On the Burnside problem on periodic groups

Small cancellation theory and Burnside problem

In these notes we detail the geometrical approach of small cancellation theory used by T. Delzant and M. Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

Permutation groups of prime degree, a quick proof of Burnside's theorem

A transitive permutation group of prime degree is doubly transitive or solvable. We give a direct proof of this theorem by Burnside which uses neither S-ring type arguments, nor representation theory.

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 $d_{\mathrm{crit}}\in(1/3,1/2)$, for densities $d\in(1/3,d_{\mathrm{crit}})$ and for $n$ large enough, the model produces \emph{infinite} $n$-periodic groups. As an application, we obtain, for every fixed large enough $n$, for every $p\in (1,\infty)$ an infinite $n$-periodic group with fixed points for all isometric actions on $L^p$-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindricall…

A criterion for detecting trivial elements of Burnside groups

In this article we give a sufficient and necessary condition to determine wether or not an element of the free group induces a non-trivial element of the free Burnside group of sufficiently large odd exponent. This criterion can be stated without any knowledge about Burnside groups, in particular about the proof of its infiniteness. Therefore it provides a useful tool that we will use later to study outer automorphisms of Burnside groups. We also state an analogue result for periodic quotients of torsion-free hyperbolic groups.

Solution to the Burnside Problem

The Burnside Problem asks whether a finitely generated group of exponent n is finite. We present a solution for 2-generator groups of prime power exponent. Results of P. Hall and G. Higman extends the finiteness conclusion to groups having composite exponents. Our main result, called the Generalized Burnside Theorem, is a solvability theorem that applies to a family of groups called GB (Generalized Burnside) groups that contain infinite as well as finite groups. The final section discusses the extension to k-generator groups although details are left for another time.

Splitting automorphisms of prime power orders of free Burnside groups

We prove that if the order of a splitting automorphism of free Burnside group~$B(m,n)$ of odd period~$n\ge1003$ is a prime power, then the automorphism is inner. Thus, we give an affirmative answer to the question on the coincidence of splitting and inner automorphisms of free Burnside groups $B(m,n)$ for automorphisms of orders~$p^k$ ($p$ is a prime number). This question was posed in the Kourovka Notebook in 1990 (see 11th ed., Question 11.36.~b).

A sample iterated small cancellation theory for groups of Burnside type

We develop yet another technique to present the free Burnside group $B(m,n)$ of odd exponent $n$ with $m\ge2$ generators as a group satisfying a certain iterated small cancellation condition. Using the approach, we provide a reasonably accessible proof that $B(m,n)$ is infinite with a moderate bound $n > 2000$ on the odd exponent $n$.

On subgroups of free Burnside groups of large odd exponent

We prove that every noncyclic subgroup of a free $m$-generator Burnside group $B(m,n)$ of odd exponent $n \gg 1$ contains a subgroup $H$ isomorphic to a free Burnside group $B(\infty,n)$ of exponent $n$ and countably infinite rank such that for every normal subgroup $K$ of $H$ the normal closure $<K >^{B(m,n)}$ of $K$ in $B(m,n)$ meets $H$ in $K$. This implies that every noncyclic subgroup of $B(m,n)$ is SQ-universal in the class of groups of exponent $n$.

Spectrum of the Burnside Tambara functor on a cyclic $p$-group

For a finite group $G$, a Tambara functor on $G$ is regarded as a $G$-bivariant analog of a commutative ring. In this analogy, previously we have defined an ideal of a Tambara functor. In this article, we will demonstrate a calculation of the prime spectrum of the Burnside Tambara functor, when $G$ is a cyclic $p$-group for a prime integer $p$.

Embedding free Burnside groups in finitely presented groups

We construct an embedding of a free Burnside group $B(m,n)$ of odd $n > 2^{48}$ and rank $m >1$ in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent $n \gg 1$ which are not locally finite.

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.

Uniform non--amenability of free Burnside groups

We show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann -- Day problem concerning amenability of groups without non--abelian free subgroups. As another consequence, we obtain that the above--mentioned groups are of uniform exponential growth.

A straightforward solution to the Burnside Problem

We present a solution to the Burnside Problem for 2 generator groups of prime-power exponent that does not rely on induced maps as in [2]. As before, we construct a surjective map of a rank 2 free group to a solvable group G and finish by showing that the Burnside group is an image of G. Theorem B in the paper with H. A. Heilbronn and H. Y. Mochizuki [9] is indispensable in the proof.

