Past Junior Topology and Group Theory Seminar

I will give a self-contained introduction to the theory of cross ratios on boundaries of Gromov hyperbolic and CAT(-1) spaces, focussing on the connections to the following two questions. When are two spaces with the 'same' Gromov boundary isometric/quasi-isometric? Are closed Riemannian manifolds completely determined (up to isometry) by the lengths of their closed geodesics?

When groups may be built up as graphs of 'simpler' groups, it is often 
of interest to study how good residual finiteness properties of simpler 
groups can imply residual properties of the whole. The essential case of 
this theory is the study of residual properties of finite groups. In 
this talk I will discuss the question of when a graph of finite 
$p$-groups is residually $p$-finite, for $p$ a prime. I describe the 
previous theorems in this area for one-edge and finite graphs of groups, 
and their method of proof. I will then state my recent generalisation of 
these theorems to potentially infinite graphs of groups, together with 
an alternative and more natural method of proof. Finally I will briefly 
describe a usage of these results in the study of accessibility -- 
namely the existence of a finitely generated inaccessible group which is 
residually $p$-finite.

I will give a description of a method introduced by N. Ivanov to study the abstract commensurator of a group by using a rigid action of that group on a graph. We will sketch Ivanov's theorem regarding the abstract commensurator of a mapping class group. Time permitting, I will describe how these methods are used in some of my recent work with Horbez on outer automorphism groups of free groups.

Knot theory investigates the many ways of embedding a circle into the three-dimensional sphere. The study of these embeddings is not only important for understanding three-dimensional manifolds, but is also intimately related to many new and surprising phenomena appearing in dimension four. I will discuss how four-dimensional interpretations of some invariants can help us understand surfaces that bound a given link (embedding of several disjoint circles).

In 1982, Gromov introduced bounded cohomology to give estimates on the minimal volume of manifolds. Since then, bounded cohomology has become an independent and active research field. In this talk I will give an introduction to bounded cohomology, state many open problems and relate it to other fields. 

I will discuss the basics of normal surface theory, and how they were used to give an algorithm for deciding whether a given diagram represents the unknot. This version is primarily based on Haken's work, with simplifications from Schubert and Jaco-Oertel.
 

Polycyclic groups either have polynomial growth, in which case they are virtually nilpotent, or exponential growth. I will give two interesting examples of "small" polycyclic groups which are extensions of $\mathbb{R}^2$ and the Heisenberg group by the integers, and attempt to justify the claim that they are small by sketching an argument that every exponential growth polycyclic group contains one of these.

You’re an amateur investigator hired to uncover the mysterious goings on of a dark cult. They call themselves Geometric Group Theorists and they’re under suspicion of pushing humanity’s knowledge too far. You’ve tracked them down to their supposed headquarters. Foolishly, you enter. Your mind writhes as you gaze unwittingly upon the Eldritch horror they’ve summoned… Group Theory! You think fast; donning the foggy glasses of quasi-isometry, you prevent your mind shattering from the unfathomable complexity of The Beast. You spy a weak spot and the phrase `Gromov Hyperbolicity’ flashes across your mind. You peer deeper, further, forever… only to find yourself somewhere rather familiar, strange, but familiar… no, self-similar! You’ve fought with fractals before, this weirdness can be tamed! Your insight is sufficient and The Beast retreats for now.
In other words, given an infinite group, we associate to it an infinite graph, called a Cayley graph, which gives us a notion of the ‘geometry’ of a group. Through this we can ask what kind of groups have hyperbolic geometry, or at least an approximation of it called Gromov hyperbolicity. Hyperbolic groups are quite a nice class of groups but a large one, so we introduce the Gromov boundary of a hyperbolic group and explain how it can be used to distinguish groups in this class.

A core problem in the study of manifolds and their topology is that of telling them apart. That is, when can we say whether or not two manifolds are homeomorphic? In two dimensions, the situation is simple, the Classification Theorem for Surfaces allows us to differentiate between any two closed surfaces. In three dimensions, the problem is a lot harder, as the century long search for a proof of the Poincaré Conjecture demonstrates, and is still an active area of study today.
As an early pioneer in the area of 3-manifolds Seifert carved out his own corner of the landscape instead of attempting to tackle the entire problem. By reducing his scope to the subclass of 3-manifolds which are today known as Seifert fibred spaces, Seifert was able to use our knowledge of 2-manifolds and produce a classification theorem of his own.
In this talk I will define Seifert fibred spaces, explain what makes them so much easier to understand than the rest of the pack, and give some insight on why we still care about them today.
 

"Fibre theorems" in the style of Quillen's fibre lemma are versatile tools used to study the topology of partially ordered sets. In this talk, I will formulate two of them and explain how these can be used to determine the homotopy type of the complex of (conjugacy classes of) free factors of a free group.
The latter is joint work with Radhika Gupta (see https://arxiv.org/abs/1810.09380).

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

Cubulating a group means finding a proper cocompact action on a CAT(0) cube complex. I will describe how cubulating a group tells us some nice properties of the group, and explain a general strategy for finding cubulations.

