On the degree of groups of polynomial subgroup growth

Let be a finitely generated residually finite group and let denote the number of index subgroups of . If for some and for all , then is said to have polynomial subgroup growth (PSG, for short). The degree of is then defined by .

Very little seems to be known about the relation between and the algebraic structure of . We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if is a finite index subgroup, then .

A large part of the paper is devoted to the structure of groups of small degree. We show that is bounded above by a linear function of if and only if is virtually cyclic. We then determine all groups of degree less than , and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval .

Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.

     

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.     

Some properties of polynomial subgroup growth groups

A PSG group is one in which the number of subgroups of given index is bounded by a fixed power of this index. The finitely generated PSG groups are known. Here we prove some properties of such groups which need not be finitely generated. We derive, e.g., restrictions on the chief factors (Theorem 1) and on the number of generators of subgroups (Theorem 5). To Wolf Prize laureate John Thompson Partially supported by a BSF grant.     

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.     

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).     

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.     

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.     

On subgroup functors of finite soluble groups

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups. We prove that they form a complemented and non-modular lattice containing two relevant sublattices. This is the answer to a question (Question 1.2.12) proposed by Skiba (1997) in the finite soluble universe.     

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.     

Relative cohomology of finite groups and polynomial growth

This article investigates the cohomology of a finite group relative to a collection of sub-groups. In particular a new spectral sequence abutting to relative cohomology is given and is used to deduce that relative cohomology has 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].     

On the Swan subgroup of certain periodic groups

Ohne Zusammenfassung