Spectral gap of scl in free products

Chen, Lvzhou

doi:10.1090/proc/13899

Cited by 18 publications

(12 citation statements)

References 16 publications

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“…We remark that Corollary 1 is similar to a recent result of Chen [, Corollary 3.7], however, neither Corollary 1 follows from [, Corollary 3.7] nor [, Corollary 3.7] follows from Corollary 1.…”

Section: Introductionsupporting

confidence: 68%

Quasiperiodic and mixed commutator factorizations in free products of groups

Ivanov

Klyachko

2018

Bull. London Math. Soc.

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It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation [x1, y1] . . . [x k , y k ] = z n , where n 2k, in the free product F of groups without nontrivial elements of order n implies that z is conjugate to an element of a free factor of F . If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct. m j=1 n j . Culler [8] discovered that, in the free group F (a, b) with free generators a, b, the cube [a, b] 3 of the commutator [a, b] := a −1 b −1 ab is a product of two commutators, [a, b] 3 = [a −1 ba, a −2 bab −1 ][bab −1 , b 2 ].Moreover, [a, b] n is a product of k commutators whenever n 2k − 1, see [8].

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Section: Introductionsupporting

confidence: 68%

Quasiperiodic and mixed commutator factorizations in free products of groups

Ivanov

Klyachko

2018

Bull. London Math. Soc.

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“…• Sometimes, one may control scl on certain generic group elements. If G = G 1 G 2 is the free product of two torsion-free groups G 1 and G 2 and g ∈ G does not conjugate into one of the factors, then scl(g) ≥ 1/2; see [Che18] and [IK17]. Similarly, if G = A C B and g ∈ G does not conjugate into one of the factors and such that CgC does not contain a copy of any conjugate of g −1 then scl(g) ≥ 1/12.…”

Section: Quasimorphisms and Bavard's Duality Theorem For What Followsmentioning

confidence: 99%

Gaps in scl for Amalgamated Free Products and RAAGs

Heuer

2019

Geom. Funct. Anal.

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We develop a new criterion to tell if a group G has the maximal gap of 1/2 in stable commutator length (scl). For amalgamated free products G = A C B we show that every element g in the commutator subgroup of G which does not conjugate into A or B satisfies scl(g) ≥ 1/2, provided that C embeds as a left relatively convex subgroup in both A and B. We deduce from this that every non-trivial element g in the commutator subgroup of a right-angled Artin group G satisfies scl(g) ≥ 1/2. This bound is sharp and is inherited by all fundamental groups of special cube complexes.We prove these statements by constructing explicit extremal homogeneous quasimorphisms φ : G → R satisfyingφ(g) ≥ 1 and D(φ) ≤ 1. Such maps were previously unknown, even for non-abelian free groups. For these quasimorphismsφ there is an action ρ : G → Homeo + (S 1 ) on the circle such that [δ 1φ ] = ρ * eu R b ∈ H 2 b (G, R), for eu R b the real bounded Euler class.

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“…The answer is negative in general, but we have an algorithmic criterion (Theorem 6. 20) in the special case of Baumslag-Solitar groups. For any reduced rational chain c, the criterion for the existence of extremal surfaces is expressed in terms of branched surfaces canonically built from the result of the linear programming problem computing scl BS(M,L) (c).…”

Section: Extremal Surfacesmentioning

confidence: 99%