it to right-angled Artin groups in order to find many examples of such subgroups. The aim of this talk is to give an introduction to Bestvina-Brady's Morse theory and its original applications and, time permitting, discuss some new ideas on finding interesting subgroups of hyperbolic groups.
Title: Convex geometry over ordered hyperfields
Abstract: We will introduce hyperfields as structures with multi-valued addition and understand how orderings over this domain behave. We define the notion of convexity and present properties of convex sets over hyperfields. We will then explore generalisations of several classical results in convex geometry, including Carathédory's theorem. To finish we will then discuss how separation in this setting differs from the classical setting, in particular requiring the use hemispaces rather than half spaces. This is joint work with Ben Smith (Manchester).
28-02-23 - Shaked Bader (University of Oxford)
Title: CAT(0) polygonal complexes are 2 median
Abstract: A (1-)median space is a space in which for every three points the intersection of the three intervals between them is a unique point. Having this in mind, in my talk I will define a 2-median space which is a 2 dimensional variation of the median space. I will then present some ideas in the proof of the theorem in the title. This talk is based on my Master Thesis done under the supervision of Nir Lazarovich.
21-02-23 - Stefanie Zbinden (Heriot-Watt University)
Title: Introduction to the Morse boundary
Abstract: We will introduce the algorithmic problem of splitting detection in groups, and survey some results. We will then present an algorithm which can decide (in 'most' cases) whether a given quasiconvex subgroup of a hyperbolic group is associated to a splitting.
Title: Will it split? Recognising quasiconvex splittings in hyperbolic groups
Abstract: We will introduce the algorithmic problem of splitting detection in groups, and survey some results. We will then present an algorithm which can decide (in 'most' cases) whether a given quasiconvex subgroup of a hyperbolic group is associated to a splitting.
Title: How to show that some groups are one-ended?
Abstract: I will give a brief introduction to ends of groups, with a particular focus on one-ended groups. Stallings’ theorem on ends of groups states that a group has more than one end if and only if it splits as a free amalgamated product or an HNN extension over a finite subgroup. This suggests that it is worth looking at the case where a group is one-ended. I will give two examples of methods to show that a group has one end. These include 1) right-angled Artin groups (when the defining graph is connected), and 2) random few relator groups.
Title: Subgroup spaces and actions on trees
Abstract: Let G be a countable group. The subgroup space of G is the set of all its subgroups equipped with a natural topology, known as the Chabauty topology. There has been recent interest in the action by conjugation of groups on their space of subgroups and in particular on the probability measures which are invariant under this action, so called invariant random subgroups (IRS’). Any non-atomic IRS is supported on the perfect kernel of the subgroup space (I.e. the largest closed subspace with no isolated points) so it is natural to ask what this perfect kernel is for a given group. I will present a method, developed in joint work with Damien Gaboriau, to show that certain subgroups of groups acting on trees are in the perfect kernel. As a corollary, we can show that the perfect kernel of a group with infinitely many ends has maximal perfect kernel and classify the perfect kernels of surface groups.
13-12-22 - Harsh Patil (University of Bristol)
Title: Braids, links and mapping class groups
Abstract: The first definition of the Braid group was given by Artin in the 1920’s. Since then, several different (but equivalent!) definitions have been given. We discuss a few of these and prove their equivalence. We discuss how knots/links arise naturally as ‘closure’ of braids. The celebrated Markov theorem tells us exactly when two different braids give rise to equivalent knots/links. We also discuss how a certain representation (Burau representation) of the Braid group can be used to calculate the Jones polynomial of its closure. We also discuss how braid groups arise as mapping class groups of the n-times punctured disc. We discuss the solution to the word problem on Braid groups.
Title: Knot projections in 3-manifolds other than the 3-sphere
Abstract: Knot projections for knots in the 3-sphere allow us to easily describe knots, compute invariants, enumerate all knots, manipulate them under Reidemister moves and feed them into a computer. One might hope for a similar representation of knots in general 3-manifolds. We will survey properties of knots in general 3-manifolds and discuss a proposed diagram-esque representation of them, in which there are no crossings.
