Embedability between right-angled Artin groups

Kim, Sang-Hyun; Koberda, Thomas

doi:10.2140/gt.2013.17.493

Cited by 98 publications

(150 citation statements)

References 31 publications

(60 reference statements)

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“…Equivalently, these are its Morse elements, its elements with contracting axes, its elements that act as rank-one isometries on the CAT(0) space associated to A(Γ), and its elements with cyclic centralizers [Ser89,BC12,BF09]. In [KK14a], loxodromic elements are characterized as those with unbounded orbit in the action of A(Γ) on its extension graph Γ e , a hyperbolic space introduced by the first author and Kim [KK13] to study embeddings between rightangled Artin groups. Here we study purely loxodromic subgroups of A(Γ): those in which every non-trivial element is loxodromic.…”

Section: Resultsmentioning

confidence: 99%

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The geometry of purely loxodromic subgroups of right-angled Artin groups

Koberda

Mangahas

Taylor

2017

Trans. Amer. Math. Soc.

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Abstract. We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Γ. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups.

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Section: Resultsmentioning

confidence: 99%

“…Rightangled Artin groups also play a key role in the study of three-manifold topology, culminating in Agol's resolution of the virtual Haken conjecture [Ago13, KM12,Wis11]. Right-angled Artin groups are also a prototypical class of CAT(0) groups, and have figured importantly in the study of mapping class groups of surfaces [CW04,CLM12,Kob12,KK13,KK14b,MT13].…”

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confidence: 99%

The geometry of purely loxodromic subgroups of right-angled Artin groups

Koberda

Mangahas

Taylor

2017

Trans. Amer. Math. Soc.

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“…The group A(P 4 ) contains a copy of (F 2 × Z) * Z, which cannot embed in Diff 1+bv + (M ) by Corollary 1.4. An explicit embedding of (F 2 × Z) * Z into A(P 4 ) is given by a, b, c, dad −1 A(P 4 ) (see [28] for a discussion on this fact). The program completed by Corollary 1.7 fully answers a question raised in a paper of Kapovich (attributed to Kharlamov) as to which right-angled Artin groups admit faithful C ∞ actions on the circle [27].…”

Section: Notes and Referencesmentioning

confidence: 99%

Free products and the algebraic structure of diffeomorphism groups

Kim

Koberda

2018

Journal of Topology

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Let M be a compact one-manifold, and let Diff 1+bv + (M ) denote the group of C 1 orientation preserving diffeomorphisms of M whose first derivatives have bounded variation. We prove that if G is a group which is not virtually metabelian, then (G × Z) * Z is not realized as a subgroup of Diff 1+bv + (M ). This gives the first examples of finitely generated groups G, H Diff ∞ + (M ) such that G * H does not embed into Diff 1+bv + (M ). By contrast, for all countable groups G, H Homeo + (M ) there exists an embedding G * H → Homeo + (M ). We deduce that many common groups of homeomorphisms do not embed into Diff 1+bv + (M ), for example the free product of Z with Thompson's group F . We also complete the classification of right-angled Artin groups which can act smoothly on M and in particular, recover the main result of a joint work of the authors with Baik [3]. Namely, a right-angled Artin group A(Γ) either admits a faithful C ∞ action on M , or A(Γ) admits no faithful C 1+bv action on M . In the former case,where Gi is a free product of free abelian groups. Finally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on S 1 .for the relevant manifolds (cf. [26,30])?Returning to the original motivation, Theorem 1.1 combined with results of Farb-Franks and Jorquera completes the classification of right-angled Artin groups admitting actions of various regularities on compact one-manifolds. Before stating the result, we define some terminology. We write Γ for a finite simplicial graph with vertex set V (Γ) and edge set E(Γ). The right-angled Artin group (or RAAG, for short) on Γ is defined as SANG-HYUN KIM AND THOMAS KOBERDAAfter some set theoretic computation, one sees the following:Note that we used bd ⊆b ∪d, and alsoc ∩d = ∅. It now suffices for us to prove the following claim:Claim. We have the following:cTo see the first part of the claim, let us consider x ∈ X satisfying cb(x) ∈c ∩ cbd.Then we have x ∈d and cb(x) ∈c. Sincec ∩d = ∅ and b(x) ∈ c −1c =c, we see x = b(x). In particular, we have x ∈b and cb(x) ∈ cb(b ∩d). This proves the first part of the claim. The second part follows by symmetry. Lemma 3.7. If b, c, d ∈ Diff 1 + (I) are given such that supp c ∩ supp d = ∅,Proof. As in the proof of Lemma 3.4, we let B = π 0 supp b and C = π 0 supp c. Letfor each B ∈ B. By Lemma 3.6, we have thatMoreover, for each B ∈ B we note thatClaim. The following set is a finite collection of intervals:We will employ the C 1 -hypothesis for this claim. Let us write

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“…For example, if

π

is the mapping class group of closed hyperbolic surface and

scriptA

is the collection of abelian subgroups generated by Dehn twists, then

K

is the curve complex of the underlying surface. If

π

is a right‐angled Artin group and

scriptA

is the collection standard abelian subgroups and their conjugations, then

K

is the flag complex of the ‘extension graph’, which was introduced by Kim and Koberda .…”

Section: Configurations Of Standard Abelian Subgroupsmentioning

confidence: 99%

Determining the action dimension of an Artin group by using its complex of abelian subgroups

Davis

Huang

2017

Bull. London Math. Soc.

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Suppose that (W,S) is a Coxeter system with associated Artin group A and with a simplicial complex L as its nerve. We define the notion of a ‘standard abelian subgroup’ in A. The poset of such subgroups in A is parameterized by the poset of simplices in a certain subdivision L⊘ of L. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right‐angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for BA. (This is the ‘action dimension’ of A denoted actdimA.) If Hdfalse(L;double-struckZ/2false)≠0, where d=dimL, then actdimA⩾2d+2. Moreover, when the K(π,1)‐Conjecture holds for A, the inequality is an equality.

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Embedability between right-angled Artin groups

Cited by 98 publications

References 31 publications

The geometry of purely loxodromic subgroups of right-angled Artin groups

The geometry of purely loxodromic subgroups of right-angled Artin groups

Free products and the algebraic structure of diffeomorphism groups

Determining the action dimension of an Artin group by using its complex of abelian subgroups

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