THE BURNSIDE PROBLEM FOR GROUPS OF LOW QUADRATIC GROWTH

1996 ◽  
Vol 06 (03) ◽  
pp. 369-377
Author(s):  
ROBERTO INCITTI
Keyword(s):  
Infinite Order ◽  
Polynomial Growth ◽  
Growth Function ◽  
Infinite Group ◽  
Quadratic Growth ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Combinatorial Argument ◽  
Groups Of Polynomial Growth ◽  
Generating System

We show with a combinatorial argument that a finitely generated infinite group whose growth function relative to some finite generating system is less or equal to [Formula: see text], r<2, contains an element of infinite order. This result is aimed at investigating the combinatorial nature of M. Gromov’s theorem on groups of polynomial growth.

scholarly journals Growth of linear semigroups

1996 ◽  
Vol 60 (1) ◽  
pp. 18-30 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Finite Rank ◽  
Polynomial Growth ◽  
Growth Function ◽  
Unipotent Radical ◽  
Maximal Degree ◽  
Finitely Generated ◽  
Group Of Fractions

AbstractWe show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.

scholarly journals A MEALY MACHINE WITH POLYNOMIAL GROWTH OF IRRATIONAL DEGREE

2008 ◽  
Vol 18 (01) ◽  
pp. 59-82 ◽  
Author(s):  
LAURENT BARTHOLDI ◽  
ILLYA I. REZNYKOV
Keyword(s):  
Polynomial Growth ◽  
Growth Function ◽  
Finitely Generated ◽  
Mealy Machine ◽  
A Chain

We consider a very simple Mealy machine (two nontrivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like ℓα, for [Formula: see text]; that the semigroup satisfies the identity g6 = g4; and that its lattice of two-sided ideals is a chain.

scholarly journals Besov algebras on Lie groups of polynomial growth

2012 ◽  
Vol 212 (2) ◽  
pp. 119-139 ◽  
Author(s):  
Isabelle Gallagher ◽  
Yannick Sire
Keyword(s):  
Lie Groups ◽  
Polynomial Growth ◽  
Groups Of Polynomial Growth

scholarly journals Group Cohomology and Lp-Cohomology of Finitely Generated Groups

2003 ◽  
Vol 46 (2) ◽  
pp. 268-276 ◽  
Author(s):  
Michael J. Puls
Keyword(s):  
Banach Space ◽  
Cohomology Group ◽  
Nilpotent Groups ◽  
Group Cohomology ◽  
Infinite Group ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Cohomology Space ◽  
First Cohomology Group

AbstractLet G be a finitely generated, infinite group, let p > 1, and let Lp(G) denote the Banach space . In this paper we will study the first cohomology group of G with coefficients in Lp(G), and the first reduced Lp-cohomology space of G. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

scholarly journals The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

2017 ◽  
Vol 166 (1) ◽  
pp. 83-121
Author(s):  
NEHA GUPTA ◽  
ILYA KAPOVICH
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Growth Function ◽  
Residual Finiteness ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Finitely Generated ◽  
Closed Curves ◽  
Index Function ◽  
Finite Covers

AbstractMotivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

scholarly journals Weighted group algebras on groups of polynomial growth

2003 ◽  
Vol 245 (4) ◽  
pp. 791-821 ◽  
Author(s):  
G. Fendler ◽  
K. Gr�chenig ◽  
M. Leinert ◽  
J. Ludwig ◽  
C. Molitor-Braun
Keyword(s):  
Polynomial Growth ◽  
Group Algebras ◽  
Groups Of Polynomial Growth

IDENTITIES AND A BOUNDED HEIGHT CONDITION FOR SEMIGROUPS

2003 ◽  
Vol 13 (05) ◽  
pp. 565-583 ◽  
Author(s):  
L. M. SHNEERSON
Keyword(s):  
Polynomial Growth ◽  
Nilpotent Groups ◽  
The Other ◽  
Finitely Generated ◽  
Different Types ◽  
Associative Rings ◽  
Nontrivial Identity

We consider two different types of bounded height condition for semigroups. The first one originates from the classical Shirshov's bounded height theorem for associative rings. The second which is weaker, in fact was introduced by Wolf and also used by Bass for calculating the growth of finitely generated (f.g.) nilpotent groups. Both conditions yield polynomial growth. We give the first two examples of f.g. semigroups which have bounded height and do not satisfy any nontrivial identity. One of these semigroups does not have bounded height in the sense of Shirshov and the other satisfies the classical bounded height condition. This develops further one of the main results of the author's paper (J. Algebra, 1993) where the first examples of f.g. semigroups of polynomial growth and without nontrivial identities were given.

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