Title: The geometry of random hyperbolic surfaces
Abstract: The aim of this talk is to present a model of random hyperbolic surfaces, the Weil--Petersson model. I will explain a few tools developed by Mirzakhani to study these random surfaces, and describe their geometry.
Title: The outer automorphism group of a free product
Abstract: Given a free product G=G_1*…*G_n, we can construct a space whose points are trees on which G acts (nicely). Then Out(G) acts on this space, and using a theorem of K. Brown, we can extract a presentation for Out(G). We will explore this construction (and explain how to extract a presentation) by considering some examples when n is small.
Title: How many contacts can exist between oriented squares of various sizes?
Abstract: A homothetic packing of squares is any set of various-size squares with the same orientation where no two squares have overlapping interiors. If all n squares have the same size then we can have up to roughly 4n contacts by arranging the squares in a grid formation. The maximum possible number of contacts for a set of n squares will drop drastically, however, if the size of each square is chosen more-or-less randomly. In my talk I will describe a necessary and sufficient condition for determining if a set of n squares with fixed sizes can be arranged into a homothetic square packing with more than 2n−2 contacts. I will then discuss how any similar analogues will fail for cube packings.
Title: Combining quasiconvex subgroups in hyperbolic groups
Abstract: In this talk, we will introduce the notions of hyperbolic groups and quasiconvex subgroups, and explore some of their nice properties. We then outline a method to (up to finite index) combine any two quasiconvex subgroups to get another one, with a view towards explaining a recent joint work with Minasyan generalising this method to the much wider class of "relatively hyperbolic" groups.
Title: Spin parities and how to calculate them
Abstract: In this talk, I will introduce the notion of the parity of a spin structure on a Riemann surface with a particular focus on the setting of translation surfaces. I will then discuss various methods for how to calculate the spin parity for a given translation surface with a particular focus on work in progress with Tarik Aougab, Adam Freidman-Brown and Jason Ma in which we are interested in calculating the spin parities of families of single-cylinder square-tiled surfaces.
25-10-22 - Vladimir Vankov (University of Bristol)
Title: Hyperbolic groups and cohomology
Abstract: Hyperbolic groups are important objects in geometric group theory, both in terms of being a rich playground for developing tools of negatively curved geometry and a source of many deep and difficult problems. We will discuss how one can use homological algebra to derive group-theoretic and geometric results relating to hyperbolic groups.
Title: Cube complexes and braid groups
Abstract: The braid group of a space X is the fundamental group of its configuration space, which tracks the motion of some number of particles as they travel through X. When X is a graph, the configuration space turns out to be a cube complex -- a space constructed by gluing cubes of different dimensions to each other along their faces. I show how these cube complexes are constructed and use this to provide methods for computing braid groups of various graphs, as well as criteria for a graph braid group to split as a free product. This has various applications, such as characterising various forms of hyperbolicity in graph braid groups and determining when a graph braid group is isomorphic to a right-angled Artin group.
Title: Curves counting on surfaces
Abstract: The study of the asymptotic growth of the number of closed geodesics on a hyperbolic surface dates back to Huber (1961) and has implications in various fields of mathematics. In her thesis, Mirzakhani proved that for an orientable hyperbolic surface of finite area, the number of simple closed geodesics of length less than L is asymptotically equivalent to a polynomial in L, whose degree only depends on the Euler characteristic.
In this talk I will review these results and talk a bit about what fails in the case of non-orientable surfaces.
Title: Mapping class group orbit closures for non-orientable surfaces
Abstract: The study of the asymptotic growth of the number of closed geodesics on a hyperbolic surface dates back to Huber (1961) and has implications in various fields of mathematics. In her thesis, Mirzakhani proved that for an orientable hyperbolic surface of finite area, the number of simple closed geodesics of length less than L is asymptotically equivalent to a polynomial in L, whose degree only depends on the Euler characteristic.
