Right-angled Artin groups in the C ∞ diffeomorphism group of the real line

Baik, Hyungryul; Kim, Sang-Hyun; Koberda, Thomas

doi:10.1007/s11856-016-1307-8

Cited by 13 publications

(18 citation statements)

References 21 publications

(30 reference statements)

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“…We deduce that many common groups of homeomorphisms do not embed into

{prefixDiff}_{+}^{1 + bv} false(M false)

, for example the free product of

double-struckZ

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 . Namely, a right‐angled Artin group

A (Γ)

either admits a faithful

C^{\infty}

action on

M

, or

A (Γ)

admits no faithful

C^{1 + bv}

action on

M

.…”

supporting

confidence: 62%

“…Our main result implies that there is a large class A 0 of finitely generated subgroups of Diff ∞ + (M ) such that for all G, H ∈ A 0 , the free product G * H can never be realized as a subgroup of Diff 1+bv + (M ); see Corollary 1.7. As a corollary, we complete a program initiated by Baik and the authors in [2,3] to decide which right-angled Artin groups admit faithful C ∞ actions on a compact one-manifold, and exhibit many classes of finitely generated subgroups of Homeo + (M ) which cannot be realized as subgroups of Diff 1+bv + (M ).…”

Section: Introductionmentioning

confidence: 99%

“…As a corollary, we complete a program initiated by Baik and the authors in to decide which right‐angled Artin groups admit faithful

C^{\infty}

actions on a compact one‐manifold, and exhibit many classes of finitely generated subgroups of

{prefixHomeo}^{+} false(M false)

which cannot be realized as subgroups of

{prefixDiff}_{+}^{1 + bv} false(M false)

.…”

Section: Introductionmentioning

confidence: 99%

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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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“…We deduce that many common groups of homeomorphisms do not embed into

{prefixDiff}_{+}^{1 + bv} false(M false)

, for example the free product of

double-struckZ

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 . Namely, a right‐angled Artin group

A (Γ)

either admits a faithful

C^{\infty}

action on

M

, or

A (Γ)

admits no faithful

C^{1 + bv}

action on

M

.…”

supporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Free products and the algebraic structure of diffeomorphism groups

Kim

Koberda

2018

Journal of Topology

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“…Let Y 0 be the collection of intervals Y P Y such that for some I a P J a contained in Y and some I d P J d , we have cbI a X I d X pMzYq ‰ ∅; in this case, Lemma 5.5 and the fact supp a X Jpc, dq " ∅ together imply that I a ‰ I d and I d Ď Jpc, dq. Assume either (1)…”

Section: Lemma 53 the Following Statements Holdmentioning

confidence: 99%

Unsmoothable group actions on compact one-manifolds

2019

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We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by C 1`bv diffeomorphisms on the circle, which generalizes a result of Farb-Franks, and which parallels a result of Ghys and Burger-Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by C 1`bv diffeomorphisms on a compact one-manifold, then its defining graph has no subpath of length three. As a corollary, we also show that no finite index subgroup of AutpF n q and OutpF n q for n ě 3, the Torelli group for genus at least 3, and of each term of the Johnson filtration for genus at least 5, can act faithfully by C 1`bv diffeomorphisms on a compact one-manifold.

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“…For example, we do not even know if the 2-dimensional cube I 2 is Hölder. 1 On the other hand, it follows from the Lefshetz fixed-point theorem that even-dimensional spheres S 2n , n ≥ 1 (and more generally, even-dimensional rational homology spheres) are trivially Hölder. Interestingly, the circle S 1 is also a (non-trivial) Hölder manifold, that is any freely acting subgroup of orientation preserving circle homeomorphisms is necessarily Abelian (see [9]).…”

Section: Introductionmentioning

confidence: 99%