Past Geometric Group Theory

The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms g and h between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. 

While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. In this talk I will give an overview of what is known about the PCP in hyperbolic groups, nilpotent groups and beyond (joint work with Alex Levine and Alan Logan).

One of the more common ways to study a residually finite group (or its profinite completion) is via breaking it down into a graph of groups in some way. The descriptions of this theory generally found in the literature are highly algebraic and difficult to digest. I will present alternative, more geometric, definitions and perspectives on these theories based on properties of virtually free groups and their profinite completions.

The celebrated Kirchberg-Phillips classification theorem classifies so-called Kirchberg algebras by K-theory. Many examples of Kirchberg algebras can be constructed via the crossed product construction starting from a group action on a compact space. One might ask: When exactly does the crossed product construction produce a Kirchberg algebra? In joint work with Gardella, Geffen, and Naryshkin, we obtained a dynamical answer to this question for a large class of nonamenable groups which we call "groups with paradoxical towers". Our class includes many non-positively curved groups such as acylindrically hyperbolic groups and lattices in Lie groups. I will try to advertise our notion of paradoxical towers, outline how we use it, and pose some open questions.

For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.

Given a group $G$, finding a geometric action of $G$ on a CAT(0) cube complex can be used to say some rather strong things about $G$. Such actions are not always easy to find, however, which makes it useful to have sufficient conditions, both for existence and for non-existence. This talk concerns the latter: we shall see a coarse geometric obstruction to a group admitting a cocompact cubulation. Based on joint work with Zach Munro.

Let $SO(3,R)$ be the $3D$-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Confirming a conjecture by Breuillard and Green, we show that if $A$ is an open subset of $SO(3,R)$ with sufficiently small measure, then $\mu(A^2) > 3.99 \mu(A)$. This is joint work with Chieu-Minh Tran (NUS) and Ruixiang Zhang (Berkeley). 

The Dehn function of a finitely presented group provides a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. By work of Gersten, Holt and Riley the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$. However, we are still far from determining the precise Dehn functions of all nilpotent groups. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every $k>2$, we obtain many pairs of finitely presented $k$-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups.  This talk is based on joint works with García-Mejía, Pallier and Tessera.

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest. 

Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.

We begin with a seemingly simple question: how can one distinguish a surface group from other cyclic amalgamations of two free groups? This question will prompt a (geometrically flavoured) investigation of virtual homological properties of graphs of free groups amalgamated along cyclic edge groups, where surface subgroups play a key role. 

We next turn to study limit groups and residually free groups through their finite quotients, and apply our findings to the study of profinite rigidity within these classes of groups. In particular, we will sketch out why a direct product of free and surface groups cannot have the same finite quotients as any other finitely presented residually free group.

If time permits, we will discuss other possible characterizations of surface groups among limit groups. The talk is based on joint work with Ismael Morales.

Generating n-tuples of a group G, or in other words epimorphisms Fₙ→G are usually studied up to the natural right action of Aut(Fₙ) on Epi(Fₙ,G); here Fₙ is the free group of n generators. The orbits are then called Nielsen classes. It is a classic result of Nielsen that for any n ≥ k there is exactly one Nielsen class of generating n-tuples of Fₖ. This result was generalized to surface groups by Louder.

In this talk the case of Fuchsian groups is discussed. It turns out that the situation is much more involved and interesting. While uniqueness does not hold in general one can show that each class is represented by some unique geometric object called an "almost orbifold covers". This can be thought of as a classification of Nielsen classes. This is joint work with Ederson Dutra.

The study of the fundamental group of an affine manifold has a long history that goes back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact Euclidian affine manifold has a subgroup of a finite index such that every element of this subgroup is translation. The motivation was the study of the symmetry groups of crys- talline structures which are of fundamental importance in the science of crystallography. A natural way to generalize the classical problem is to broaden the class of allowed mo- tions and consider groups of affine transformations. In 1964, L. Auslander in his paper ”The structure of complete locally affine manifolds” stated the following conjecture, now known as the Auslander conjecture: The fundamental group of a compact complete locally flat affine manifold is virtually solvable.

In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”, J.Milnor asked if a free group can be the fundamental group of complete affine flat mani- fold.
The purpose of the talk is to recall the old and to talk about new results, methods and conjectures which are important in the light of these questions .

The talk is aimed at a wide audience and all notions will be explained 1

After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.This a joint work with Dr Jan Spakula.

A parabolic representation of the free group  is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the moduli space of parabolic, discrete and faithful representations. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first  substantive properties of the nondiscrete representations using ergodic properties of the action of a polynomial semigroup and identifying the Riley slice as the ``Julia set’’ of this dynamical system. We prove a supergroup density theorem: given any irreducible parabolic representation of $F_2$ whatsoever, {\em any}  non-discrete parabolic representation has an arbitrarily small perturbation which contains that group as a conjugate.  Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.

