On the spread of infinite groups

Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated. Thompson's group $V$ was the first finitely presented infinite simple group to be discovered. The Higman--Thompson groups $V_n$ and the Brin--Thompson groups $mV$ are two families of finitely presented groups that generalise $V$. In this paper, we prove that all of the groups $V_n$, $V_n'$ and $mV$ are $\frac{3}{2}$-generated. As far as the authors are aware, the only previously known examples of infinite noncyclic $\frac{3}{2}$-generated groups are the pathological Tarski monsters. We conclude with several open questions motivated by our results.

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

Let W W be an infinite irreducible Coxeter group with ( s 1 , … , s n ) (s_1, \ldots , s_n) the simple generators. We give a short proof that the word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s n ⋯ s 1 s 2 ⋯ s n s_n \cdots s_1 s_2 \cdots s_n is reduced for any number of repetitions of s 1 s 2 ⋯ s n s_1 s_2 \cdots s_n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 w_0 which is obtained from the semi-infinite word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s n ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots by interchanging commuting elements and taking a prefix.

On Thompson's Theorem

Variants of some of the Brauer-Fowler theorems

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O’Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional $$\mathbb {C}[G]$$ -modules where the McKay–Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on $$\varGamma _0({{\mathrm{ord}}}(g))$$ . As there are only finitely many normalized Hauptmoduln, groups whose McKay–Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level $${{\mathrm{ord}}}(g)$$ as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include $$(\mathbb {Z}/ 5 \mathbb {Z})^5$$ and $$(\mathbb {Z}/ 7 \mathbb {Z})^4$$ , which are not subgroups of the Monster.

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

This thesis presents recent developments concerning two open problems related to algebra and discrete geometry. In 1934, Harold S. M. Coxeter introduced Coxeter groups as abstractions of groups generated by reflections in a vector space [Cox34]. The present work lays out how a study of geometric and combinatorial properties of Coxeter groups contributed to the comprehension of the two open problems. Furthermore, this thesis presents some progress in proving the first problem, and an application of the approach introduced in this thesis for the second problem. Open Problem (Lattice for infinite Coxeter groups, Dyer [Dye11]). Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? In Chapter 2, we study the asymptotical behaviour of roots of infinite Coxeter groups, which is part of a joint work with Christophe Hohlweg and Vivien Ripoll [HLR13]. In particular, we show that the directions of roots tend to the isotropic cone of the geometric representation of the root system. Moreover, using this framework, this thesis presents a proof that there is a complete ortholattice structure enclosing the weak order of infinite Coxeter groups of rank at most 3. Open Problem (Existence of multiassociahedra, Jonsson [Jon05]). Let k be an integer such that k >= 1. Is there a polytope whose boundary complex corresponds to the simplicial complex of sets of diagonals of a convex polygon not containing k + 1 mutually crossing diagonals? In Chapter 3 we introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. This family generalizes the concept of multitriangulations of type A and B to arbitrary finite Coxeter groups. For k = 1, this simplicial complex coincides with the finite cluster complex of the given type. We study combinatorial and geometric properties of multi-cluster complexes. In particular, we show that every spherical subword complex is the link of a face of a multi-cluster complex. This work was realized jointly with Cesar Ceballos and Christian Stump [CLS13]. Finally, this approach allows us to exhibit formulas counting the number of common vertices of permutahedra and generalized associahedra for arbitrary finite Coxeter groups and Coxeter elements.

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition. In fact, express order of a group and order of an element of a group in real numbers. Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element am∈G in which an= e= identity element, then find the least common multiple (i.e.(LCM))= λ) of m and n. The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers. So O(am )= λ/m. Also, if G is a finite group, n is a positive integer, and a∈G then the order of the products na. When G is a finite group, every element must have finite order, but the converse is false. There are infinite groups where each element has finite order. Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.

We construct an explicit infinite family of pairwise non‐isomorphic infinite simple groups of type (in particular, they are finitely presented) that act faithfully on the circle by orientation‐preserving homeomorphisms, but that admit neither non‐trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest‐skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras.