On groups of polynomial subgroup growth

Summary Let be a finitely generated group anda _n ()=the number of its subgroups of indexn. We prove that, assuming is residually nilpotent (e.g., linear), thena _n () grows polynomially if and only if is solvable of finite rank. This answers a question of Segal. The proof uses a new characterization ofp-adic analytic groups, the theory of algebraic groups and the Prime Number Theorem. The method can be applied also to groups of polynomial word growth.Oblatum 1-VII-1989 & 7-VI-1990     

Finitely generated groups of polynomial subgroup growth

We determine the structure of finitely generated residually finite groups in which the number of subgroups of each finite indexn is bounded by a fixed power ofn. To John Thompson, an inspiration to group theory, on his being awarded the Wolf Prize Partially supported by BSF and GIF grants. Partially supported by a BSF grant.     

Variations on polynomial subgroup growth

A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ ^a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).     

On the degree of polynomial subgroup growth in class 2 nilpotent groups

We use the theory of zeta functions of groups to establish a lower limit for the degree of polynomial normal subgroup growth in class two nilpotent groups.     

Metric properties on Lie groups of polynomial volume growth

It is well known that we have an algebraic characterization of connected Lie groups of polynomial volume growth: a Lie group G has polynomial volume growth if and only if it is of type R. In this paper, we shall give a geometric characterization of connected Lie groups of polynomial volume growth in terms of the distance distortion of the subgroups. More precisely, we prove that a connected Lie group G has polynomial volume growth if and only if every closed subgroup has a polynomial distortion in G.     

Discrete groups of slow subgroup growth

It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least $n^{\frac{{\log n}}{{\log \log n}}} $ . In this paper we prove the existence of a finitely generated group whose subgroup growth is of type $n^{\frac{{\log n}}{{(\log \log n)^2 }}} $ . This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen ^logn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostn ^{c logn} subgroups of indexn.     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

The subgroup growth spectrum of virtually free groups

For a finitely generated group Γ denote by μ(Γ) the growth coefficient of Γ, that is, the infimum over all real numbers d such that s _n (Γ) < n! ^d . We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group.     

Weighted group algebras on groups of polynomial growth

Let G be a compactly generated group of polynomial growth and a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L ¹ (G,). In particular, if the weight is sub-exponential, then the algebra L ¹ (G,) is symmetric. For these weights we develop a functional calculus on a total part of L ¹ (G,) and use it to prove the Wiener property. Mathematics Subject Classification (2000):43A20, 22D15, 22D12.Supported by the Austrian Science Foundation project FWF P-14485.Supported by the research grants MEN/CUL/98/007 and CUL/01/014.     

Hardy spaces on Lie groups of polynomial growth

We give several characterizations of Hardy spaces associated with complex, second-order,subelliptic operators on Lie groups with polynomial growth.     

Distances on Lie groups of polynomial volume growth

For any simply connected solvable Lie group Q of polynomial volume growth, we introduce the notion of nil-shadow of Q. We shall give an explicit formula for the distance to the origin of an element q ∈ Q in terms of its exponential coordinates of the second kind taken in an appropriate basis. This result extends a previous result for nilpotent Lie groups [6, Theorem DN] and [7, Theorem 1].     

图的特征多项式的若干性质

图的特征多项式有许多性质,本文给出了特征多项式的指数表达式,特征多项式的导数的几个不同表达式以及高阶导数的图论意义。     

Centered densities on Lie groups of polynomial volume growth

 We study the asymptotic behavior of the convolution powers of a centered density on a connected Lie group G of polynomial volume growth. The main tool is a Harnack inequality which is proved by using ideas from Homogenization theory and by adapting the method of Krylov and Safonov. Applying this inequality we prove that the positive -harmonic functions are constant. We also characterise the -harmonic functions which grow polynomially. We give Gaussian estimates for , as well as for the differences and . We give estimates, similar to the ones given by the classical Berry-Esseen theorem, for and . We use these estimates to study the associated Riesz transforms. Received: 5 July 1999 / Revised version: 8 April 2002 / Published online: 22 August 2002     

Normal subgroup growth in free class-2-nilpotent groups

Let F_2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.     

灰群的性质

在文「２」的基础上继续研究灰子群的性质,得到若干重要结果。     

Some properties of optimal spacing in polynomial estimation

Summary It is shown that the choice of x values that minimizes the generalized variance of the estimates of the coefficients in polynomial regression also minimizes the maximum of the generalized variance of the estimates of an arbitrary number of ordinates on the regression curve. This optimum spacing is also optimal with respect to the estimation of the coefficients of the first derivative of the polynomial regression function. This research was partially supported by mean of an Air Force Research Grant while the author was on sabbatical leave in Japan.