Periodic products of groups

Адян, Сергей Иванович; Atabekyan, V. S.

doi:10.3103/s1068362317030013

Cited by 11 publications

(7 citation statements)

References 17 publications

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“…Bass-Serre theory also provides examples of groups acylindrically on a tree for which our approach works (Theorem 5.14). For instance if G = A * B is a free product, the corresponding quotient Q corresponds to the n-periodic product of A and B, see for instance [1]. Note that the same strategy could also be used to study the outer automorphism group of free Burnside groups of even exponents, extending some other work of the author [8].…”

Section: Historical Backgroundmentioning

confidence: 96%

“…see for instance [10,Proposition 3.34] or Proposition 3.4. Consequently, in order to make sure that for every k ∈ N, the relations defining G k+1 from G k satisfy a suitable small cancellation condition, it suffices to control (among others) the values of A(G k , X k ) and ν(G k , X k ) all along the approximation sequence (1). This was done in [10] in the absence of even torsion.…”

Section: A Geometrical Approachmentioning

confidence: 99%

“…As we mentioned earlier, if n is odd, then every maximal finite subgroup of B r (n) is isomorphic to the cyclic group Z n . Moreover finite subgroups "stabilize" along the approximation sequence (1). More precisely we have the following property: if F 0 is a finite subgroup of some G k , then for every k, every finite subgroup of G containing the image of F 0 actually comes from a finite subgroup F of G k which already contains F 0 .…”

Section: A Geometrical Approachmentioning

confidence: 99%

“…In Section 2 we make a short review of hyperbolic geometry. We define in Section 3 all the geometric and algebraic invariants needed to control the small cancellation parameters when building the approximation sequence (1). In Section 4 we first review the main properties of small cancellation theory.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Section 5 collects all the previous work. We first state and prove the induction hypothesis used to produce the approximation sequence (1), see Proposition 5.1. Then we apply our main result (Theorem 5.4) to various examples (Section 5.3) such as free Burnside groups, periodic quotients of hyperbolic groups, etc.…”

Section: Outline Of the Papermentioning

confidence: 99%

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Infinite periodic groups of even exponents

Coulon

2018

Preprint

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Section: Historical Backgroundmentioning

confidence: 96%

Section: A Geometrical Approachmentioning

confidence: 99%

Section: A Geometrical Approachmentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

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Infinite periodic groups of even exponents

Coulon

2018

Preprint

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Small cancellation theory over Burnside groups

Coulon

Gruber

2019

Advances in Mathematics

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We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent n0 such that for every odd integer n n0, the well-known classical C ′ (1/6)-small cancellation theory, as well as its graphical generalization and its version for free products, produce examples of infinite n-periodic groups. Our result gives a powerful tool for producing (uncountable collections of) examples of nperiodic groups with prescribed properties. It can be applied without any prior knowledge in the subject of n-periodic groups.As applications, we show the undecidability of Markov properties in classes of n-periodic groups, we produce n-periodic groups whose Cayley graph contains an embedded expander graphs, and we give an n-periodic version of the Rips construction. We also obtain simpler proofs of some known results like the existence of uncountably many finitely generated nperiodic groups and the SQ-universality (in the class of n-periodic groups) of free Burnside groups.The small cancellation theorem. Let us present now a simplified version of our main result. We use the usual definition of the classical C ′ (λ)-condition [31, Chapter V]; see also Definition 2.1. Roughly speaking, this condition on a presentation requires that whenever two relators r = r ′ have a common subword u, then |u| < λ min{|r|, |r ′ |}.Theorem 1.6. Let p ∈ N * . There exists n p ∈ N such that for every odd integer n n p the following holds. Let G = S | R be a non-cyclic group given by a classical C ′ (1/6)-presentation. Assume that no p-th power of a word is a subword of an element of R, and no r ∈ R is a proper power.Our proof, in fact, yields the much more general Theorems 2.4 and 6.3, encompassing Gromov's graphical small cancellation theory, as well as classical and graphical small cancellation theory over free products. The philosophy is always similar to the one of Theorem 1.6: if the small cancellation presentation defines a non-elementary group, and if some restrictions on proper powers are satisfied, then some of the standard conclusions of small cancellation theory hold. For example, nperiodic graphical small cancellation produces infinite n-periodic groups with prescribed subgraphs in their Cayley graphs, and n-periodic free product small cancellation produces infinite n-periodic quotients of free products of n-periodic groups in which each of the generating free factors survives as subgroup.Remark 1.7. Even in the case that both S and R are finite, the statement of Theorem 1.6 is not covered by prior results. It is known [39,17] that, given a torsion-free Gromov hyperbolic group G, there exists n G ∈ N such that for all odd integers n n G , the quotient G/G n is infinite. Note here that n G depends on the specific group G and, in fact, given an exponent n ∈ N, the proof only applies to finitely many hyperbolic groups G of a given rank. In our result, on the other hand, the constant n p is independent of the presentation S | R . We shall see in the following how this eas...

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On free groups in the infinitely based varieties of S. I. Adian

Адян¹,

Atabekyan²

2017

Izv. Math.

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Periodic products of groups

Cited by 11 publications

References 17 publications

Infinite periodic groups of even exponents

Infinite periodic groups of even exponents

Small cancellation theory over Burnside groups

On free groups in the infinitely based varieties of S. I. Adian

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