When looking at non-orientable surfaces, the situation is very different. One of the main differences in this framework is the behaviour of the action of the mapping class group on the space of measured laminations.
In a joint work with Erlandsson, Gendulphe and Souto, we characterised mapping class group orbit closures of measured laminations, projective measured laminations and points in Teichmueller space.
23-11-21 - Gemma Crowe (Heriot-Watt University)
Title: On the conjugacy problem in virtual RAAGs
Abstract: The conjugacy problem in RAAGs was first solved by Green in the 90s, and even has a quick linear time solution provided by Crisp, Godelle and Wiest. This decision problem remains open however for a closely related group, namely virtual RAAGs. In my talk I will give an introduction to the methods used to solve the conjugacy problem in RAAGs, and some of my ideas so far to tackle this problem in virtual RAAGs.
16-11-21 - Beth Branman (University of Wisconsin–Madison)
Title: Pants complexes on surfaces of infinite type
Abstract: Let S be an orientable surface without boundary. The pants graph P(S) has one vertex for each homotopy class of pants decompositions for S. Vertices of P(S) are connected by an edge if they differ by an “elementary move.” When S is a finite-type surface (that is, when S has finitely generated fundamental group), the pants graph is well-studied. In particular, its automorphism group is the (extended) mapping class group: the group of isotopy classes of homeomorphisms from S to itself.
In this talk, we discuss two different generalizations of the pants graph to surfaces of infinite type. The usual definition of the pants graph P(S) yields a graph with infinitely many connected-components. In the first part of the talk, we discuss the properties of this disconnected graph. In particular, we show that the extended mapping class group is isomorphic to a proper subgroup of Aut(P(S)).
In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow P(S) with additional structure. To this end, we define a coarser topology on P(S) than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group of S.
Title: Groups, equations, and languages
Abstract: Ever since Makanin proved that the satisfiability of systems equations in free groups was decidable, there has been significant interest in the study of equations in groups. There have been a few attempts to describe solutions to specific equations using languages since then, however, Ciobanu, Diekert and Elder made a leap by using EDT0L languages to describe the sets of solutions to systems of equations in free groups. Following this there have been several attempts use languages for systems of equations in other classes of groups. We give a brief introduction to this area.
Title: Quasiisometries and CAT(0) cube complexes
Abstract: CAT(0) cube complexes are very nice metric spaces that have attracted a lot of attention in geometric group theory in recent years. Quasiisometries are often thought of as the "natural maps" of geometric group theory. In this talk, we ask the question: When is a metric space quasiisometric to a CAT(0) cube complex? We shall discuss some recent work that gives a technique for finding such a quasiisometry.
26-10-21 - Xabier Legaspi Juanatey (Université de Rennes 1)
Title: On growth rate of quasi-convex subgroups and constricting elements.
Abstract: Let G be a group acting properly on a geodesic metric space X. Let H be a subgroup of G. Have you ever wondered how big is H inside G with respect to the geometry induced by X? It turns out that the exponential growth rate ω(H,X) is a real number that gives an estimation of this ``size''.
In this talk, we will see that if G acts on X with a constricting element -- namely, an element with ``hyperbolic-like'' manners -- then we have the strict inequality ω(H,X) < ω(G,X) provided that H is quasi-convex in X and of infinite index. If time permits, we will see that another invariant ω(H\G, X) can be defined for the quotient space H\G -- thought of as a Schreier graph -- and that, under the same hypothesis, we have the equality ω(H\G,X) = ω(G,X). In particular, the class of such G includes relatively hyperbolic groups, CAT(0) groups, and hierarchically hyperbolic groups containing a Morse element.
Title: Helicity and tangled orbits of Anosov flows
Abstract: The helicity of a volume-preserving flow is one of the basic invariants studied in fluid dynamics. Its diffeomorphism invariance makes it useful for solving certain variational problems, particularly in magnetohydrodynamics. Arnold proposed that one can characterise helicity in terms of the linking numbers of knots constructed by closing up trajectories of the flow with geodesic arcs. In this talk we will describe Arnold's characterisation, followed by a new characterisation for the case of Anosov flows, in terms of the linking of periodic trajectories. We will also discuss an application of this characterisation to a particular class of (almost) geodesic flows.
