The Tits Alternative for Groups Defined by Periodic Paired Relations

The class of groups defined by periodic paired relations, as introduced by Vinberg, includes the generalized triangle groups, the generalized tetrahedron groups, and the generalized Coxeter groups. We observe that any group defined by periodic paired relations Γ can be realized as a so-called “Pride group”. Using results of Howie and Kopteva we give necessary and sufficient conditions for this Pride group to be non-spherical. Under such conditions, we show that Γ satisfies the Tits alternative.

Communicated by A. Olshanskiy     

The Word Problem for Pride Groups

Pride groups are defined by means of finite (simplicial) graphs, and examples include Artin groups, Coxeter groups, and generalized tetrahedron groups. Under suitable conditions, we calculate an upper bound of the first order Dehn function for a finitely presented Pride group. We thus obtain sufficient conditions for when finitely presented Pride groups have solvable word problems. As a corollary to our main result, we show that the first order Dehn function of a generalized tetrahedron group, containing finite generalized triangle groups, is at most cubic.     

Extension of Some Generalized Hermite-Type Interpolation Operators to the Triangle with One Curved Side

Abstract

We construct some generalized Hermite-type interpolation operators for the case of a triangle with one curved side, and we consider their product and Boolean sum operators. We study the interpolation properties of these operators, the orders of accuracy and the remainders of the interpolation formulas. Finally, we give some numerical examples.     

Subgroups of Word Hyperbolic Groups in Dimension 2

If G is a word hyperbolic group of cohomological dimension 2,then every subgroup of G of type FP₂ is also word hyperbolic.Isoperimetric inequalities are defined for groups of type FP₂and it is shown that the linear isoperimetric inequality inthis generalized context is equivalent to word hyperbolicity.A sufficient condition for hyperbolicity of a general graphis given along with an application to ‘relative hyperbolicity’.Finitely presented subgroups of Lyndon's small cancellationgroups of hyperbolic type are word hyperbolic. Finitely presentedsubgroups of hyperbolic 1-relator groups are hyperbolic. Finitelypresented subgroups of free Burnside groups are finite in thestable range.     

On one-relator products induced by generalized triangle groups I

In this paper and a sequel, we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz.     

On one-relator products induced by generalized triangle groups II

This is the second of two papers in which we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz.     

Triangle subgroups of Kleinian groups

We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF ²=∏/Γ _∏ is always embedded in the hyperbolic 3-orbifoldM ³=ℍ³/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup. We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation of co-volumes of Kleinian groups containing these triangle subgroups. for J. A. Kalman on the occasion of his 65th birthday Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science and Technology and the U.K. Scientific and Engineering Research Council.     

Words and Almost Nilpotent Varieties of Groups #

For a word of a free group of rank n , the author obtains an invariant called its standard exponent, and shows that if any residually finite group satisfying the law defined by such a word is almost nilpotent, then the standard exponent of the word equals 1 .

Conversely, if the standard exponent of a word ω is 1 , then any residually finite or soluble group and any locally finite or soluble group satisfying the group law ω≡ 1 is nilpotent-of-bounded-class-by-bounded-exponent.     

Triangle centres: some questions in Euclidean geometry

The circumcentre E of a triangle ABC is defined, as in figure 1, by the two relations EA = EB EB = EC The other centres (such as the incentre, the centroid, etc.) can be defined by two similar relations. This note is an elaboration on the simple fact that if two centres of a triangle coincide then it is equilateral. We take a certain centre of a given triangle and investigate what can be deduced from the assumption that it satisfies one of the two defining relations of another centre. This is done for each pair of, what one may think of as, the seven most natural centres.     

Refinements of generalized triangle inequalities

In this paper, we discuss refinements of the well-known triangle inequality and it is reverse inequality for strongly integrable functions with values in a Banach space X. We also discuss refinement of a generalized triangle inequality of the second kind for Lp functions. For both cases, the attainability of the equality is also investigated.     

