Burnside Problem Research Articles

Combinatorial Estimations on Burnside Type Problems

The Burnside problem, formulated by W. Burnside in 1902, is one of the most well-known and important open questions in the field of Group Theory. Despite significant progress made in the past century towards solving this problem, its complete solution remains unknown. In this paper, we investigate one of the approaches to solving the Burnside problem based on the application of an iterative theory of small cancellations and canonical forms developed by E. Rips in recent years. We present a novel self-contained exposition of this theory and utilize it to obtain new estimates on the infiniteness of initial approximations of Burnside groups where only a finite number of periodic relations is used for relatively small odd exponents (n>120).

Groups of small period growth

AbstractWe construct finitely generated groups of small period growth, i.e. groups where the maximum order of an element of word length n grows very slowly in n. This answers a question of Bradford related to the lawlessness growth of groups and is connected to an approximative version of the restricted Burnside problem.

On upper bounds for the first ℓ²-Betti number

This article presents a method for proving upper bounds for the first ℓ 2 \ell ^2 -Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first ℓ 2 \ell ^2 -Betti number. Our approach extends to generalizations of ℓ 2 \ell ^2 -Betti numbers that are defined using characters. We illustrate this flexibility by generalizing results of Peterson-Thom on q-normal subgroups to this setting.

Equivariant Burnside groups and toric varieties

We study G-equivariant birational geometry of toric varieties, where G is a finite group.

On families of nilpotent subgroups and associated coset posets

We study some properties of the coset poset associated with the family of subgroups of class \(\le 2\) of a nilpotent group of class \(\le 3\). We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the group of \(4\times 4\) upper unitriangular matrices over \(\mathbb {F}_p\), and for the Burnside groups of exponent 3.

Equivariant Burnside groups and representation theory

We apply the equivariant Burnside group formalism to distinguish linear actions of finite groups, up to equivariant birationality. Our approach is based on De Concini–Procesi models of subspace arrangements.

POWERS OF SUBSETS IN FREE PERIODIC GROUPS

It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 658n^2$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|>4\cdot 2.9^{[\frac{t}{658s^2}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s\ge1039$.

Graham Higman. 19 January 1917—8 April 2008

Graham Higman was a highly talented mathematician whose principal field of research was group theory. He played a key role in the development of this field, and is regarded as one of the three most significant British group theorists of the 20th century, alongside William Burnside (FRS 1893) and Philip Hall (FRS 1951). He studied at Oxford, gaining a doctorate under the supervision of Henry Whitehead (FRS 1944). After working for the Meteorological Office during the Second World War, he was appointed to a position at the University of Manchester in 1946, and subsequently at Oxford. In 1958 he was elected FRS, and in 1960 he was appointed to the Waynflete professorship in mathematics at Oxford, a position he held until his retirement in 1984. His research was significant and influential. Highlights include his work on embeddings of groups, including ‘HNN-extensions’ and their application in the famous Higman Embedding Theorem, and his constructions of a finitely-presented group that is isomorphic to a proper factor of itself, and a finitely-presented infinite simple group. Some of his work also played an important role in the proof of the Odd Order Theorem and a solution to the restricted Burnside problem. He and his wife Ivah were dedicated Methodists, and Graham was a keen bird-watcher. He was also well known for his view that pure mathematics should be studied for its beauty and its life-affirming properties, for the non-standard yet highly supportive way he supervised his students and for his somewhat mischievous sense of humour.

Equivariant birational types and Burnside volume

We introduce equivariant Burnside groups, new invariants in equivariant birational geometry, generalizing birational symbols groups for actions of finite abelian groups, due to Kontsevich, Pestun, and the second author, and study their properties. We establish a specialization map for the equivariant birational type of a smooth algebraic variety with an action of a finite group.

Combinatorial Burnside groups

We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties.

Product set growth in Burnside groups

Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.

Invariants for metabelian groups of prime power exponent, colorings, and stairs

AbstractWe study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$ . In particular, we improve bounds found by Newman for the order of $M(2,2^k)$ . We study identities in $M(2,n)$ , which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$ .

ON INDEPENDENT SYSTEMS OF DEFINING RELATIONS FOR FREE BURNSIDE GROUPS OF PERIOD 3

We construct systems of independent defining relations for free Burnside groups $B(m,3)$ of ranks $m=2, 3$. The proof for the case $m=2$ is established using the matrix representation of $B(2,3)$. For the case $m=3$ the approach is based on the natural embedding of $B(2,3)$ into $B(3,3)$.

General Burnside problem for locally compact groups

A locally compact group G is called compactly covered (respectively compactly ruled) if it is an union of compact subgroups (resp. if it is a directed union of compact subgroups). The class of all compactly ruled groups is a proper subclass of the class of all compactly covered groups. In this paper we shall study the following question: For which classes of locally compact groups do the following holds:

The restricted Burnside problem for Moufang loops

AbstractWe prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

Random triangular Burnside groups

We introduce a model for random groups in varieties of n-periodic groups as n-periodic quotients of triangular random groups. We show that for an explicit dcrit ∈ (1/3, 1/2), for densities d ∈ (1/3, dcrit) and for n large enough, the model produces infinite n-periodic groups. As an application, we obtain, for every fixed large enough n, for every p ∈ (1, ∞) an infinite n-periodic group with fixed points for all isometric actions on Lp-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindrically hyperbolic.

Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian

This is a survey of results on the Burnside problem and properties of Burnside groups, the finite basis problem for group identities, periodic products of groups and Malcev’s problem, construction of groups with special properties (Tarski monsters), constructive bounds in the Burnside- Magnus problem, and algorithmic problems: the problem of recognition of group properties, the word problem for semigroups with one relation, and semi-Thue systems. The focus is on the most important results obtained in papers of Adian and his students. Bibliography: 81 titles.

The fibered Burnside group of a fusion system

We apply Reeh’s methods of determining fusion-stable finite sets together with Boltje’s integrality conditions to determine stable elements in the [Formula: see text]-fibered Burnside group of a [Formula: see text]-group under the action of a saturated fusion system [Formula: see text].

Faltings-Serre method on three dimensional selfdual representations

We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

Collectives of Automata in Finitely Generated Groups

The present paper is devoted to traversing a maze by a collective of automata. This part of automata theory gave rise to a fairly wide range of diverse problems ([1], [2]), including those related to problems of the theory of computational complexity and probability theory. It turns out that the consideration of complicated algebraic objects, such as Burnside groups, can be interesting in this context. In the paper, we show that the Cayley graph a finitely generated group cannot be traversed by a collective of automata if and only if the group is infinite and its every element is periodic.