16-03-21 - Amina Ladjali (University of Southampton)
Title: An introduction to coarse median spaces
Abstract: Coarse median spaces were introduced by Bowditch in 2013 and informally are coarsened versions of CAT(0) cube complexes. These spaces provide a unified view of interesting classes, such as geodesic hyperbolic spaces and mapping class groups. In this talk, I’ll define what they are along with outlining some known results. I’ll also briefly mention our current work on showing polynomial growth of intervals of these spaces.
Title: Big mapping class groups
Abstract: Traditionally the study of mapping class groups has been concerned with surfaces of finite type. Recently interest has arisen in the mapping class groups of infinite-type surfaces; that is, surfaces with "infinite topological complexity" or, more formally, ones whose fundamental groups are not finitely generated. I'll give an introduction to the field and talk about recent results and open questions. If time permits I'll touch on some of my own work on the abelianization of these groups.
Title: A signature for some subgroups of the permutation group of [0,1[
Abstract: For every infinite set X we define S(X) as the group of all permutations of X. On its subgroup consisting of all finitely supported permutations, there exists a natural homomorphism, called the signature. However, thanks to an observation of Vitali in 1915, we know that this group homomorphism does not extend to S(X). In the talk we extend the signature to the subgroup of S([0,1[) consisting of all piecewise isometric elements (strongly related to the Interval Exchange Transformation group). This allows us to list all of its normal subgroups and gives also information about an element of the second cohomology group of some groups.
Title: Braid groups on graphs and their cubical geometry
Abstract: Consider a finite collection of particles lying on a finite graph. The configuration space of these particles is the collection of all possible ways the particles can be arranged on the graph with no two particles at the same point. As we move through the configuration space, the particles move along the graph, without colliding.The braid group on our graph is then defined to be the fundamental group of this configuration space. By discretising the motion of the particles, we obtain a combinatorial version of the configuration space, which can be shown to be a special cube complex. Moreover, this cube complex deformation retracts onto the original configuration space, meaning the braid group is unchanged. This allows for cubical techniques to be employed in studying the geometry of braid groups on graphs. In ongoing work, I use this structure to obtain a classification of relative hyperbolicity for graph braid groups.
Title: Meanders, filling pairs, and quadratic differentials
Abstract: A meander is a special kind of planar graph and the enumeration of meanders with a fixed number of vertices continues to be an extremely difficult task. Such enumerations are of interest in many areas of study. For example, meanders are related to polymer chain folding and certain matrix models in physics, to the study of particular examples of sorting algorithms in computer science, and, in mathematics, have cropped up regularly in various areas of geometry. Indeed, they can be found in some work of Poincaré from 1912.
In this talk, I will introduce meanders and discuss their relation to pairs of filling curves on punctured spheres and to special cases of genus zero quadratic differentials. In particular, I will discuss a problem that becomes equivalent to that of constructing meanders of a specific combinatorial type using a minimal number of vertices.
Title: Random groups at density d < 3/14 act on CAT(0) cube complexes
Abstract: For random groups in the Gromov density model at d < 3/14, we construct walls in the Cayley complex which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities d < 1/5 and d < 5/24, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density d < 1/4.
Title: The flip graph
Abstract: I will introduce the flip graph, a graph measuring "distance" between different ways of triangulating a surface. We'll discuss basic properties and fundamental results about the flip graph, and explore ways the flip graph has been used as a tool for studying other objects, such as the mapping class group and cluster algebras.
Title: An introduction to random groups
Abstract: Random groups were introduced by Gromov to answer the question of what a 'generic' group looks like. As in other areas of maths, models of random groups are a useful way to produce groups with interesting properties, and they display striking 'sharp phase transitions'. In this talk we introduce two common models of random groups, and discuss some theorems concerning them.
