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     

Finitely presented monoids with linear Dehn function need not have regular cross-sections

This paper shows that a finitely presented monoid with linear Dehn function need not have a regular cross-section, strengthening the previously-known result that such a monoid need not be presented by a finite complete string rewriting system, and contrasting with the fact that finitely presented groups with linear Dehn function always have regular cross-sections.     

Finiteness and Dehn functions of automatic monoids having directed fellow traveller property

A left-cancellative automatic monoid having directed fellow traveller property is finitely presented, and the first order Dehn functions of such automatic monoids are bounded above by a quadratic function. These results coincide with those of automatic groups. The research of X. Wang was partially supported by China National Science Funds (No:10771077 and 10671114).     

The Dehn Function of Stallings’ Group

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F₃{\mathcal{F}_3} is quadratic.     

On Dehn Functions of Infinite Presentations of Groups

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups. In particular, we prove that the (standard) Dehn function of a 2-group Γ of intermediate growth, defined by a system of defining relators due to Lysenok, is bounded from above by C₁x² log₂ x, where C₁ > 1 is a constant. We also show that the (standard) Dehn function of a free m-generator Burnside group B(m, n) of exponent n ≥ 2⁴⁸, where n is either odd or divisible by 2⁹, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x^19/12. Received: September 2007, Revision: March 2008, Accepted: March 2008     

双曲群,自动机群与群的Dehn函数

本文第一部分对组台群论的发展史作了一个简要的回顾．然后重点介绍近20年来组合群论领域中的三个热点课题：即双曲群，自动机群和群的Dehn函数的有关概念，并对相关的重要研究成果给予概述，其中包括作者本人在群的二阶Dehn函数的研究工作中的若干成果．最后提出9个公开问题。     

The Dehn function of Baumslag’s metabelian group

Baumslag??s group is a finitely presented metabelian group with a ${\mathbb Z \wr \mathbb Z}$ subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes ${C_m \wr \mathbb Z}$ . We prove that Baumslag??s group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions.     

On Presentations of Bruck–Reilly Extensions

We first consider the class of monoids in which every left invertible element is also right invertible, and prove that if a monoid belonging to this class admits a finitely presented Bruck–Reilly extension then it is finitely generated. This allow us to obtain necessary and sufficient conditions for the Bruck–Reilly extensions of this class of monoids to be finitely presented. We then prove that thes 𝒟-classes of a Bruck–Reilly extension of a Clifford monoid are Bruck–Reilly extensions of groups. This yields another necessary and sufficient condition for these Bruck–Reilly extensions to be finitely generated and presented. Finally, we show that a Bruck–Reilly extension of a Clifford monoid is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid, and that this property cannot be generalized to Bruck–Reilly extensions of arbitrary inverse monoids.     

Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function n^l+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic. Supported by RFBR grant No. 04-01-00489. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007.     

Euler Characteristics for Generalized Tetrahedron Groups

This paper is concerned with a class of groups which generalize the ordinary tetrahedron groups. This class of groups was introduced by Fine and Rosenberger and independently by Vinberg. Some conditions for the generalized tetrahedron groups to have a rational Euler characteristic are shown here and the values of the Euler characteristics in these cases are calculated. In addition here are some geometrical applications of the Euler characteristic of generalized tetrahedron groups.^{1 2}     

Positively finitely related profinite groups

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.     

Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of . Received: October 7, 1999.     

Generating the surface mapping class group by two elements

Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.

     

Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

Isolated exceptional Dehn surgeries on hyperbolic knots

For a hyperbolic knot in the 3-sphere, at most finitely many Dehn surgeries yield non-hyperbolic manifolds. Such exceptional surgeries are classified into four types, lens space surgery, small Seifert fibered surgery, toroidal surgery and reducing surgery, according to the resulting manifolds. For each of the three types except reducing surgery, we give infinitely many hyperbolic knots with integral exceptional Dehn surgeries of the given type, whose adjacent integral surgeries are not exceptional.     

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n²log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n²log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem. Both authors were supported in part by the NSF grants DMS 0245600 and DMS 0455881. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 05-01-00895, the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298. Received: February 2005; Revision: September 2005; Accepted: September 2005     

First and Second Order Necessary Conditions for Stochastic Optimal Control Problems

In this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state.     

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF^∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF^∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes-Moscovici and Dru?u-Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF^∞ and isocohomological.     

李群表示论和Schubert条件

本文将Grassmann流形上的Schubert子簇所满足的经典的Schubert条件推广到一般的复半单李群G的广义旗流形.利用复半单李群的表示理论,我们首先在李群的权空间上引入自然的Ehresman偏序.这一偏序可以导出李群的最高权表示的权系、Weyl群及其陪集空间上的Ehresman偏序.然后我们对一般的复表示定义了相应的射影空间,Grassmann流形和旗流形.这使得能够像经典的情形一样来分析广义旗流形的Schubert子簇满足的Schubert条件.在讨论中,我们还给出了李群G的Weyl群及其陪集空间中的Bruhat-Chevalley偏序的简单判别条件.我们的结果应用到例外群,给出了Fulton提出的关于例外群的Schubert分析的问题的部分回答.     

Characterization of generalized Chernikov groups among groups with involutions

The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct productA of quasi-cyclicp-groups with finitely many factors for each primep such that each of its elements does not commute elementwise with only finitely many Sylow subgroups ofA. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved. Translated fromMatematicheskie Zametki, Vol. 62, No 4, pp. 577–587, October, 1997. Translated by A. I. Shtern