Automorphism tower problem and semigroup of endomorphisms for free Burnside groups

We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower…

Twisted Burnside Theory for the Discrete Heisenberg Group and for Wreath Products of Some Groups

The RP-property of Fel'shtyn and Troitsky is proved for wreath products of finitely generated Abelian groups with the group of integers. Such wreath products become the first known example of finitely generated RP-groups being not almost polycyclic.

The functor of units of Burnside rings for p-groups

In this note I describe the structure of the biset functor $B^\times$ sending a $p$-group $P$ to the group of units of its Burnside ring $B(P)$. In particular, I show that $B^\times$ is a rational biset functor. It follows that if $P$ is a $p$-group, the structure of $B^\times(P)$ can be read from a genetic basis of $P$: the group $B^\times(P)$ is an elementary abelian 2-group of rank equal to the number isomorphism classes of rational irreducible representations of $P$ whose type is trivial, cyclic of order 2, or dihedral.

Random Burnside Groups

We show that there exists a positive number $M_0$ such that for any odd $M\geq M_0$ a random group of exponent $M$ with overwhelming probability is infinite in the few relator model and in the density $d$ model for small $d$.

Burnside Rings of Fusions Systems and their Unit Groups

For a saturated fusion system $\mathcal F$ on a $p$-group $S$, we study the Burnside ring of the fusion system $B(\mathcal F)$, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring $B(S)$. We give criteria for an element of $B(S)$ to be in $B(\mathcal F)$ determined by the $\mathcal F$-automorphism groups of essential subgroups of $S$. When $\mathcal F$ is the fusion system induced by a finite group $G$ with $S$ as a Sylow $p$-group, we show that the restriction of $B(G)$ to $B(S)$ has image equal to $B(\mathcal F)$. We also show that for $p=2,$ we can gain infor…

On the unit groups of partial Burnside rings

We describe the generalized Matsuda's theorem, and some results of a Burnside ring extend a partial Burnside ring. In particular, we give isomorphism between partial Burnside rings of different groups. Moreover, we consider the relationship between image of Frobenius-Wielandt homomorphism, a partial Burnside ring, and structure of a group.

The Generalized Burnside Theorem

All groups have 2 generators. For every prime power q, the Generalized Burnside Theorem (Theorem GB) produces an infinite number of solvable groups, Some, such as groups of a prime power exponent, have only elements of finite order and are therefore finite groups. Others have elements of infinite order and are thus infinite groups. All these groups, even when infinite, are closely related to groups of exponent q. They have at least one generator of order q, and their commutator subgroup has exponent q.

The ideals of the slice Burnside p-biset functor

Let G be a finite group and K be a field of characteristic zero. Our purpose is to investigate the ideals of the slice Burnside functor KΞ. It turns out that they are the subfunctors F of KΞ such that for any finite group G, the evaluation F(G) is an ideal of the algebra KΞ(G). This allows for a determination of the full lattice of ideals of the slice Burnside p-biset functor KΞp.

Simple biset functors and double Burnside rings

Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(-,H) by the radical of the bilinear form is a semi-simple functor. This allows for a description of the evaluation of simple functors, hence of simple modules for the double Burnside ring.

Subgroups of groups finitely presented in Burnside varieties

For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a presentation $G=\langle A\mid R\rangle$ with a finite set of generators $A$ and a recursively enumerable set $R$ of defining relations if and only if it is a subgroup of a group $H$ finitely presented in the variety ${\cal B}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\cal B}_n$ group containing isomorphic copies of all fin…

Burnside problem for groups of homeomorphisms of compact surfaces

A group $Γ$ is said to be periodic if for any $g$ in $Γ$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $\mathrm{O}(3,\R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose …

Yet another solution to the Burnside problem for matrix semigroups

We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

Profinite completions of Burnside-type quotients of surface groups

Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.

Connected reversible Mealy automata of prime size cannot generate infinite Burnside groups

The simplest example of an infinite Burnside group arises in the class of automaton groups. However there is no known example of such a group generated by a reversible Mealy automaton. It has been proved that, for a connected automaton of size at most~3, or when the automaton is not bireversible, the generated group cannot be Burnside infinite. In this paper, we extend these results to automata with bigger stateset, proving that, if a connected reversible automaton has a prime number of states, it cannot generate an infinite Burnside group.

Mazurov关于自由Burnside群自同构不动点的问题

Mazurov's question on fixed points of automorphisms of free Burnside groups

本文否定回答了Mazurov关于自由Burnside群自同构不动点的问题，证明对任意奇数m≥3及足够大的素数周期q，循环置换生成元的m阶自同构无不动点。

Local-global invariants of finite and infinite groups: around Burnside from another side

This expository essay is focused on the Shafarevich-Tate set of a group $G$. Since its introduction for a finite group by Burnside, it has been rediscovered and redefined more than once. We discuss its various incarnations and properties as well as relationships (some of them conjectural) with other local-global invariants of groups.

A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, there are some classes of automata that do not contain such examples. For instance, all infinite Burnside automaton groups in the literature are generated by non reversible Mealy automata and it was recently shown that 2-state invertible-reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state invertible-reversible Mealy automata, using new original techniques. The results provide the first uniform method to construct elements …

Solvable matrix groups and the Burnside problem

All groups are 2-generator. For any prime-power q, Theorem 1 constructs a solvable matrix group over a quotient of a Laurent polynomial ring. This group is closely related to a group of exponent q as shown in Theorems 2 & 3 . Theorem 4 in section 5 shows that a group of prime-power exponent contains the relations of a solvable group. It follows that the Burnside groups of exponent q are solvable, and it is easy to deduce that the solvability class of these groups tends to infinity with q. Crucial to this work, especially for precise bounds on the solvability class, is an earlier paper with Hei…

The Dixmier problem, lamplighters and Burnside groups

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.

A note on the lambda-structure on the Burnside ring

Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an explicit formula, expressing these images as linear combinations of classes of G-sets.

Certain Applications of the Burnside Rings and Ghost Rings in the Representation Theory of Finite Groups (II)

Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent $A(G)$ of finite $p$-groups was obtained in a previous article of the first-named author. In this paper, we compute $A(G)$ for any finite group $G$ -- hence providing the global version of that result.

The slice Burnside ring and the section Burnside ring of a finite group

This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of $G$-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural structure of Green biset functor. The functorial structure of unit groups of these rings is also discussed.

概率 Burnside 群

Probabilistic Burnside groups

本文构造了一个有限生成群，其随机元素以趋于 1 的概率满足某个固定群律，但该群包含非交换自由子群，因而不满足任何群律，从而回答了 Amir 等人提出的两个问题。

Surjective word maps and Burnsides $p^aq^b$ theorem

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map sending (x,y) to the product of the Nth powers of x and y is surjective on every finite non-abelian simple group; If is an odd integer, then the word map sending the triple (x,y,z) to the product of the Nth powers is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit-Thompson. We also prove asymptotic results about the surjectivity of the word map sending (x,y) to the …

Understanding Wall's theorem on dependence of Lie relators in Burnside groups

G.E. Wall gave two different proofs of a remarkable result about the multilinear Lie relators satisfied by groups of prime power exponent $q$. He showed that if $q$ is a power of the prime $p$, and if $f$ is a multilinear Lie relator in $n$ variables where $n\neq1\operatorname{mod}(p-1)$, then $f=0$ is a consequence of multilinear Lie relators in fewer than $n$ variables. For years I have struggled to understand his proofs, and while I still have not the slightest clue about his first proof published in the Journal of Algebra, I finally have some understanding of his second proof published in …

非交换限制伯恩赛德问题

The anabelian restricted Burnside problem

本文证明了由非平凡群律定义的簇中相对自由群存在最大非交换有限商，并利用有限单群律长度的显式界估计其大小。

The Burnside groups and small cancellation theory

In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. The present article gives a brief introduction to the methods and techniques involved in the proofs. Many of the ideas originate with the recent work of A. Yu. Olshanskii and S. V. Ivanov. Some familiarity with their work on the Burnside problem is assumed.

The Burnside problem for locally compact groups

Using topological notions of translation-like actions introduced by Schneider, we give a positive answer to a geometric version of Burnside problem for locally compact group. The main theorem states that a locally compact group is non-compact if and only if it admits a translation-like action by the group of integers $\mathbf Z$. We then characterize the existence of cocompact translation-like actions of $\mathbf Z$ or non-abelian free groups on a large class of locally compact groups, improving on Schneider's results and generalising Seward's.