Title: Virtually unipotent curves in some non-NPC graph manifolds
Abstract: A graph manifold is a compact aspherical 3-manifold all of whose JSJ pieces are Seifert. It follows from work of Leeb that if a graph manifold cannot be endowed with a nonpositively curved (NPC) Riemannian metric, then its fundamental group admits no discrete embedding into GL(n,C) whose image consists entirely of diagonalizable matrices. For at least some of these manifolds M, it is even true that there is a non-trivial element of pi_1(M) whose image under any (possibly non-discrete) representation into GL(n,C) has a unipotent power. In particular, and in contrast to the fundamental groups of their NPC cousins, the fundamental groups of such M admit no faithful finite-dimensional unitary representations. We discuss a special case of this phenomenon.
Title: An introduction to solvable groups
Abstract: I will give a brief introduction to the world of solvable groups, motivated by a range of examples of both discrete and Lie groups.
Title: Introduction to stable subgroups
Abstract: Stable subgroups of finitely generated groups are a generalization of quasi-convex subgroups of hyperbolic groups. They were originally defined by Durham and Taylor as an alternate characterization of convex cocompact subgroups of surface mapping class groups. In this talk, I will introduce the notion of a stable subgroup and provide an overview of what is currently known about stable subgroups in various contexts, including mapping class groups, Right-angled Artin groups, and hierarchically hyperbolic groups.
Title: Horocyclic products of Gromov hyperbolic spaces
Abstract: Gromov hyperbolicity is a property of metric spaces that generalises the notion of negative curvature for manifolds. After an introduction about these spaces, we will explain the construction of horocyclic products, which are related to lamplighter groups, Baumslag-Solitar groups and Sol geometry.
We will describe the shape of geodesics in these products, and present rigidity results on their quasi-isometries due to Farb, Mosher, Eskin, Fisher and Whyte.
Title: Statistics on Coxeter groups
Abstract: Coxeter groups are a family of groups that are easily defined using generators and relations but have many interesting geometric and combinatorial properties.
After giving a short introduction to this family of groups, I will talk about how one can use statistical methods for studying them. The leading question for this will be: What does a "random" element in a (finite) Coxeter group look like?
Title: Minimising entropy on translation surfaces
Abstract: Translation surfaces can be viewed as a collection of polygons in the plane, where each edge is glued to another edge via a translation. Such an identification yields a closed surface with a flat metric except at a finite number of singularities. These singularities may be thought of as points of high negative curvature which motivates us to consider which geometrical properties of negatively curved surfaces carry over to translation surfaces. In the first half of this talk I will give an overview of some initial results which show that this analogy does carry some weight. In the second half, I will look at the problem of determining which translation surfaces have the smallest volume entropy.
Title: Finiteness properties of subgroups of Borel groups
Abstract: Given a finitely generated group G, the Σ-invariants of G consist of geometrically defined subsets Σ^k(G) of the set S(G) of all homomorphisms χ : G --> R. These invariants were introduced independently by Bieri-Strebel and Neumann for k=1 and generalized by Bieri-Renz to the general case in the late 80's in order to determine the finiteness properties of all subgroups H of G that contain the commutator subgroup [G,G].
In this talk we determine the Σ-invariants of certain S-arithmetic subgroups of Borel groups in Chevalley groups. In particular we will determine the finiteness properties of every subgroup of the group of upper triangular matrices B_n( Z[1/p] ) < SL_n( Z[1/p] ) that contains the group U_n( Z[1/p] ) of unipotent matrices where p is any sufficiently large prime number.
Title: Variants of Leighton's graph covering theorem
Abstract: Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will explain how this theorem can be generalised to various different kinds of decorated graphs. I will also discuss applications to 3-manifolds and graphs of groups.
Title: Non-uniquely ergodic laminations on surfaces
Abstract: Geodesic laminations are important objects in the study of surface homeomorphisms. It is a curious fact that there exist (minimal, filling) geodesic laminations that support two projectively distinct transverse measures. Such laminations are called non-uniquely ergodic. In this talk, I will define geodesic lamination, transverse measure and present a construction of non-uniquely ergodic (minimal, filling) lamination on a surface due to Gabai. I will briefly mention the existence of similar phenomena for R-trees in the boundary of Outer space.
