Growth of Algebras and Gelfand-Kirillov Dimension

Krause, Günter; Lenagan, T. H.

doi:10.1090/gsm/022

Cited by 410 publications

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“…As a corollary we obtain the p c (G) < 1 for almost solvable and linear group, which are not almost Z, as well as for Burnside group B(n, p), where p > 661 (see [21,28]). Let us also mention here a different result of Grigorchuk that if γ(n) e √ n , and G is residually finite p-group, then G has polynomial growth (see [10]).…”

Section: Theorem 3 (Lyons)mentioning

confidence: 74%

“…It is known that for any two finite sets of generators S 1 , S 2 of a group G, the corresponding two growth functions are equivalent (see e.g. [21,28]). Note also that if…”

Section: Growth Of Groups and Percolationmentioning

confidence: 99%

“…We add that by the Tits alternative all linear groups must be either almost solvable or contain a free group F 2 . Finally, the Milnor-Wolf theorem states that all solvable groups have either polynomial or exponential growth (see [21,28]). …”

Section: Growth Of Groups and Percolationmentioning

confidence: 99%

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Percolation on Grigorchuk Groups

Muchnik¹,

Pak²

2001

Communications in Algebra

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Let p c (G) be the critical probability of the site percolation on the Cayley graph of group G. In [2] of Benjamini and Schramm conjectured thatp c < 1, given the group is infinite and not a finite extension of Z. The conjecture was proved earlier for groups of polynomial and exponential growth and remains open for groups of intermediate growth.In this note we prove the conjecture for a special class of Grigorchuk groups, which is a special class of groups of intermediate growth. The proof is based on an algebraic construction. No previous knowledge of percolation is assumed.

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Section: Theorem 3 (Lyons)mentioning

confidence: 74%

“…It is known that for any two finite sets of generators S 1 , S 2 of a group G, the corresponding two growth functions are equivalent (see e.g. [21,28]). Note also that if…”

Section: Growth Of Groups and Percolationmentioning

confidence: 99%

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Percolation on Grigorchuk Groups

Muchnik¹,

Pak²

2001

Communications in Algebra

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“…Every somewhat commutative algebra A is a Noetherian finitely generated finitely partitive algebra of finite GelfandKirillov dimension, the Gelfand-Kirillov dimension of every finitely generated A-modules is an integer, and (Quillen's lemma): the ring End A (M) is algebraic over K (see [17], Ch. 8 or [14] for details). If, in addition, the algebra A is a domain, then we denote by D = D A its quotient division ring (i.e.…”

Section: Theorem 13 (Bernstein's Inequality) Gk (M)mentioning

confidence: 99%

Gelfand-Kirillov dimension of commutative subalgebras of simple inﬁnite dimensional algebras and their quotient division algebras

Bavula¹

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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“…For this reason, GK dimension has seen great use over the years as a useful tool for obtaining noncommutative analogues of results from classical algebraic geometry. For more information about GK dimension we refer the reader to Krause and Lenagan [5].…”

Section: Introductionmentioning

confidence: 99%

Extended Centres of Finitely Generated Prime Algebras

Bell

Smoktunowicz

2009

Communications in Algebra

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Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) − 2 over K. As a consequence, we are able to show that if A is a prime K-algebra of quadratic growth, then either the extended centre is a finite extension of K or A is PI. Finally, we give an example of a finitely generated non-PI prime K-algebra of GK dimension 2 with a locally nilpotent ideal such that the extended centre has infinite transcendence degree over K. * The first author thanks NSERC for its generous support.

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Growth of Algebras and Gelfand-Kirillov Dimension

Cited by 410 publications

References 0 publications

Percolation on Grigorchuk Groups

Percolation on Grigorchuk Groups

Gelfand-Kirillov dimension of commutative subalgebras of simple inﬁnite dimensional algebras and their quotient division algebras

Extended Centres of Finitely Generated Prime Algebras

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