Infinitely presented<i>C</i>(6)-groups are SQ-universal

Gruber, Dominik

doi:10.1112/jlms/jdv022

Cited by 5 publications

(18 citation statements)

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“…We glean from [28] the following: Recall that a bi-infinite path graph α in Cay(G, S) is periodic if there is a cyclic subgroup of G that stabilizes α and acts cocompactly on it. In the proof we have surjections S H G. Let | · | H denote the word length in H with respect to the image of S. Similarly, let | · | G denote the word length in G with respect to the image of S.…”

Section: 1mentioning

confidence: 99%

“…Assume there exists a C 0 as in the second possibility. Then we have the following property by [28, Proof of Theorem 4.2 in Case 2a] and [28,Lemma 4.17]: Whenever v is a geodesic word representing g n for some n, then the equation v = w n already holds in G(Γ <6C 0 ), where Γ <6C 0 denotes the subgraph of Γ that is the union of all components with girth less than 6C 0 . Therefore, the epimorphism G(Γ <6C 0 )…”

Section: 1mentioning

confidence: 99%

“…Proof. Let x be an infinite order element of G. By [28,Section 4], there exists g ∈ G such that x is conjugate to g N for some N and such that, if w is a shortest word representing g, we have the following possibilities: Any bi-infinite periodic path graph γ labelled by the powers of w is a convex geodesic (Case 1 in [28, Section 4]), or, for each n and any shortest word g n representing g n , there exists a diagram B n over Γ whose boundary word is g n w −n , such that B n is a combinatorial geodesic bigon with respect to the obvious decomposition of ∂B n (Cases 2a, 2b in [28,Section 4]). In particular, every disk component of B n is a single face, or has shape I 1 .…”

Section: 1mentioning

confidence: 99%

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Negative curvature in graphical small cancellation groups

et al. 2019

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We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical Gr ( 1 /6) small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically.We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group G containing an element g that is strongly contracting with respect to one finite generating set of G and not strongly contracting with respect to another. In the case of classical C ( 1 /6) small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting.We show that many graphical Gr ( 1 /6) small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups.In the course of our analysis we show that if the defining graph of a graphical Gr ( 1 /6) small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.

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Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Negative curvature in graphical small cancellation groups

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“…If Γ satisfies the C ′ (1/6)-condition or the C ′ * (1/6)-condition, then any closed path γ in Cay(G(Γ), S) admits a Γ-reduced diagram D over Γ, see [23,Lemma 2.13] or [25,Theorem 1.23] for the free group case and [24,Lemma 3.8] or [25,Theorem 1.35] for the free product case.…”

Section: Convexity Of Geodesics Inẋmentioning

confidence: 99%

“…The first 3 bullets are stated explicitly in [24], and the last bullet is deduced as follows: if both Π and Π ′ do not satisfy the claim, then |Π ∩ ω n | > |Π|/2 and |Π ′ ∩ ω n | > |∂Π ′ |/2 by the small cancellation hypothesis. Therefore, both |∂Π| 2|w| and |∂Π ′ | 2|w| by the minimality hypothesis on w. Hence, [24,Lemma 4.10] shows that both Π and Π ′ are special in the sense of [24,Lemma 4.11], whence we may apply [24,Lemma 4.11] as follows: if a is the arc in the intersection of Π and Π ′ , then |Π ∩ ω n | + |a| < |w| + |∂Π|/6 2|∂Π|/3. Apart from a, Π has at most one additional interior arc, and this arc must have length less than |∂Π|/6.…”

Section: Characterization Of Elliptic Elementsmentioning

confidence: 99%

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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The geometry of generalized loxodromic elements

Abbott¹,

Hume²

2018

Preprint

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We explore geometric conditions which ensure a given element of a finitely generated group is, or fails to be, generalized loxodromic; as part of this we prove a generalization of Sisto's result that every generalized loxodromic element is Morse. We provide a sufficient geometric condition for an element of a small cancellation group to be generalized loxodromic in terms of the defining relations and provide a number of constructions which prove that this condition is sharp.

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Infinitely presentedC(6)-groups are SQ-universal

Cited by 5 publications

References 26 publications

Negative curvature in graphical small cancellation groups

Negative curvature in graphical small cancellation groups

Small cancellation theory over Burnside groups

The geometry of generalized loxodromic elements

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