Title: Hierarchies for one-relator groups via covering spaces
Abstract: A group splits as an HNN-extension if and only if the rank of its abelianisation is strictly positive. If we fix a class of groups, one may ask a few questions about these splittings: What form can vertex and edge groups take? If they remain in our fixed class, do they also split? If so, under iteration will we terminate at something nice?
In this talk we will first see how the class of finitely presented groups is too general to answer these questions in any nice form. If we restrict our attention to the much simpler class of one-relator groups, we will show that everything is (sort of) as nice as possible and go through an example or two. Time permitting, we will also discuss possible generalisations to groups with staggered presentations and applications.
Title: Groups of type FP via graphical small cancellation theory
Abstract: Until the early 1990's it was unknown whether or not a group of type FP could be non-finitely presented. Bestvina and Brady used Morse Theory on CAT(0) cubical complexes to construct groups that were both of type FP and non-finitely presented. Since then there have been more constructions of such groups, all of them either use Morse Theory on cube complexes directly or they rely on a previously constructed non-finitely presented group of type FP, so are using it indirectly.
We have constructed an uncountable family of groups of type FP using completely new methods. In this talk I will give an overview of the construction, focusing on the use of graphical small cancellation theory and the secret ingredient - our suitable complexes.
26-11-19 - Monika Kudlinksa (University of Bristol)
Title: Automorphisms of free groups and train tracks
Abstract: Given an automorphism of a free group, one can associate to it a topological representative: a graph G, and a map f : G → G, from which we can recover most of the relevant data about the automorphism. The choice of topological representative for an automorphism is far from unique. Handel and Bestvina showed that ‘nice’ automorphisms admit a special type of topological representative, called a train track map, whose dynamics can be well understood.
In this talk, I will outline the definition and motivation for train track maps. Time permitting, I will also give a sketch of Handel-Bestvina's algorithm for finding train tracks.
Title: Boundaries of hyperbolic groups
Abstract: A core philosophy of Geometric Group Theory is building connections between disparate areas of mathematics to gain new insight. For instance, by turning a group into a graph, called a Cayley graph, one can study the large-scale geometry of this graph to say something about the algebraic properties of the original group.
I will talk about another of these connections, which links the large-scale geometry of Gromov-hyperbolic spaces and the analytical properties of bounded, complete metric spaces. I will say something about how this link has been used to study groups using concepts like Hausdorff dimension, and present some of my contributions to this program of studying groups.
Title: Generalising Mirzakhani’s curve counting result
Abstract: On any hyperbolic surface, the number of curves of length at most L is finite. However, it is not immediately clear how quickly this number grows with L. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.
Title: Cutting up surfaces and commutators in free groups
Abstract: When is an element g of a group G a product of commutators [a,b]? What is the minimal number of commutators n such that g=[a_1,b_1]...[a_n,b_n]? What are all possible solutions (a_1,b_1,...,a_m,b_m) to the equation g=[a_1,b_1]...[a_m,b_m]? Very difficult, yet important, questions --- but ones whose answers seem to lie in the darkest recesses of combinatorial group theory.
Not so! At least for G a free group, these questions can all be answered elegantly and beautifully by cutting up and colouring surfaces. In this talk I shall present solutions to these problems with an emphasis on drawing nice pictures.
Title: Rigidity of curve graphs and Ivanov's Metaconjecture
Abstract: Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every simplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.
During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. Lastly we will talk about generalisation of the curve complex, and highlight some results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".
Title: Enumerating 1-vertex quadrangulations of surfaces using bi-Eulerian paths
Abstract: A bi-Eulerian path is a path around a graph which visits each edge exactly twice, once in each direction. In the case that the graph is n-regular, certain bi-Eulerian paths are in direct correspondence with n-angulations of once-punctured compact orientable surfaces. We develop methods to count such paths in the special case that n = 4, hence enumerating the quadrangulations of once-punctured surfaces.
