Solvable groups with polynomial Dehn functions

Given a finitely presented group , finitely generated subgroup of , and a monomorphism , we obtain an upper bound of the Dehn function of the corresponding HNN-extension in terms of the Dehn function of and the distortion of in . Using such a bound, we construct first examples of non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol'shanskii, Rips, and Sapir.

     

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。     

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     

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 PSL2(Z[1/p])

We show that PSL₂(Z[1/p]) admits a combing with bounded asynchronous width, and use this combing to show that PSL₂(Z[1/p]) has an exponential Dehn function. As a corollary, PSL₂(Z[1/p]) has solvable word problem and is not an automatic group.     

The Selmer Groups of Elliptic Curves

Let E/K be an elliptic curve with K-rational p-torsion points.The p-Selmer group of E is described by the image of a map λk and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.     

On integration of vector functions with respect to vector measures

We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S*-integral. Our main result states that S*-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).     

A generalized Henstock-Stieltjes integral involving division functions

We can consider the Riemann-Stieltjes integral dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.      

Bartholdi zeta functions of digraphs

We give a determinant expression for the Bartholdi zeta function of a digraph which is not symmetric. This is a generalization of Bartholdi’s result on the Bartholdi zeta function of a graph or a symmetric digraph.     

Monopoles, particles and rational functions

We prove a recent conjecture of Manton and Murray: if a polynomialp(z) of degreek — 1 is given, then anSU (2) monopole corresponding to a rational functionp(z)/q(z) with well-separated poles \₁,...,\_k is approximately made up from charge 1 monopoles located at points (1/2 In p(\_i), \_i). We show how the rate of approximation changes with the numeratorp(z) with the result that, as long as the values of the numerator remain close together relative to the distances between poles, the above statement remains true and ceases to be so otherwise.We also show that the spectral curve of the monopole approaches the union of curves of charge 1 monopoles exponentially fast. This remains true forSU (N) monopoles.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

Pointwise and integral estimates for the B-Riesz potential in terms of B-maximal and B-fractional maximal functions

We study the maximal and fractional maximal functions and Riesz potentials that are generated by the generalized shift operator associated with the Laplace-Bessel operator. We obtain some pointwise and integral estimates that give a relation between the B-maximal and B-fractional maximal functions and B-Riesz potentials and extend the available results to the objects of a more general nature. Basing on these results, we prove interpolation theorems for the B-fractional maximal functions and B-Riesz potentials.     

Left loops, bipartite graphs with parallelism and bipartite involution sets

We describe a representation of any semiregularleft loop by means of asemiregular bipartite involution set or, equivalently, a 1-factorization (i.e., a parallelism) of a bipartite graph, with at least one transitive vertex. In these correspondences,Bol loops are associated on one hand toinvariant regular bipartite involution sets and, on the other hand, totrapezium complete bipartite graphs with parallelism; K-loops (or Bruck loops) are further characterized by a sort of local Pascal configuration in the related graph. Research partially supported by the Research Project of M.I.U.R. (Italian Ministry of Education, University and Research) “Strutture geometriche, combinatoria e loro applicazioni” and by the Research group G.N.S.A.G.A. of INDAM.     

On the ranks of bent functions

The rank of a bent function is the 2-rank of the associated symmetric 2-design. In this paper, it is shown that it is an invariant under the equivalence relation among bent functions. Some upper and lower bounds of ranks of general bent functions, Maiorana–McFarland bent functions and Desarguesian partial spread bent functions are given. As a consequence, it is proved that almost every Desarguesian partial spread bent function is not equivalent to any Maiorana–McFarland bent function.     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

The Groups LS and Morphisms of Quadratic Extensions

For a morphism of quadratic extensions of antistructures, groups similar to the groups of obstructions to splitting along one-sided submanifolds are defined. These groups are a natural generalization of the splitting obstruction groups. The results obtained open additional possibilities for constructing groups and natural maps in L-theory.     

Essentially finite algorithms for minimizing a class of convex piecewise polynomial functions

We consider minimax problems of the type min _t[a,b] max _i p _t (t), where thep _i (t) are real polynomials which are convex on an interval [a, b]. Here, the function max _i p _i (t) is a convex piecewise polynomial function. The main result is an algorithm yielding the minimum and the minimizing point, which is efficient with respect to the necessary number of computed polynomial zeros. The algorithms presented here are applicable to decision problems with convex loss. While primarily of theoretical interest, they are implementable for quadratic loss functions with finitely many states of nature. An example of this case is given. The algorithms are finite when polynomial zeros need not be approximated.The author is indebted to T. E. S. Raghavan for many stimulating discussions.     

Miura opers and critical points of master functions

Critical points of a master function associated to a simple Lie algebra come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population. Supported in part by NSF grant DMS-0140460 Supported in part by NSF grant DMS-0244579