Golod–shafarevich Groups: A Survey

Ershov, Mikhail

doi:10.1142/s0218196712300010

Cited by 40 publications

(42 citation statements)

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“…In the context of free groups, Lie algebras and enveloping algebras of Lie algebras the notion of noncommutative derivation has been investigated since the late 1940's by several authors. In [10], Ralph Fox introduced derivatives over free groups to study invariants of group presentations (see [9]). In [19] Umirbaev introduced derivatives for enveloping algebras of Lie algebras, and in [24] Woronowicz introduced differential calculus on quantum groups (see [15] for more details).…”

Section: Introductionmentioning

confidence: 99%

Golod–Shafarevich-Type Theorems and Potential Algebras

Iyudu

Smoktunowicz

2018

International Mathematics Research Notices

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Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential.We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [22], related to the geometric argument of Toda [18]. We derive from the improved version of the Golod-Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree n ⩾ 3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class P n of potential algebras with homogeneous potential of degree n + 1 ⩾ 4, the minimal Hilbert series is H n = 1 1−2t+2t n −t n+1 , so they are all infinite dimensional. Moreover, growth could be polynomial (but non-linear) for the potential of degree 4, and is always exponential for potential of degree starting from 5.For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invariants.MSC: 16A22, 16S37, 14A22

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Section: Introductionmentioning

confidence: 99%

Golod–Shafarevich-Type Theorems and Potential Algebras

Iyudu

Smoktunowicz

2018

International Mathematics Research Notices

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“…Note that the usual notion of a Golod-Shafarevich group corresponds to the degree function which assigns to all x ∈ X the degree 1. We use the the same terminology as in [3]. The following is contained in [3, Theorem 4.3 and Theorem 4.4].…”

Section: Virtual Golod-shafarevich Groupsmentioning

confidence: 99%

Large normal subgroup growth and large characteristic subgroup growth

Barnea

Schlage‐Puchta

2019

Journal of Group Theory

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The maximal normal subgroup growth type of a finitely generated group is n log n . Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let Γ be a group and ∆ a subgroup of finite index. Suppose ∆ has normal subgroup growth of type n log n , does Γ has normal subgroup growth of type n log n ? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod-Shafarevich group, then G has normal subgroup growth of type of n log n . We also use our methods to show that one can find a group with characteristic subgroup growth of type n log n .2010 Mathematics subject classification. 20E07, 20E18, 20F05.

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“…As we have mentioned before, recursively presented finitely generated residually finite groups may have undecidable word problem [15,21,38]. See also the survey [16] where it is shown how to construct complicated residually finite groups using the method of Golod-Shafarevich.…”

Section: The "Yes" and "No" Parts Of The Mckinsey Algorithmmentioning

confidence: 99%