Title: Fibrations of 3-manifolds over the circle and their corresponding veering triangulations
Abstract: My aim is to give a gentle introduction to veering triangulations and their connection to fibrations of 3-manifolds over the circle.
Any 3-manifold fibred over the circle can be obtained as a mapping torus of a surface homeomorphism. However, typically there are infinitely many ways in which we can present a manifold in this way.
Given a fibration of a fibred oriented hyperbolic 3-manifold we can build a canonical triangulation which is veering and whose 2-skeleton carries the foliation by fibres. This is an ideal triangulation of the cusped 3-manifold obtained from the initial one by drilling out a link determined by the gluing homeomorphism. It turns out that different fibrations of the same 3-manifold can share the same veering triangulation and we can tell exactly when this happens.
Worth noting is that not every veering triangulation carries a fibration, but we will restrict ourselves to triangulations which do. The talk should be accessible to anyone with a basic topology background - I will only assume familiarity with manifolds, orientability, triangulations and homeomorphisms.
Title: What does a Cayley graph look like?
Abstract: Every Cayley graph of a finitely generated group has some basic properties: they are locally finite, connected, and vertex-transitive. These are not sufficient conditions, there are some well known examples of graphs that have all these properties but are non-Cayley. These examples do however "look like" Cayley graphs, which led to Woess asking in the 1980s if there exist any vertex-transitive graphs that do not look like Cayley graphs. I plan to give some of the history of this question, as well as the construction of the example that finally answered it.
Title: Some properties of right-angled Artin groups
Abstract: Free groups and free abelian groups have been deeply studied and so many important properties are well-known. In this talk, I will define a class of groups that 'generalize' them, known as right-angled Artin groups. We know that subgroups of free groups or free abelian groups are again of the same type, and trivially they are coherent groups (meaning that all the finitely generated subgroups are finitely presented). I will explain that right-angled Artin groups keep these properties under certain conditions.
Moreover, Baumslag and Roseblade showed that the only finitely presented subgroups of a direct product of two free groups are either free or are virtually a direct product of two free groups. I will explain that a similar situation happens in the case of right-angled Artin groups.
Title: Special cube complexes
Abstract: This will be a basic introduction to a certain kind of cube complex which has its fundamental group embedding into a Right-Angled Artin Group. I'll talk about why this is useful, and briefly present a new virtually special group in my research, coming from a generalisation of the Bestvina-Brady Morse theory construction.
Title: Orderable groups in topology and Farrell's theorem
Abstract: A group G is called left-orderable if it has an ordering relation (<) satisfying:
(a < b) ⟹ (ga < gb) for all a, b, g in G.
For such a simple definition, this property has led to a surprising number of insights and results, especially concerning groups that appear naturally in topology. After explaining some basic properties of orderable groups, I will try to give an idea of some of these reasons why topologists might care about orderability, in particular focussing on a nice result of Farrell:
The fundamental group of a (generically "well-behaved") space X is left-orderable if and only if its universal covering map X' → X lifts to an embedding of X' into the product of X with the real line.
Title: Decision problems for groups
Abstract: As group theorists, we are in the (perhaps unique) situation that our day-to-day might not involve working with problems where groups are the obvious object of interest.
This talk will be on decision problems, of which the word problem is an example. Although this can be a highly technical area of maths, I will try and take a light-touch approach, partly by linking back to some previous BRIJGES seminars - we’ll see examples using hyperbolic groups and residually finite groups - but also by trading some formal Turing Machine definitions for pictures and analogy. I will also introduce Higman’s Embedding Theorem in order to construct a nice example.
Title: Fuchsian Groups and Geometry of Surfaces
Abstract: Topologically, orientable closed compact surfaces can be classified by their genus. We can, however, say much more about their geometry. We will investigate Fuchsian groups (discrete subgroups of SL_2(R)) acting on the hyperbolic plane. We will use them along with a black box version of the uniformization theorem to construct every closed surface as the quotient of either C, H^2, or the sphere S^2.
