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Details van artikel 10 van 26 gevonden artikelen
| Titel: |
Growth series of central amalgamations |
| Auteur: |
Billington, Nicholas |
| Verschenen in: |
Communications in algebra |
| Paginering: | Jaargang 21 (1993) nr. 2 pagina's 371-397 |
| Jaar: |
1993 |
| Inhoud: |
Evaluation at 1 of the growth series of a finitely generated group G of rational growth often yields the reciprocal of the ordinary Euler characteristic of G. This leads one to suspect that the power series reciprocal of the growth series might behave like a series-valued Euler characteristic: this is indeed the case and the easiest way to see it is to pass to a graded algebra associated with the finitely generated group and then utilise rhe corresponding results on Hilbert-Poincare' series of graded algebras. In this paper we are principally concerned with amalgams G *c H where C is a central subgroup of borh G and H and with power-series analogue of the Euler-characteristic formula [image omitted] Due principally to the fact that a subgroup of a finitely generated group G need not be finitely generated, but nonetheless still comes equipped with a notion of length inherited from G, we broaden our poinr of view and deal with a class of groups which we calls stratified: in the case of finitely generated group G with a specified generating set T we can think of the nth stratum, for n ≥ as consisting of those elements of G which can be written a product of n, but no fewer, elements from T ∪ T1. In this first section we discuss stratified groups in general. In the second section we pass from stratified groups to graded algebras and by using results of Lemaire [4] on the Hilbert-Poincare' series of graded algebras we deduce formulae for the growth of direct products and free products of stratified groups. Next we deal with the problem of extending a stratification from a normal subgroup and then we apply this to the growth series of a central amalgamation. Finally we illustrate our results by calculating certain growth series of the torus knot groups. The author wishes to point out that results which appear in this paper were established in his PhD thesis [2]. The term “stratified group” was coined in [1] where the propenies of algebraic versus non-algebraic growth functions were investigated. |
| Uitgever: |
Taylor & Francis |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
Details van artikel 10 van 26 gevonden artikelen
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