2-chains and square roots of Thompson’s group

Koberda, Thomas; Lodha, Yash

doi:10.1017/etds.2019.14

Cited by 2 publications

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“…Examples of countable simple groups in G 0 pIqzG 1 pIq turn out to be abundant in isomorphism types. For us, a continuum means a set that has the cardinality of R. In joint work of the authors with Lodha [41] and in joint work of the second author with Lodha [43], the existence of a continuum of isomorphism types of finitely generated groups and of countable simple groups in G 0 pIqzG 1 pIq is established. These results relied on work of Bonatti-Lodha-Triestino [7].…”

Section: Sang-hyun Kim and Thomas Koberdamentioning

confidence: 99%

Diffeomorphism groups of critical regularity

Kim

Koberda

2020

Invent. math.

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Let M be a circle or a compact interval, and let α " k`τ ě 1 be a real number such that k " tαu. We write Diff ὰ pMq for the group of orientation preserving C k diffeomorphisms of M whose k th derivatives are Hölder continuous with exponent τ. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G ď Diff ὰ pMq with the property that G admits no injective homomorphisms into Ť βąα Diff β pMq. We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of Ş βăα Diff β pMq with the property that G admits no injective homomorphisms into Diff ὰ pMq. The groups G are constructed so that their commutator groups are simple. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if α ě 1 is a real number not equal to 2, then there is no nontrivial homomorphism Diff ὰ pS 1 q Ñ Ť βąα Diff β pS 1 q. Finally, we obtain an independent result that the class of finitely generated subgroups of Diff 1 pMq is not closed under taking finite free products.

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Section: Sang-hyun Kim and Thomas Koberdamentioning

confidence: 99%

Diffeomorphism groups of critical regularity

Kim

Koberda

2020

Invent. math.

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“…As for other classes of groups of homeomorphisms which cannot be realized as groups of C 1`bv diffeomorphisms, Corollary 1.5 is a complement to a result of the second author with Lodha [33], in which they show that certain "square roots" of Thompson's group F may fail to act faithfully by C 1`bv diffeomorphisms on a compact one-manifold, even though they are manifestly groups of homeomorphisms of these manifolds. Thus, the Ghys-Sergiescu Theorem appears to place F and T at the cusp of smoothability in the sense that even relatively minor algebraic variations on F and on T fail to be smoothable.…”

Section: Introductionmentioning

confidence: 85%

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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2-chains and square roots of Thompson’s group

Cited by 2 publications

References 17 publications

Diffeomorphism groups of critical regularity

Diffeomorphism groups of critical regularity

Free products and the algebraic structure of diffeomorphism groups

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