Title: Surface homeomorphisms and pants decompositions
Abstract: In geometric group theory, it is common practice to study properties of a group by finding a meaningful action on a metric space. This talk is about the mapping class group of a compact surface and its action on the pants graph.
The mapping class group is the group of homeomorphisms up to isotopy. I will define this group and state some properties and open questions. After this motivation, I will introduce the pants graph and its natural mapping class group action. Time permitting, I will discuss certain subgraphs, some of which are obstructions to Gromov hyperbolicity.
Title: The eight 3-manifold geometries
Abstract: In 1982 Thurston conjectured that any closed 3-manifold can be cut up in a simple way so that each of the resulting pieces admits one of exactly eight geometric structures. This conjecture determined the direction of research in low dimensional geometry until its proof by Perelman in 2003. In this talk I will explain what it means for a manifold to admit a particular geometric structure in Thurston's formalism. I will then describe how the eight Thurston geometries arise, and—time permitting—explain why there are only eight of them.
Title: Introduction to hierarchically hyperbolic spaces and groups
Abstract: Hierarchically hyperbolic spaces were introduced in 2015 by J. Behrstock, M. Hagen and A. Sisto. These spaces generalise the notion of (Gromov-) hyperbolic spaces and give a common language to work with mapping class groups of surfaces and CAT(0) cube complexes. Furthermore, a finitely generated group is called hierarchically hyperbolic if it acts in a nice way on a hierarchically hyperbolic space. In this talk we will see main examples and classes of hierarchically hyperbolic groups and motivate the definition of hierarchically hyperbolic groups. We will take a look at main results established for this class of groups and present some open questions.
Title: A beginner's guide to 3-manifolds and the Poincaré Conjecture
Abstract: The Poincaré Conjecture was first formulated over a century ago and states that there is only one closed simply connected 3-manifold, hinting at a link between 3-manifolds and their fundamental groups. This seemingly basic fact went unproven until the early 2000s when Perelman proved Thurston's much more powerful Geometrisation Conjecture, providing us with a powerful structure theorem for understanding all closed 3-manifolds.
In this talk I will introduce the results developed throughout the 20th century that led to Thurston and Perelman's work. Then, using Geometrisation as a black box, I will present a proof of the Poincaré Conjecture. Throughout we shall follow the crucial role that the fundamental group plays and hopefully demonstrate the geometric and group theoretical nature of much of the modern study of 3-manifolds.
As the title suggests, no prior understanding of 3-manifolds will be expected.
Title: A beginner’s guide to 'Stable' Homotopy Theory
Abstract: At some point most of us will attend a talk on 'Stable' Homotopy Theory. This seminar aims to help us get something more from the experience than just a feeling of drowning in a whirlpool of abstraction. The questions I would like to answer are:
- What is homotopy theory?
- What does ‘stable’ mean?
- What are spectra?
I will assume that everyone knows the definition of the fundamental group, and what it means for a sequence to be exact, but will assume little more than that. I will focus on motivation over rigor.
Title: Finite quotients of surface groups
Abstract: It is often fruitful to study an infinite discrete group via its finite quotients. For this reason, conditions that guarantee many finite quotients can be useful. One such notion is residual finiteness.
A group is residually finite if for any non-identity element g, there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and if time allows I’ll show that it is in fact enough to consider homomorphisms onto alternating groups in this case.
Title: Groups acting on trees (or are they...?)
Abstract: The main idea behind geometric group theory is to think about groups by thinking about their actions. That is, to understand your favourite group better, make it act on a useful space, mix around and see what comes out. Trees are very useful spaces, it turns out. If your group acts on one, you get lots of information about it: this is the central observation behind Bass-Serre theory. So much so in fact, that (loosely speaking) not acting on trees gets its own property name - FA. We'll take a tour through some of the results and tools in this area, including a question about when certain automorphism groups act on trees.