The Burnside problem for $\text{Diff}_{\text{Vol}}(\mathbb{S}^2)$

Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every periodic group of bounded exponent $G \subset \text{Diff}_{\text{Vol}}(S)$ is a finite group.

部分伯恩赛德环的张量积与单位元

Tensor products and units of partial Burnside rings

研究有限群子群集合相关的部分伯恩赛德环张量积，给出典范同态为满射的充要条件，描述可约有限考克斯特群符号单位分解。

关于流形同胚群的Burnside问题的注记

A note on the Burnside problem for homeomorphism groups of manifolds

本文研究紧连通流形同胚群的Burnside问题，证明表面非例外时同胚群恒等分量torsion-free，扩展映射类群Tits猜想，恢复并扩展Guelman和Liousse定理至非定向表面，证明圆同胚群有限生成周期子群为有限cyclic，并讨论有边界的流形及双柄体的开放问题。

Burnside's Problem, spanning trees, and tilings

In this paper we study geometric versions of Burnside's Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric version of the von Neumann Conjecture by showing that a finitely generated group is non-amenable if and only if it admits a translation-like action by any (equivalently every) non-abelian free group. We strengthen Whyte's result by proving that this translation-like action can be chosen to be transitive …

Unexpected connections between Burnside Groups and Knot Theory

In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our approach uses a new concept, Burnside groups of links, which establishes unexpected relationship between Knot Theory and Group Theory. Our method has the potential to be used in computational biology in the analysis of DNA via tangle embedding theory, as developed by D.W.Sumners.

Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this note we explain these flaws and prove that every cyclic group of composite order is a B-grou…

An algorithm for the unit group of the Burnside ring of a finite group

In this note we present an algorithm for the construction of the unit group of the Burnside ring $Ω(G)$ of a finite group $G$ from a list of representatives of the conjugacy classes of subgroups of G.

Central extensions of free periodic groups

It is proved that any countable abelian group $D$ can be embedded as a centre into a $m$-generated group $A$ such that the quotient group $A/D$ is isomorphic to the free Burnside group $B(m,n)$ of rank $m>1$ and of odd period $n\ge665$. The proof is based on some modification of the method which was used by S.I.Adian in his monograph in 1975 for a positive solution of Kontorovich's famous problem from Kourovka Notebook on the existence of a finitely generated non-commutative analogue of the additive group of rational numbers. More precisely, he proved that the desired analogues in which the in…

辫群与伯恩赛德群

Braid groups and Burnside groups

研究辫群$\text{Br}_4(d)$到$\text{Br}_3(d)$的特殊商群，用纯群论方法找到截断商映射核的表示形式为“本原”伯恩赛德群，并给出其有限性结果，还对相关解释作了评论。

Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms

A group $G$ is said to be periodic if for any $g\in G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $μ$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.

A new look at the Burnside-Schur theorem

The famous Burnside-Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latters we introduce Schur rings over a finite commutative ring and prove an analog of this statement for them. Besides, the finite local commutative rings are characterized in the permutation group terms.

Prime ideals of the Burnside ring of a saturated fusion system

In a 1969 article, A. Dress described the prime ideals of the Burnside ring of a finite group G and the inclusion relations between them. One may ask whether similar results exist for the Burnside ring of a saturated fusion system F on a finite p-group S. The present paper answers positively, providing a typology for the prime ideals of A(F) and for those of A(F) (p) as a consequence.

Orthogonal units of the bifree double Burnside ring

The bifree double Burnside ring $B^Δ(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B^Δ_\circ(G,G)$ of orthogonal units in $B^Δ(G,G)$. It is shown that this group is always finite and contains a subgroup isomorphic to $B(G)^\times\rtimes \Out(G)$, where $B(G)^\times$ denotes the unit group of the Burnside ring of $G$ and $\Out(G)$ denotes the outer automorphism group of $G$. Moreover it is shown that if $G$ is nilpotent then $B^Δ_\circ(G,G)\cong B(G)^\times\rtimes \Out(G)$. The results can be interpreted as positive answers to questions on equivalences of $p$-bl…