Gaps in scl for Amalgamated Free Products and RAAGs

Heuer, Nicolaus

doi:10.1007/s00039-019-00477-5

Cited by 10 publications

(19 citation statements)

References 24 publications

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“…We also note that Heuer, very recently, obtained the same lower bound of 1/2 for scl in any rightangled Artin group [Heu18]. His method is based on constructing quasimorphisms, as was the previous general lower bound of 1/24 established in [FFT16].…”

Section: Introductionsupporting

confidence: 55%

Genus bounds in right-angled Artin groups

Forester¹,

Soroko²,

Tao³

2020

Publicacions Matemàtiques

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Section: Introductionsupporting

confidence: 55%

Genus bounds in right-angled Artin groups

Forester¹,

Soroko²,

Tao³

2020

Publicacions Matemàtiques

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“…In the previous example, we already have seen that non-abelian free groups have a gap in stable commutator length of 1/2. This result has recently been generalised to right-angled Artin groups [40]. Many classes of non-positively curved groups have a gap in scl, though this gap may not be uniform in the whole class of groups.…”

Section: Gaps and Groups Of Non-positive Curvaturementioning

confidence: 78%

“…Many classes of non-positively curved groups have a gap in scl, though this gap may not be uniform in the whole class of groups. Prominent examples include hyperbolic groups [20], mapping class groups [10], free products of torsion-free groups [22] and amalgamated free products [23,28,40].…”

Section: Gaps and Groups Of Non-positive Curvaturementioning

confidence: 99%

The spectrum of simplicial volume

Heuer

Löh

2020

Invent. math.

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New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in $$\mathbb {R}_{\ge 0}$$ R ≥ 0 . In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the $$l^1$$ l 1 -semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.

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“…While this paper was being considered for publication, Heuer [Heu18] announced a sharp lower bound of 1/2 for scl in RAAGs, using a very different family of quasimorphisms.…”

Section: Introductionmentioning

confidence: 99%

Effective quasimorphisms on right-angled Artin groups

Fernós¹,

Forester²,

Tao³

2019

Annales De l'Institut Fourier

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We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24.These results make use of some new tools that we develop for the study of group actions on CAT(0) cube complexes: the essential characteristic set and equivariant Euclidean embeddings. arXiv:1602.05637v2 [math.GR] 27 Jun 2018Theorem A. Let X be a CAT(0) cube complex with a RAAG-like action by G. Then scl(g ) ≥ 1/24 for every hyperbolic element g ∈ G.Since the standard action of a right-angled Artin group on its associated CAT(0) cube complex is RAAG-like, with all non-trivial elements acting hyperbolically, the following corollary is immediate.Corollary B. Let G be a right-angled Artin group. Then scl(g ) ≥ 1/24 for every nontrivial g ∈ G.What is perhaps surprising about this result is that there is a uniform gap for scl, independent of the dimension of X . Note that in Theorem A we do not assume that X is either finite-dimensional or locally finite; thus Corollary B applies to right-angled Artin groups defined over arbitrary simplicial graphs.The defining properties of RAAG-like actions arose naturally while working out the arguments in this paper. It turns out, however, that RAAG-like actions are closely related to the special cube complexes of Haglund and Wise [HW08]. That is, if G acts freely on X , then the action is RAAG-like if and only if the quotient complex X /G is special. See Section 7 and Remark 7.4 for the precise correspondence between these notions.Corollary C. Let G be the fundamental group of a special cube complex. Then scl(g ) ≥ 1/24 for every non-trivial g ∈ G.This follows from Theorem A since the action of G on the universal cover is RAAG-like, with every non-trivial element acting hyerbolically. Alternatively, it follows from Corollary B and monotonicity, since every such group embeds into a right-angled Artin group. Related resultsThere are other gap theorems for stable commutator length in the literature, though in some cases the emphasis is on the existence of a gap, rather than its size. The first such result was Duncan and Howie's theorem [DH91] that every non-trivial element of a free group has stable commutator length at least 1/2. In [CFL16] it was shown that in Baumslag-Solitar groups, stable commutator length is either zero or at least 1/12. A different result in [CFL16] states that if G acts on a tree, then scl(g ) ≥ 1/12 for every "well-aligned" element g ∈ G. There are also gap theorems for stable commutator length in hyperbolic groups [Gro82, CF10] and in mapping class groups (and their finite-index subgroups) [BBF16b], w...

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Gaps in scl for Amalgamated Free Products and RAAGs

Cited by 10 publications

References 24 publications

Genus bounds in right-angled Artin groups

Genus bounds in right-angled Artin groups

The spectrum of simplicial volume

Effective quasimorphisms on right-angled Artin groups

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