GROWTH OF FINITELY GENERATED POLYNILPOTENT LIE ALGEBRAS AND GROUPS, GENERALIZED PARTITIONS, AND FUNCTIONS ANALYTIC IN THE UNIT CIRCLE

1999 ◽  
Vol 09 (02) ◽  
pp. 179-212 ◽  
Author(s):  
V. M. PETROGRADSKY
Keyword(s):  
Unit Circle ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Precise Asymptotics ◽  
Finitely Generated ◽  
Asymptotic Growth ◽  
Enveloping Algebra ◽  
Finitely Generated Groups ◽  
Kirillov Dimension

Recently, the author has suggested a series of dimensions of algebras which includes as first terms dimension of a vector space, Gelfand–Kirillov dimension, and superdimension. These dimensions enabled us to describe the change of the growth in the transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper, we obtain more precise asymptotics for generalized partitions. As a main application, we obtain more precise asymptotics for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of the lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert–Poincaré series and some facts on the growth of functions analytic in the unit circle. By the growth of such functions, we mean their growth when the variable tends to 1. Finally, we study two kinds of p-central series for free polynilpotent finitely generated groups. We obtain asymptotics for the ranks of these series, in one case we have an example of a polynomial, but not rational growth.

On the Structure of Q2(G) for Finitely Generated Groups

1973 ◽  
Vol 25 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Lower Central Series ◽  
Integral Group Ring ◽  
Symmetric Power ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finitely Generated ◽  
Finitely Generated Groups

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

scholarly journals Finitely generated residually torsion-free nilpotent groups. I

1999 ◽  
Vol 67 (3) ◽  
pp. 289-317 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Kirillov Dimension ◽  
Torsion Free

AbstractThe object of this paper is to study the sequence of torsion-free ranks of the quotients by the terms of the lower central series of a finitely generated group. This gives rise to the introduction into the study of finitely generated, residually torison-free nilpotent groups of notions relating to the Gelfand-Kirillov dimension. These notions are explored here. The main result concerning the sequences alluded to is the proof that there are continuously many such sequences.

scholarly journals Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

scholarly journals On lower central series quotients of finitely generated algebras over Z

2015 ◽  
Vol 423 ◽  
pp. 559-572 ◽  
Author(s):  
Katherine Cordwell ◽  
Teng Fei ◽  
Kathleen Zhou
Keyword(s):  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated

scholarly journals Dimension quotients of metabelian Lie rings

2017 ◽  
Vol 27 (02) ◽  
pp. 251-258
Author(s):  
Inder Bir S. Passi ◽  
Thomas Sicking
Keyword(s):  
Lower Central Series ◽  
Universal Enveloping Algebra ◽  
Central Series ◽  
Augmentation Ideal ◽  
Lie Ring ◽  
Ring Of Integers ◽  
Enveloping Algebra ◽  
Lie Rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].

scholarly journals Dimension and lower central subgroups of metabelian p-groups

1985 ◽  
Vol 100 ◽  
pp. 127-133 ◽  
Author(s):  
Narain Gupta ◽  
Ken-Ichi Tahara
Keyword(s):  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated

It is a well-known result due to Sjogren [9] that if G is a finitely generated p-group then, for all n ≦ − 1, the (n + 2)-th dimension sub-group Dn+2(G) of G coincides with γn+2(G), the (n + 2)-th term of the lower central series of G. This was earlier proved by Moran [5] for n ≦ p − 2. For p = 2, Sjogren’s result is the best possible as Rips [8] has exhibited a finite 2-group G for which D4G) ≠ γ4(G) (see also Tahara [10, 11]). In this note we prove that if G is a finitely generated metabelian p-group then, for all . It follows, in particular, that, for p odd, for all n ≦ p and all metabelian p-groups G.

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