Tag - Hyperbolic groups

Sahana Balasubramanya: Acylindrically and relatively hyperbolic groups

28th June, 2021

The class of acylindrically hyperbolic groups has been of immense interest in recent times. It is an extremely large class of groups, containing many interesting examples. Yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context. The goal of this mini course is to provide an introduction to this class of groups, and focus on some important techniques.

In the first lecture, we will define acylindrical actions and talk about the motivation to study them. We will then define acylindrically hyperbolic groups and discuss some examples and properties of this class of groups. The second lecture will focus on the notion of hyperbolically embedded subgroups and discuss relative hyperbolicity in this context. In the last lecture, we will discuss the concept of group theoretic Dehn filling and some of its applications. Time permitting, I will also talk a bit about my own research.

Eduardo Oregón-Reyes: Cubulated groups and virtual specialness

21st June, 2021

CAT(0) cube complexes were introduced by Gromov merely as examples of metric spaces of non-positive curvature, but now they play a prominent role in geometric group theory. One reason for this is that many interesting groups are known to act nicely on these spaces, including free and surface groups, small cancellation groups, 1-relator groups with torsion, and many 3 manifold groups. Another reason is that some of these groups are, in addition, virtually special, notion defined by Haglund and Wise that implies being (up to finite index) the subgroup of some right-angled Artin group.

In the first lecture, we will define CAT(0) cube complexes, explore some of their combinatorial structure, and discuss some examples of cubulated groups. For the second lecture, we will introduce the class of virtually special groups, review some of their properties, and mention some criteria for virtual specialness. We will end the mini-course with a discussion of the main techniques for studying cubulated hyperbolic groups, focusing on some theorems of Wise and Agol. If time permits, I will mention a few things about the relatively hyperbolic case.

Benjamin Fine: Elementary and universal theories of group rings

6th May, 2021

In a series of papers the above authors examined the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R. Here we assume that R is a commutative ring with identity 1 ≠ 0. Of course, these are relative to an appropriate logical language L0, L1, L2 for groups, rings and group rings respectively. Axiom systems for these were provided. Kharlampovich and Myasnikov, as part of the proof of the Tarskii theorems, prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. We examined and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers ℤ then R[G] is universally equivalent to ℤ[F] again with respect to an appropriate language. It was proved that if R[G] is elementarily equivalent to S[H] with respect to L2, then simultaneously the group G is elementarily equivalent to the group H with respect to L0, and the ring R is elementarily equivalent to the ring S with respect to L1.

The structure of group rings is related to the Kaplansky zero-divisor conjecture. A Kaplansky group is a torsion-free gorup which satisfies the Kaplansky conjecture. We next show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We have that 𝒦 the class of Kaplansky groups is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in 𝒦 or more generally two torsion-free groups are universally equivalent.

Finally we consider F to be a rank 2 free group and ℤ be the ring of integers. we call ℤ[F] a free group ring. Examining the universal theory of the free group ring ℤ[F] the hazy conjecture was made that the universal sentences true in ℤ[F] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in ℤ[F] modified appropriately for group theory. We prove that this conjecture is true in terms of axiom systems for ℤ[F].

Ilya Kapovich: Non-linear words and free groups

29th April, 2021

An important theme in the study of combinatorics of words involves looking for models of nonlinear words, that is words that are not indexed by segments of integers. We discuss one such model arising from the theory of Stallings subgroup graphs. This model naturally leads to the notion of subset currents on free groups (and on other word-hyperbolic groups) which are measure-theoretic analogs of conjugacy classes of finitely generated subgroups. Many new features manifest themselves in this context, including connections with the Hanna Neumann Conjecture and Whitehead's algorithm for subgroups.