We will study the l1-homology of the 2-class in one relator groups. We will see that there are many qualitative and quantitive similarities between the l1-norm of the top dimensional class and the stable commutator length of the defining relation. As an application we construct manifolds with small simplicial volume.

This work in progress is joint with Clara Loeh.

One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse PD_n subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This  generalises work of Whyte and Mosher--Sageev--Whyte.

In the 80s, Hatcher and Thurston introduced the pants graph as a tool to prove that the mapping class group of a closed, orientable surface is finitely presented. The pants graph remains relevant for the study of the mapping class group, sitting between the marking graph and the curve graph. More precisely, there is a sequence of natural coarse lipschitz maps taking the marking graph via the pants graph to the curve graph.

A second motivation for studying the pants graph comes from Teichmüller theory. Brock showed that the pants graph can be interpreted as a combinatorial model for Teichmüller space with the Weil-Petersson metric.

In this talk I will introduce the pants graph, discuss some of its properties and state a few open questions.

Elements of a finitely generated group have a natural notion of length: namely the length of a shortest word over the generators that represents the element. This allows us to study the growth of such groups by considering the size of spheres with increasing radii. One current area of interest is the rationality or otherwise of the formal power series whose coefficients are the sphere sizes. I will describe a combinatorial way to study this series for the class of virtually abelian groups, introduced by Benson in the 1980s, and then outline its applications to other types of growth series.

Thompson's group F is a group of homeomorphisms of the unit interval which exhibits a strange mix of properties; on the one hand it has some self-similarity type properties one might expect of a really big group, but on the other hand it is finitely presented. I will give a proof of finite generation by expressing elements as pairs of binary trees.

When dealing with geometric structures one natural question that arise is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometric invariant. This motivates the notion of Morse subsets. In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups (HHG), and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell.

In 2012 Eskin, Fisher and Whyte proved there was a locally finite vertex transitive graph which was not quasi-isometric to any connected locally finite Cayley Graph. This motivates the study of vertex transitive graphs from a geometric group theory point of view. We will discus how concepts and problems from group theory generalise to this setting. Constructing one framework in which problems can be framed so that techniques from group theory can be applied. This is work in progress with Agelos Georgakopoulos.

tba

TBC

 The $n$-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus $g$ with $n$-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the $n=1$ case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively.  In this talk, I will describe the upper and lower bounds I have proved as a function of $g$ and $n$.

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.
 

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.

Cube Complexes with Coupled Links (CLCC) are a special family of non-positively curved cube complexes that are constructed by specifying what the links of their vertices should be. In this talk I will introduce the construction of CLCCs and try to motivate it by explaining how one can use information about the local geometry of a cube complex to deduce global properties of its fundamental group (e.g. hyperbolicity and cohomological dimension). On the way, I will also explain what fly maps are and how to use them to deduce that a CAT(0) cube complex is hyperbolic.

We will discuss topological and algebraic aspects of splittings of free groups. In particular we will look at the core of two splittings in terms of CAT(0) cube complexes and systems of surfaces in a doubled handlebody.

CAT(0) spaces are defined as having triangles that are no fatter than Euclidean triangles, so it is no surprise that under special conditions  you find pieces of the Euclidean plane appearing in CAT(0) spaces. What is surprising though is how weak these special conditions seem to be. I will present some well known results of this phenomenon, along with detailed sketch proofs.

I will talk about the properties of algebraic integers that can arise as stretch factors of pseudo-Anosoc maps. I will mention a conjecture of Fried on which numbers supposedly arise and Thurston’s theorem that proves a similar result in the context of automorphisms of free groups. Then I will talk about recent developments on the Fried conjecture namely, every Salem number has a power arising as a stretch factor. 

I will discuss a wonderful structure theorem for finitely generated group containing a codimension one polycyclic-by-finite subgroup, due to Martin Dunwoody and Eric Swenson. I will explain how the theorem is motivated by the torus theorem for 3-manifolds, and examine some of the consequences of this theorem.

Stable commutator length (scl) is a well established invariant of group elements g  (write scl(g)) and  has both geometric and algebraic meaning.

It is a phenomenon that many classes of non-positively curved groups have a gap in stable commutator length: For every non-trivial element g, scl(g) > C for some C>0. Such gaps may be found in hyperbolic groups, Baumslag-solitair groups, free products, Mapping class groups, etc. 
However, the exact size of this gap usually unknown, which is due to a lack of a good source of “quasimorphisms”.

In this talk I will construct a new source of quasimorphisms which yield optimal gaps and show that for Right-Angled Artin Groups and their subgroups the gap of stable commutator length is exactly 1/2. I will also show this gap for certain amalgamated free products.

 In a recent paper Friedl, Zentner and Livingston asked when a sum of torus knots is concordant to an alternating knot. After a brief analysis of the problem in its full generality, I will describe some effective obstructions based on Floer type theories.

Warped cones are infinite metric spaces that are associated with actions by homeomorphisms on metric spaces. In this talk I will try to explain why the coarse geometry of warped cones can be seen as an invariant of the action and what it can tell us about the acting group.