To a group theorist ribbon groups look like knot groups, except  that we know everything about knot groups and next to nothing about ribbon groups.

I will talk about an old paper of mine with Peter Teichner where several questions on ribbon groups naturally arise.

Given a group G, one can seek to understand (some of) its subgroups. Centralisers of elements are easy to define, but maybe not so easy to understand: even in such well studied groups as Out(Fn) they are not yet understood in general. I'll discuss recent work with Armando Martino where we extend what is known in Out(Fn), involving a (surprising?) connection to free-by-cyclic groups and their automorphisms as well as working with actions on trees. The strategies seem like they should apply in many more cases, and if time allows I'll discuss ongoing work (with Gilbert Levitt and Armando Martino) exploring these possibilities.

This talk will be an invitation to the study of cubulated groups and their quotients via the tools of cubical small cancellation theory. Non-positively curved cube complexes are a class of cell-complexes whose geometry and combinatorial structure is closely related to the structure of the groups that act nicely on their universal covers. I will tell you a bit about what we know and don’t know about these groups and spaces, and about the tools we have to study their quotients. I will explain some applications of the study of these quotients to producing a large variety of examples of large-dimensional hyperbolic (and non-hyperbolic) groups.

I will explain that all affine isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that box space expanders constructed from higher rank lattices are superexpanders. This is joint work with Mikael de la Salle.

Bounded cohomology is a functional-analytic analogue of ordinary cohomology that has become a fundamental tool in many fields, from rigidity theory to the geometry of manifolds. However it is infamously hard of compute, and the lack of very basic examples makes the overall picture still hard to grasp. I will report on recent progress in this direction, and draw attention to some natural questions that remain open.

In 1977, Milnor formulated the following conjecture: every discrete group of affine transformations acting properly on the affine space is virtually solvable. We now know that this statement is false; the current goal is to gain a better understanding of the counterexamples to this conjecture. Every group that violates this conjecture "lives" in a certain algebraic affine group, which can be specified by giving a linear group and a representation thereof. Representations that give rise to counterexamples are said to be non-Milnor. We will talk about the progress made so far towards classification of these non-Milnor representations.

The talk is based on joint work with Uri Bader. We prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank and avoids any automorphic machinery. If time permits, we describe applications to the cohomology and stability of arithmetic groups (the latter being joint work with Alex Lubotzky and Shmuel Weinberger).

There is a deep analogy between Kleinaian groups and subgroups of the mapping class group. Inspired by this, Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study, producing surprising applications to the geometry of surface group extension and surface bundles.  In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex cocompact if and only if the corresponding surface group extension is Gromov hyperbolic.

Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged.  Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in the mapping class group might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the stabilizer of a multicurve is hierarchically hyperbolic.

An example of a uniformly proper action is the action of a group (or any of its subgroups) on its Cayley graph. A natural question appearing in a paper of Coulon and Osin, is whether the class of groups acting uniformly properly on hyperbolic spaces coincides with the class of subgroups of hyperbolic groups. In joint work with Vladimir Vankov we construct an uncountable family of finitely generated groups which act uniformly properly on hyperbolic spaces. This gives the first examples of finitely generated groups acting uniformly properly on hyperbolic spaces that are not subgroups of hyperbolic groups. We also give examples that are not virtually torsion-free.

In recent years, duality approaches have yielded new results about the high-dimensional cohomology of several groups and moduli spaces, such as $\operatorname{SL}_n(\mathbb{Z})$ and $\mathcal{M}_g$. I will explain the general strategy of these approaches and survey results that have been obtained so far. To give an example, I will first explain how Borel-Serre duality can be used to show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes near its virtual cohomological dimension. This is based on joint work with Miller-Patzt-Sroka-Wilson and builds on results by Church-Farb-Putman. I will then put this into a more general context by giving an overview of analogous results for mapping class groups of surfaces, automorphism groups of free groups and further arithmetic groups such as $\operatorname{SL}_n(\mathcal{O}_K)$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$.

A metric is said to be (globally) median,  if any three points have a unique “median” which  lies  between  any  two  points  from  the  triple.  
Such  spaces  arise  naturally  in  many different contexts.  The property of being locally median can be viewed as a kind of
non-positive curvature condition.  We show that a complete uniformly locally median space is
globally median if and only if it is simply connected.  This is an analogue of the well known Cartan-Hadamard Theorem for non-positively curved manifolds, or more generally CAT(0) spaces.  However it leaves open a number of interesting questions.