Tag - Group theory

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.

Florian Breuer: Realizing general linear groups as Galois groups

23rd November, 2020

I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed.

William Hautekiet: Automorphism groups of transcendental field extensions

23rd November, 2020

It is well-known that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made.

Sean Eberhard: Diameter of high-rank classical groups with random generators

22nd November, 2020

Babai's conjecture asserts that the diameter of the Cayley graph of any finite simple group G is bounded by (log |G|)O(1). This conjecture has been resolved for groups of bounded rank, but for groups of unbounded rank such as SLn(2) it is wide open. Even for random generators, only the case of alternating groups is resolved. In this talk we sketch the proof of Babai's conjecture for SLn(p), p = O(1), with at least three random generators. The proof extends to other classical groups over 𝔽q if we have at least q100 random generators. The heart of the proof consists of showing that the Schreier graph of SLn(q) acting on 𝔽qn with respect to q100 random generators is an expander graph.

George Domat: Free products from spinning and rotating families

19th November, 2020

A natural goal of geometric group theory is to understand the algebraic properties of a group via geometry. The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup in a large ambient group. In particular, their work gives conditions under which the normal closure is a free product. I will talk about recent work that aims to unify their results and gives a significantly shorter proof of the theorem of DGO. This is joint work with M. Bestvina, R. Dickmann, S. Kwak, P. Patel, and E. Stark.

Alex Evetts: Introduction to growth in groups

19th November, 2020

An LMS online lecture course in growth in groups.

For a finitely generated group, the number of elements that can be spelled with words of length n, for any integer n>0, is called the growth function. This can be interpreted as a measure of the size of the group and is a powerful quasi-isometry invariant which has links to many areas of geometric group theory.

In the first lecture I will present the fundamental properties of the growth function and explore some key examples illustrating what kinds of functions can arise. I will also discuss Gromov's important theorem on groups of polynomial growth.

In the second lecture I will discuss the formal power series associated to the growth function, which is known as the growth series. I will explain some ways in which the behaviour of the growth series can provide insight into the asymptotics, and demonstrate this with examples.

Doron Puder: Random permutations sampled by free words

16th November, 2020

Fix a word w in a free group on r generators. A w-random permutation in the symmetric group SN is obtained by sampling r independent uniformly random permutations σ1, . . .,σr ∈ SN and evaluating w(σ1, . . .,σr). Such w-random permutations have surprisingly rich structure with relation to deep results in geometric group theory. I'll survey some of this structure, state some conjectures, and explain how it is related to evaluating the spectral gap of random Schreier graphs of SN.

Wenhao Wang: Dehn Functions of Finitely Presented Metabelian Groups

12th November, 2020

The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group G is 2n2k, where k is the torsion-free rank of the abelianization Gab, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.

Marcos Mazari-Armida: Model-theoretic stability in classes of modules

12th November, 2020

Dividing lines in complete first-order theories were introduced by Shelah in the early seventies. A dividing line is a property such that the classes satisfying such a property have some nice behaviour while those not satisfying it have a bad one. Two of the best understood dividing lines are those of stability and superstability.

In this talk, I will study the notion of stability and superstability in abstract elementary classes of modules with respect to pure embeddings, i.e., classes of the form (K,≤p) where K is a class of R-modules for a fixed ring R and ≤p is the pure submodule relation. In particular, using that the class of p-groups with pure embeddings is a stable AEC, I will present a solution to Problem 5.1 in page 181 of Abelian Groups by László Fuchs. Moreover, I will show how the notion of superstability can be used to give new characterizations of noetherian rings, pure-semisimple rings, and perfect rings.