Spectral gap of scl in graphs of groups and $3$-manifolds

Chen, Lvzhou; Heuer, Nicolaus

doi:10.48550/arxiv.1910.14146

Cited by 5 publications

(10 citation statements)

References 27 publications

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“…A new method using the idea of linear programming duality is used in [20] to obtain lower bounds in free products. It actually applies to graphs of groups and proves the so-called spectral gap properties with sharp estimates, which is discussed in detail in a paper with Nicolaus Heuer [21]. Here we simply describe this method in our setting.…”

Section: Lower Bounds From Dualitymentioning

confidence: 99%

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Scl in graphs of groups

Chen

2020

Invent. math.

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Section: Lower Bounds From Dualitymentioning

confidence: 99%

“…Due to our limited understanding of the scl spectrum, it is not clear if this is a proper subset and how big the difference is. We do know that the smallest positive elements in the two spectra exist and agree if M and L are odd; See [21,Corollary 5.10] and [20,Remark 3.6].…”

Section: Simple Normal Formmentioning

confidence: 99%

Scl in graphs of groups

Chen

2020

Invent. math.

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“…This invariant has seen a vast development in recent years, most prominently by Calegari and others [Cal09a,Heu19b,HL19,CH19].…”

Section: Stable Commutator Lengthmentioning

confidence: 99%

Computing commutator length is hard

Heuer¹

2020

Preprint

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The commutator length clG(g) of an element g ∈ [G, G] in the commutator subgroup of a group G is the least number of commutators needed to express g as their product.If G is a non-abelian free groups, then given an integer n ∈ N and an element g ∈ [G, G] the decision problem which determines if clG(g) ≤ n is NP-complete. Thus, unless P=NP, there is no algorithm that computes clG(g) in polynomial time in terms of |g|, the wordlength of g. This statement remains true for groups which have a retract to a non-abelian free group, such as non-abelian right-angled Artin groups.We will show these statements by relating commutator length to the cyclic block interchange distance of words, which we also show to be NPcomplete.

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“…The largest such 𝐶 is called the optimal spectral gap of 𝐺 for elements (respectively, chains). Various kinds of groups are known to have a gap for elements: word-hyperbolic groups [10], finite index subgroups of mapping class groups [4], subgroups of right-angled Artin groups (RAAGs; defined below) [26] and 3-manifold groups [14]; see Theorem 2.17. The spectral gap property can be used to obstruct group homomorphisms since the scl is non-increasing under homomorphisms.…”

Section: Introductionmentioning

confidence: 99%

Stable commutator length in right‐angled Artin and Coxeter groups

Chen

Heuer

2022

Journal of London Math Soc

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We establish a spectral gap for stable commutator length (scl) of integral chains in right-angled Artin groups (RAAGs). We show that this gap is not uniform, that is, there are RAAGs and integral chains with scl arbitrarily close to zero. We determine the size of this gap up to a multiplicative constant in terms of the opposite path length of the defining graph. This result is in stark contrast with the known uniform gap 1∕2 for elements in RAAGs. We prove an analogous result for right-angled Coxeter groups. In a second part of this paper, we relate certain integral chains in RAAGs to the fractional stability number of graphs. This has several consequences: First, we show that every rational number 𝑞 ⩾ 1 arises as the scl of an integral chain in some RAAG. Second, we show that computing scl of elements and chains in RAAGs is NP hard. Finally, we heuristically relate the distribution of scl for random elements in the free group to the distribution of fractional stability number in random graphs. We prove all of our results in the general setting of graph products. In particular, all above results hold verbatim for right-angled Coxeter groups.

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Spectral gap of scl in graphs of groups and $3$-manifolds

Cited by 5 publications

References 27 publications

Scl in graphs of groups

Scl in graphs of groups

Computing commutator length is hard

Stable commutator length in right‐angled Artin and Coxeter groups

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