The Entropy of Graded Algebras

“…Using Lemma 3.1 with m = 2 and n 1 = 5 and n 2 = 6, we see that there exists a subspace N of G ⊗ G such that dim G = 11, dim N d 6 + d 5 + d 5,6 22 + 15 + 33 = 70 and E 4 (N, G) = 0. By Lemma 2.3, h 4 (K, 11, 51) = 0.…”

Section: Proof Of Theorem 15mentioning

confidence: 95%

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

Iyudu

Shkarin

2011

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CWcomplexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations d 2 , and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to n(n−1) 2 was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.We prove that over any infinite field, the Anick conjecture holds for d 4(n 2 +n) 9and arbitrary number of generators, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

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“…The (lower) density of subalgebras in graded restricted

F_{p}

‐Lie algebras and in graded

F_{p}

‐Lie algebras is defined as follows; see for related notions, indeed closely related for graded subalgebras. Definition Let

R = ⨁_{n = 1}^{\infty} {sans-serifR}_{n}

be a non‐zero graded restricted

F_{p}

‐Lie algebra such that each homogeneous component

R_{n}

is finite dimensional.…”

Section: Non‐abelian Free Pro‐p Groups and The Zassenhaus Seriesmentioning

confidence: 99%

Pro‐p groups of positive rank gradient and Hausdorff dimension

Ocaña

Garrido

Klopsch

2019

J. London Math. Soc.

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Using Hausdorff dimension, we show that finitely generated closed subgroups H of infinite index in a finitely generated pro-p group G of positive rank gradient never contain any infinite subgroups K that are subnormal in G via finitely generated successive quotients. This generalises similar assertions that were known to hold for non-abelian free pro-p groups and other related pro-p groups.Our main results are as follows. We show that every finitely generated pro-p group G of positive rank gradient has full Hausdorff spectrum hspec F (G) = [0, 1] with respect to the Frattini series F (or, more generally, any iterated verbal filtration). Using Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-p groups and non-soluble Demushkin groups G have full Hausdorff spectrum hspec Z (G) = [0, 1] with respect to the Zassenhaus series Z. This resolves a long-standing problem in the subject.In fact, the results hold more generally for finite direct products of finitely generated prop groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-p groups and non-soluble Demushkin groups.Finally, we determine the normal Hausdorff spectra of such direct products.

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The Entropy of Graded Algebras

Cited by 23 publications

References 12 publications

Icosahedral Symmetry: A Review

Icosahedral Symmetry: A Review

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

Pro‐p groups of positive rank gradient and Hausdorff dimension

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