Abnormal,Pronormal, Contranormal,and Carter Subgroups in Some Generalized Minimax Groups

ABSTRACT

Some properties of abnormal and pronormal subgroups in generalized minimax groups are considered. For generalized minimax groups (not only periodic) whose locally nilpotent residual is nilpotent and satisfies Min-G the existence of Carter subgroups and their conjugations have been proven. Some generalizations of results of J. Rose on abnormal and contranormal subgroups have been also obtained.     

Linear Bounds of Degrees of Alternating Quotients of the (3, q,r) Triangle Groups

Abstract

In Everitt [Everitt, B. J. (2000). Alternating quotients of Fuchsian groups. J. Algebra 223: 457–476], it was shown, in particular, that each Fuchsian triangle group Δ(p, q, r) has among its homomorphic images all but finitely many of the alternating groups. Treating p, q as fixed, the methods of Everitt (2000) give a quadratic function N(r) of r such that A _n is an image of Δ(p, q, r) for every integer n ≥ N(r). We conjecture that there is a linear function of r with this property. In this paper, we will show that the conjecture holds for the Fuchsian triangle groups Δ(3, q, r).     

Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

Abstract In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R ³ or R ² are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in R ³ or R ² are also planar equilateral triangle circular solutions. *Partially supported by the NNSF and MCME of China, the Qiu Shi Sci. and Tech. Foundation, and Edu. Comm. of Tianjin City. Associate Member of the ICTP. **Partially supported by the NNSF of China     

Free Subgroups in Certain Generalized Triangle Groups of Type (2, m, 2)

A generalized triangle group is a group that can be presented in the form where p,q,r ≥ 2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product . Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p,q,r) is one of (3, 3, 2), (3, 4, 2), (3, 5, 2), or (2, m, 2) where m=3, 4, 5, 6, 10, 12, 15 , 20, 30, 60. In this paper, we show that the Tits alternative holds in the cases (p,q,r)=(2, m, 2) where m=6, 10, 12, 15, 20, 30, 60.     

The ubiquity of generalized cluster categories

Associated with a finite-dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in Amiot (2009) [1]. It was shown to be triangulated, and 2-Calabi–Yau when it is Hom-finite. By definition, the cluster categories of Buan et al. (2006) [4] are a special case. In this paper we show that a large class of 2-Calabi–Yau triangulated categories, including those associated with elements in Coxeter groups from Buan et al. (2009) [7], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in Amiot (2009) [1].     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

Analagmatic cubics through quadratic involutive transformations (I)

We generalize the precedent study of two particular cubics, related to any given triangle, by the construction of analagmatic class six cubics under a common quadratic involutive transformation referred again to any given triangle. As resume, we point out some insights for future extensions to generalized theories about the cubics.     

Large Sets and Overlarge Sets of Triangle-Decomposition

There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even.     

Structure in the Enron Email Dataset

We investigate the structures present in the Enron email dataset using singular value decomposition and semidiscrete decomposition. Using word frequency profiles, we show that messages fall into two distinct groups, whose extrema are characterized by short messages and rare words versus long messages and common words. It is surprising that length of message and word use pattern should be related in this way. We also investigate relationships among individuals based on their patterns of word use in email. We show that word use is correlated to function within the organization, as expected. Lastly, we show that relative changes to individuals' word usage over time can be used to identify key players in major company events. Presented at theWorkshop on Link Analysis, Counterterrorism and Security at the SIAM International Conference on Data Mining, 2005. P.S. Keila is a graduate student in the School of Computing at Queen's University. His research area is data mining in text. D.B. Skillicorn is a professor in the School of Computing at Queen's University, where he heads the Smart Information Management Laboratory. His research area is data mining using matrix decompositions, particularly applied to complex datasets in areas such as biomedicine, geochemistry, counterterrorism and fraud.     

Nichols Algebras of Unidentified Diagonal Type

Abstract

The groups having exactly one normalizer are well-known. They are the Dedekind groups. All finite groups having exactly two normalizers were classified by M. D. Pérez-Ramos and, in a recent paper, S. Camp-Mora generalized that result to locally finite groups. In this paper, we will characterize arbitrary groups with a finite number of normalizers and we will investigate the properties of arbitrary groups with two, three and four normalizers.