Abstract Commensurators of Groups Acting on Rooted Trees

We show that the abstract commensurator of a nearly level transitive weakly branch group H coincides with the relative commensurator of H in the homeomorphism group of the boundary of the tree on which H acts. It is also shown that the commensurator of an infinite group which is commensurable with its own nth direct power contains a Higman–Thompson group as a subgroup. Applying these results to the Grigorchuk 2-group G we show that the commensurator of G is a finitely presented infinite simple group.     

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     

Betti numbers of finitely presented groups and very rapidly growing functions

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can the kth Betti number b_k(G)= rank H_k(G) be providing that G has length ≤N and b_k(G) is finite? We prove that for every k≥3 the maximum b_k(N) of the kth Betti numbers of all such groups is an extremely rapidly growing function of N. It grows faster that all functions previously encountered in mathematics (outside of logic) including non-computable functions (at least those that are known to us). More formally, b_k grows as the third busy beaver function that measures the maximal productivity of Turing machines with ≤N states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines.We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function.Also, we outline a construction of a finitely presented group all of whose homology groups are either or trivial such that its Betti numbers form a random binary sequence.     

Subcubic Growth of the Averaged Dehn Function for a Class 2 Nilpotent Group

We show that the averaged Dehn function with respect to each finite presentation of an arbitrary finitely generated class 2 nilpotent group is subcubic. For the finite rank 2 free class 2 nilpotent group this implies the subasymptoticity of the averaged Dehn function in the sense of M. Gromov, confirming his conjecture.Original Russian Text Copyright © 2005 Romankov V. A.The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00489) and the Scientific Program Universities of Russia of the Ministry for Education of the Russian Federation (Grant 362-05).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 663–672, May–June, 2005.     

Finite Automata of Polynomial Growth do Not Generate A Free Group

A free subgroup of rank 2 of the automorphism group of a regular rooted tree of finite degree cannot be generated by finite-state automorphisms having polynomial growth. This result is in fact proven for rooted trees of infinite degree under some natural additional conditions.     

Some Isoperimetric Inequalities for Kernels of Free Extensions

If G is a hyperbolic group (resp. synchronously or asynchronously automatic group) which can be expressed as an extension of a finitely presented group H by a finitely generated free group, then the normal subgroup H satisfies a polynomial isoperimetric inequality (resp. exponential isoperimetric inequality).     

On the dependence of the growth rate on the length of the defining relator

Let { } be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999.     

Rate of Decay of Concentration Functions on Discrete Groups

Given an irreducible probability measure on a non-compact locally compact group G, it is known that the concentration functions associated with converge to zero. In this note the rate of this convergence is presented in the case where G is a non-locally finite discrete group. In particular it is shown that if the volume growth V(m) of G satisfies V(m) cm ^D then for any compact set K we have sup _gG ⁽ⁿ⁾(Kg) Cn ^–D/2.     

Contractive Automorphisms of Locally Compact Groups and the Concentration Function Problem

Let G be a noncompact locally compact group. We show that a necessary and sufficient condition in order that G support an adapted probability measure whose concentration functions fail converge to zero is that G be the semidirect product , where is an automorphism of N contractive modulo a compact subgroup. Any adapted a probability measure whose concentration functions fail to converge to zero has the form =v×₁ where v is a probability measure on N. If G is unimodular then the concentration functions of an adapted probability measure fail to converge to zero if and only if is supported on a coset of a compact normal subgroup.     

The categoricity of the group of all computable automorphisms of the rational numbers

We prove that there is a first-order sentence ϕ such that the group of all computable automorphisms of the ordering of the rational numbers is its only model among the groups that are embeddable in the group of all computable permutations. Supported by a Scheme 2 grant from the London Mathematical Society. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 649–662, September–October, 2007.     

Automorphisms of Free Groups and Outer Space

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.     

Subharmonic Functions of Order Less than One

We study suhharmonic functions u in R^N (N3) of order at most one, growing so slowly that for some function , with M(u,r)=max_|x|=ru(x). We obtain minorisations for negative values of u(x) and estimations of the difference u(y)–u(x) for x and y on a same sphere. Mathematics Subject Classifications (2000) 31B05, 31B10; secondary: 26A12, 26D15.     

Closed Submodules of Holomorphic Functions with Two Generators

Let I be a closed submodule over a polynomial ring in a space of holomorphic functions on a domain in the complex plane. We establish sufficient conditions under which I is generated by two functions or two special submodules. As a corollary, it follows from these results that if an invariant subspace W C (a,b) (with respect to the differentiation operator) admits spectral synthesis, then it is the solution space of a system of two homogeneous convolution equations.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Structures of Boson and Fermion Fock Spaces in the Space of Symmetric Functions

We realize the Weil representation of infinite-dimensional symplectic group and spinor representation of infinite-dimensional group GL by linear operators in the space of symmetric functions in infinite number of variables.     

亚纯倒星象函数和倒凸象函数类的拟Hadamard卷积

引进了去心圆盘U~*={z:0|z|1}内的亚纯倒星象函数和倒凸象函数的某些新子类,研究了该类中函数的拟Hadamard卷积,所得结果推广了前人的某些工作.     

单叶解析函数的几个子族之间的关系及应用

在复平面单位圆盘内引入了β型螺形甬数族S<,β>的一个子类S<'β>,<,α>函数族,研究了S<'β>,<,α>族与解析函数族S*,S*(α),K,K(α)及S<,β>之间的关系,利用得到的关系式对S<'β>,<,α>族的第二项系数进行了精确估计,同时得到了K(a)族的第二、三项系数的关系式和S<,β>,族的一个积分表...