I will give a survey of known results about when two RAAGs are quasi-isometric, and will then describe a visual graph of groups decomposition of a RAAG (its JSJ tree of cylinders) that can often be used to determine whether or not two RAAGs are quasi-isometric.

If $G$ is an irreducible lattice in a semisimple Lie group, every action of $G$ on a tree has a global fixed point. I will give an elementary discussion of Y. Shalom's proof of this result, focussing on the case of $SL_2(\mathbb{R}) \times SL_2(\mathbb{R})$. Emphasis will be placed on the geometric aspects of the proof and on the importance of reduced cohomology, while other representation theoretic/functional analytic tools will be relegated to a couple of black boxes.

I will present a gentle introduction to the theory of conformal dimension, focusing on its applications to the boundaries of hyperbolic groups, and the difficulty of classifying groups whose boundaries have conformal dimension 1.

We give a construction of a boundary (the Morse boundary) which can be assigned to any proper geodesic metric space and which is rigid, in the sense that a quasi-isometry of spaces induces a homeomorphism of boundaries. To obtain a more workable invariant than the homeomorphism type, I will introduce the metric Morse boundary and discuss notions of capacity and conformal dimensions of the metric Morse boundary. I will then demonstrate that these dimensions give useful invariants of relatively hyperbolic and mapping class groups. This is joint work with Matthew Cordes (Technion).

Inspired by the theory of JSJ decomposition for 3-manifolds, one can define the JSJ decomposition of a group as a maximal canonical way of cutting it up into simpler pieces using amalgamated products and HNN extensions. If the group in question has some sort of non-positive curvature property then one can define a boundary at infinity for the group, which captures its large scale geometry. The JSJ decomposition of the group is then reflected in the treelike structure of the boundary. In this talk I will discuss this connection in the case of hyperbolic groups and explain some of the ideas used in its proof by Brian Bowditch.

Asymptotic dimension is a large-scale analogue of Lebesgue covering dimension. I will give a gentle introduction to asymptotic dimension, prove some basic propeties and give some applications to group theory. I will then define coarse homology and explain how when defined, virtual cohomological dimension gives a lower bound on asymptotic dimension.

I will compare features of (classical) cohomology theory of groups to the rather exotic features of bounded (or continuous bounded) cohomology of groups.
Besides giving concrete examples I will state classical cohomological tools/features and see how (if) they survive in the case of bounded cohomology. Such will include the Mayer-Vietoris sequence, the transfer map, resolutions, classifying spaces, the universal coefficient theorem, the cup product, vanishing results, cohomological dimension and relation to extensions. 
Finally I will discuss their connection to each other via the comparison map.

By gluing copies of a deforming polytope, we describe some deformations of complete, finite-volume hyperbolic cone four-manifolds. Despite the fact that hyperbolic lattices are locally rigid in dimension greater than three (Garland-Raghunathan), we see a four-dimensional analogue of Thurston's hyperbolic Dehn filling: a path of cone-manifolds $M_t$ interpolating between two cusped hyperbolic four-manifolds $M_0$ and $M_1$.

This is a joint work with Bruno Martelli.

This talk will introduce the notion of quasi-convex subgroups. As an application, we will prove that the intersection of two finitely generated subgroups of a free group is again finitely generated.
 

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. I shall begin by explaining why such groups are not arbitrary and then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler. Work of Bridson, Howie, Miller and Short reduces this to the case of subgroups which are not of type $\mathcal{F}_r$ for some $r$. We will give a new construction producing Kähler groups with exotic finiteness properties by mapping products of closed Riemann surfaces onto an elliptic curve. We will then explain how this construction can be generalised to higher dimensions. This talk is independent of last weeks talk on Kähler groups and all relevant notions will be explained.

A Kähler group is a group which can be realised as the fundamental group of a close Kähler manifold. We will prove that for a Kähler group $G$ we have that $G$ is residually free if and only if $G$ is a full subdirect product of a free abelian group and finitely many closed hyperbolic surface groups. We will then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler: We explain how to construct subgroups of direct products of surface groups which have even first Betti number but are not Kähler. All relevant notions will be explained in the talk.

We will explain a result of Bridson, Howie, Miller and Short on the finiteness properties of subgroups of direct products of surface groups. More precisely, we will show that a subgroup of a direct product of n surface groups is of finiteness type $FP_n$ if and only if there is virtually a direct product of at most n finitely generated surface groups. All relevant notions will be explained in the talk.

 

I will describe a method to find negatively curved structures on some groups, by manipulating metrics on piecewise hyperbolic complexes. As an example, I will prove that hyperbolic limit groups are CAT(-1).

The simplicial boundary is another way to study the boundary of CAT(0) cube complexes. I will define this boundary introducing the relevant terminology from CAT(0) cube complexes along the way. There will be many examples and many pictures, hopefully to help understanding but also to improve my (not so great) drawing skills. 

I will discuss the circumstances in which residual finiteness properties of an amalgamated free product $A\ast_c B$ may be deduced from the properties of $A$ and $B$, with particular regard to the pro-p residual properties.