Oren Becker: Stability of approximate group actions

11th April, 2021

An approximate unitary representation of a group G is a function f from G to U(n) such that f(gh) is close to f(g)f(h) for all g,h. Is every approximate unitary representation just a slight deformation of a unitary representation? The answer depends on G and on the norm on U(n). If G is amenable, the answer is positive for the operator norm on U(n) (Kazhdan '82). The answer remains positive if we use the normalized Hilbert-Schmidt norm and allow a slight change in the dimension n (Gowers-Hatami '15, De Chiffre-Ozawa-Thom '17). For both norms, the answer is negative if G is a non-abelian free group (or a non-elementary word-hyperbolic group). In this talk we shall discuss a similar notion where U(n) is replaced by Sym(n) with the normalized Hamming metric. We study the cases where G is either free, amenable or equal to SLr(ℤ), r ≥ 3. When G is finite, a slight variation of our main theorem provides an efficient probabilistic algorithm to determine whether a function f from G to Sym(n) is close to a homomorphism when |G| and n are both large.

Charlotte Hoffmann: Short words of high imprimitivity rank yield hyperbolic one-relator groups

8th March, 2021

It is a long-standing question whether a group of type F that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type F and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than 2, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most 17. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I'll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.

Anna Erschler: Ordering Ratio Function and Travelling Salesman Breakpoint for Groups and Metric Spaces

4th December, 2020

(Abstract taken from arXiv paper.) We study asymptotic invariants of metric spaces and infinite groups related to the universal Travelling Salesman Problem (TSP). We prove that spaces with doubling property (in particular virtually nilpotent groups) admit Gap for Ordering Ratio functions which holds for all orders on these spaces. We describe Travelling Salesman Breakpoint for finite graphs. We characterize groups with Travelling Salesman Breakpoint ≤ 3 as virtually free ones. We show that Ordering Ratio function is bounded (which is the best possible situation) for all uniformly discrete δ-hyperbolic spaces of bounded geometry, in particular for all hyperbolic groups. We prove that any metric space, containing weakly a sequence of arbitrarily large cubes, has infinite Travelling Salesman Breakpoint; this means that any order on such spaces satisfies OR(s)=s for all s. This is the worst possible case for Ordering Ratio functions. For a sequence of finite graphs, we provide a sufficient spectral condition for OR(s)=s. This condition is in particular satisfied for any sequence of expander graphs. Under this stronger assumption of being a family of expander graphs, we prove a stronger claim about snakes of bounded width. We show that any metric space of finite Assouad-Nagata dimension admits an order satisfying OR(s) ≤ Const ln s, and discuss general Gap Problems for Ordering Ratio functions.

Abdul Zalloum: Regularity of Morse geodesics and growth of stable subgroups

17th September, 2020

The study of groups with "hyperbolic-like directions" has been a central theme in geometric group theory. Two notions are usually used to quantify what is meant by "hyperbolic-like directions", the notion of a contracting geodesic and that of a Morse geodesic. Since the property that every geodesic ray in metric space is contracting or Morse characterizes hyperbolic spaces, being a contracting/Morse geodesic is considered a hyperbolic-like property. Generalizing work of Cannon, I will discuss a joint result with Eike proving that for any finitely generated group, the language of contracting geodesics with a fixed parameter is a regular language. This immediately implies that contracting geodesics can't exist in torsion groups.

The Morse notion is a weaker notion than that of the contracting notion, in fact, building on work of Osin, Ol’shanskii, and Sapir, Fink gave an example of a torsion group which contains an infinite Morse geodesic. This seems to contradict the claim that Morse geodesics are "hyperbolic-like" directions. As an attempt to rectify this, Russell, Spriano, and Tran introduced a class of spaces where Morse (quasi)-geodesics satisfy some local-to-global property and they showed that many interesting examples live in such a class. In these spaces, Morse (quasi)-geodesics are expected to behave more reasonably like "hyperbolic directions", therefore, such spaces/groups can be regarded as good hosts of Morse (quasi)-geodesics.

I will discuss some continuation of their work where we show that in such spaces Morse geodesics form a regular language, give a characterization of stable subgroups in terms of regular languages. Time permitting, I will discuss few other applications of these automatic structures to the growth of stable subgroups and the dynamics of the action of such groups on their Morse boundaries. This work is joint with Cordes, Russell and Spriano.

Mark Hagen: Hierarchical hyperbolicity from actions on simplicial complexes

5th June, 2020

The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition - which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes - is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups.