Diffeomorphism groups of critical regularity

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`pMq for the group of orientation preserving Ck diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent τ. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G ď Diff`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`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`pS 1q Ñ Ť βąα Diff β `pS 1q. Finally, we obtain an independent result that the class of finitely generated subgroups of Diff`pMq is not closed under taking finite free products.


Introduction
Let M be the circle S 1 " R{Z or a compact interval I. A function f : M Ñ R is Hölder continuous with exponent τ if there is a constant C such that | f pxq´f pyq| ď C|x´y| τ for all x, y P M. In the case where M " S 1 , we implicitly define |x´y| to be the usual angular distance between x and y.
For an integer k ě 1 and for a smooth manifold M, we write Diff k`τ pMq for the group of orientation preserving C k diffeomorphisms of M whose k th derivatives are Hölder continuous with exponent τ P r0, 1q. For compactness of notation, we will write Diff ὰ pMq for Diff k`τ pMq, where k " tαu and τ " α´k. By convention, we will write Diff 0 pMq " Homeo`pMq.
The purpose of this paper is to study the algebraic structure of finitely generated groups in Diff ὰ pMq, as α varies. We note that the isomorphism types of finitely generated subgroups in Diff ὰ pIq coincide with those in Diff α c pRq, the group of compactly supported C α diffeomorphisms on R; see Theorem A.3.
Let us denote by G α pMq the class of countable subgroups of Diff ὰ pMq, considered up to isomorphism. It is clear from the definition that if α ď β then G β pMq Ď G α pMq. In general, it is difficult to determine whether a given element G P G α pMq also belongs to G β pMq. A motivating question is the following: Question 1.1. Let k ě 0 be an integer.
(1) Does G k pMqzG k`1 pMq contain a finitely generated group?
(2) Does G k pMqzG k`1 pMq contain a countable simple group?
The answer to the above question is previously known only for k ď 1 in part (1), and only for k " 0 in part (2). A first obstruction for the C 1 -regularity comes from the Thurston Stability [67], which asserts that every finitely generated subgroup of Diff 1 pIq is locally indicable. An affirmative answer to part (1) of Question 1.1 follows for k " 0 and M " I; that is, G 0 pIqzG 1 pIq contains a finitely generated group. Using Thurston Stability, Calegari proved that G 0 pS 1 qzG 1 pS 1 q contains a finitely generated group; see [15] for the proof and also for a general strategy of "forcing" dynamics from group presentations. Navas [58] produced an example of a locally indicable group in G 0 pMqzG 1 pMq; see also [16].
A different C 1 -obstruction can be found in the result of Ghys [29] and of Burger-Monod [12]. That is, if G is a lattice in a higher rank simple Lie group then G R G 1 pS 1 q. This result was built on work of Witte [71]. More generally, Navas [56] showed that every countably infinite group G with property (T) satisfies G R G 1 pIq and G R G 1.5` pS 1 q for all ą 0; it turns out that G R G 1.5 pS 1 q by a result of Bader-Furman-Gelander-Monod [1]. The exact optimal bound for the regularity of property (T) groups is currently unknown.
Plante and Thurston [63] proved that if N is a nonabelian nilpotent group, then N R G 2 pMq. By Farb-Franks [28] and Jorquera [36], every finitely generated residually torsion-free nilpotent group belongs to G 1 pMq. For instance, the integral Heisenberg group belongs to G 1 pMqzG 2 pMq. So, part (1) of Question 1.1 also has an affirmative answer for the case k " 1.
Another C 2 -obstruction comes from the classification of right-angled Artin groups in G 2 pMq [2,40]. In particular, Baik and the authors proved that except for finitely many sporadic surfaces, no finite index subgroups of mapping class groups of surfaces belong to G 2 pMq for all compact one-manifolds M [2]; see also [27,62]. Mapping class groups of once-punctured hyperbolic surfaces belong to G 0 pS 1 q; see [60,33,9].
Simplicity of subgroups often plays a crucial role in the study of group actions [25,66,13,38]. 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]. In particular, part (2) of Question 1.1 has an affirmative answer for k " 0 and M " I.

Summary of results.
Recall that M P tI, S 1 u. In this article, we give the first construction of finitely generated groups and simple groups in G α pMqzG β pMq.
Main Theorem. For all α P r1, 8q, each of the sets G α pMqz ď βąα G β pMq, č βăα G β pMqzG α pMq contains a continuum of finitely generated groups, and also contains a continuum of countable simple groups.
The Main Theorem gives an affirmative answer to Question 1.1.
Remark 1.2. One has to be slightly careful interpreting the Main Theorem when α " 1. This is because the set Diff β pMq is not a group for β ă 1. Using [24], we will prove a stronger fact that G Lip pMqzG 1 pMq contains the desired continua. Here, G Lip pMq denotes the set of isomorphism types of countable subgroups of Diff Lip pMq, the group of bi-Lipschitz homeomorphisms. Remark 1.3. It is interesting to note that in the case of M " I, the simple groups guaranteed by the Main Theorem for α ą 1 are locally indicable, as follows easily from Thurston Stability. Thus, we obtain a continuum of countable, simple, locally indicable groups. The commutator subgroup of Thompson's group F is one such example.
If G ď Diff ὰ pMq and if β ą α, an injective homomorphism G Ñ Diff β pMq is called an algebraic smoothing of G. The Main Theorem implies that for each α ě 1, there exists a finitely generated subgroup G ď Diff ὰ pMq that admits no algebraic smoothings beyond α. Moreover, the finitely generated groups in the continua of the Main Theorem can always be chosen to be non-finitely-presented as there are only countably many finitely presented groups up to isomorphism.
In Section 2.1 we give the definition of concave moduli (of continuity), a strict partial order ! between them, and the symbol ą k 0. For instance, ω τ pxq " x τ is a concave modulus satisfying ω τ ą k 0 for each τ P p0, 1s and k P N. For a concave modulus ω, we let Diff k,ὼ pMq denote the group of C k -diffeomorphisms on M whose k th derivatives are ω-continuous. We also write Diff k,0 pMq :" Diff k pMq. We denote by Diff k,bv pIq the group of diffeomorphisms f P Diff k pIq such that f has bounded total variation. Note that Diff k,bv pIq contains Diff k,Lip pIq, the group of C k -diffeomorphisms whose k th derivatives are Lipschitz.
For a concave modulus ω or for ω P t0, bvu, the set of all countable subgroups of Diff k,ὼ pMq is denoted as G k,ω pMq. We will deduce the Main Theorem from a stronger, unified result as can be found below.
Theorem 1.4. For each k P N, and for each concave modulus µ " ω 1 , there exists a finitely generated group Q " Qpk, µq ď Diff k,μ pIq such that the following hold.
(i) rQ, Qs is simple and every proper quotient of Q is abelian; (ii) if ω " bv, or if ω is a concave modulus satisfying µ " ω ą k 0, then rQ, Qs R G k,ω pIq Y G k,ω pS 1 q. Theorem 1.4 will imply the Main Theorem after making suitable choices of µ above. See Section 6.4 for details.
We let F n denotes a rank-n free group. Let BSp1, 2q denote the solvable Baumslag-Solitar group of type p1, 2q; see Section 3. In the case when M " I, our construction for Theorem 1.4 builds on a certain quotient of the group G : " pZˆBSp1, 2qq˚F 2 .
Let us describe our construction more precisely. Theorem 1.5. Let k P N, and let µ be a concave modulus such that µ " ω 1 . Then there exists a representation φ k,µ : G : Ñ Diff k,μ pIq such that the following hold.
We deduce that the group φ k,µ pG : q admits no injective homomorphisms into Diff k,ὼ pIq. We will then bootstrap this construction to produce simple groups in Section 6. We define the critical regularity on M of an arbitrary group G as CritReg M pGq :" suptα | G P G α pMqu.
Here, we adopt the convention sup ∅ "´8. The critical regularity spectrum of M that is defined as C M :" tCritReg M pGq | G is a finitely generated group u Another consequence of the Main Theorem is the following.
Corollary 1.6. The critical regularity spectrum of M, which is defined as C M :" tCritReg M pGq | G is a finitely generated group u , coincides with t´8u Y r1, 8s.
Theorem 1.5 gives the first examples of groups whose critical regularities are determined (and realizable) and belong to p1, 8q. To the authors' knowledge, the critical regularities of the following three groups are previously known and finite. First, Navas proved that Grigorchuk-Machi groupH of intermediate growth has critical regularity 1, and that the critical regularity ofH can be realized [57]. Second, Castro-Jorquera-Navas proved ( [22], combined with [63]) that the integral Heisenberg group has critical regularity 2 and this critical regularity cannot be attained. Thirdly, Jorquera, Navas and Rivas [37] proved that the nilpotent group N 4 of 4ˆ4 integral lower triangular matrices with ones on the diagonal satisfies CritReg I pN 4 q " 3{2.
It is not known whether or not the critical regularity 3{2 of N 4 is realizable.
The case G P G 1 pMqzG 0 pMq requires a suitable interpretation the critical regularity. As we have mentioned in Remark 1.2, it is proved by Deroin, Kleptsyn and Navas that every countable subgroup G of Homeo`pMq is topologically conjugate to a group of bi-Lipschitz homeomorphisms [24]. Thus, it is reasonable to say that r0, 1q is missing from from the critical regularity spectrum.
The authors proved in [40] that for each integer 2 ď k ď 8, the class of finitely generated group in G k pMq is not closed under taking finite free products. From [8] and from the consideration of BSp1, 2q actions in the current paper, we deduce the following augmentation for k " 1. We are grateful to A. Navas for pointing us to the reference [8] and telling us the proof of the following corollary for M " I. See Section 3.4 for details.
Corollary 1.7. The group pZˆBSp1, 2qq˚Z does not embed into Diff 1 pMq. In particular, the class of finitely generated subgroups of Diff 1 pMq is not closed under taking finite free products.
Though we concentrate primarily on countable groups, our results have applications to continuous groups. For a smooth manifold X and for an α ě 1, we let Diff α c pXq 0 denote the group of C α diffeomorphisms of X isotopic to the identity through a compactly supported C α isotopy. If 1 ď α ă β, then there is a natural embedding Diff β c pXq 0 Ñ Diff α c pXq 0 defined simply by the inclusion. The main result (and its proof) of [47] by Mann implies that if X P tS 1 , Ru, and if 2 ă α ă β are real numbers, then there exists no injective homomorphisms Diff α c pXq 0 Ñ Diff β c pXq 0 . We generalize this to all real numbers 1 ď α ă β. Corollary 1.8. Let X " tS 1 , Ru. Then arbitrary homomorphisms of the following types have abelian images: (1) and (2), and if α ą 3 in part (3), then all the above homomorphisms have trivial images.
The Main Theorem has the following implication on the existence of unsmoothable foliations on 3-manifolds. This extends a previous result of Tsuboi [70] and of Cantwell-Conlon [21], that is originally proved for integer regularities. Corollary 1.9. Let α ě 1 be a real number. Then for every closed orientable 3manifold Y satisfying H 2 pY, Zq ‰ 0, there exists a codimension-one C α foliation pY, F q which is not homeomorphic to a Ť βąα C β foliation. Here, a homeomorphism of foliations is a homeomorphism of the underlying foliated manifolds which respects the foliated structures.

Notes and references.
1.2.1. Automatic continuity. K. Mann proved that if X is a compact manifold then the group Homeo 0 pXq of homeomorphisms isotopic to the identity has automatic continuity, so that every homomorphism from Homeo 0 pXq into a separable group is continuous [49]. She uses this fact to prove that Homeo 0 pXq has critical regularity 0 and hence has no algebraic smoothings. For discussions of a similar ilk, the reader may consult [48] and [35]. The Main Theorem implies that the critical regularity of Diff ὰ pMq is α, for M P tI, S 1 u and for α ě 1.
1.2.2. Superrigidity. Recall that Margulis Superrigidity says that under suitable hypotheses, a representation of a lattice Γ in a higher rank Lie group G is actually given by the restriction of a representation of G to Γ (see [51]). For the continuous groups Diff ὰ pMq which we consider here, there is no particularly clear analogue of a lattice. Nevertheless, some of the results proved in this paper are reminiscent of similar themes. Particularly, Corollary 1.8 is established by showing that all of the maps in question contain a countable simple group (perhaps a suitable analogue of a lattice) in their kernel, thus precluding the existence of a nontrivial homomorphism between the corresponding continuous groups.

1.2.3.
Topological versus algebraic smoothability. The smoothability issues that we consider in this paper center around algebraic smoothability of group actions. There is a stronger notion of smoothability called topological smoothability. A topological smoothing of a representation φ : G Ñ Diff ὰ pMq is a topological conjugacy of φ into Diff β pMq for some β ą α; that is, the conjugation hφh´1 of φ by some homeomorphism h on M such that we have hφpGqh´1 ď Diff β pMq. A topological smoothing of a subgroup is obviously an algebraic smoothing, but not conversely; compare [22] and [37]. By a result of Tsuboi [70], there exists a two-generator solvable group G and a faithful action ϕ k of G on the interval such that ϕ k pGq ď Diff k pIq but such that ϕ k pGq is not topologically conjugate into Diff k`1 pIq. Since ϕ k is injective, these actions are algebraically smoothable. See Section 6.5 regarding implications for foliations.

Disconnected manifolds.
It is natural to wonder whether or not the results of this paper generalize to compact one-manifolds which are not necessarily connected; these manifolds are disjoint unions of finitely many intervals and circles (cf. [2,40]). It is not difficult to see that the results generalize. Indeed, if G is a group of homeomorphisms of a compact disconnected one-manifold M, then a finite index subgroup of G stabilizes all the components of M. We build a finitely generated group G whose commutator subgroup rG, Gs is simple, and such that rG, Gs has the critical regularity exactly α with respect to faithful actions on the interval or the circle. Some finite index subgroup of G stabilizes each component of M, and since rG, Gs is infinite and simple, rG, Gs stabilizes each component of M. It follows that G has critical regularity α with respect to faithful actions on M.
1.2.5. Kernel structures. In Theorem 1.5, let us fix P p0, 1q such that ω ! µ. It will be impossible to find a finite set S Ď G : z ker φ such that for all ψ P HompG : , Diff k` `p Iqq we have S X ker ψ ‰ ∅. Indeed, Lemma 3.5 implies that for all finite set S Ď G : zt1u there exists a C 8 action of G : on R with a compact support such that S does not intersect the kernel of this action. So, one must consider an infinite set of candidates that could be a kernel element of such a ψ.

1.3.
Outline of the proof of Theorem 1.5. Given a concave modulus µ, we build a certain representation φ of the group G : into Diff k,μ pIq. For P p0, 1s satisfying ω :" ω ! µ, we also show that the group φpG : q admits no algebraic smoothing into Diff k,ὼ pIq. We remark that Diff k`1 pIq ď Diff k,ὼ pIq.
To study maps into Diff k,ὼ pIq, we use a measure of complexity of a diffeomorphism f , which is roughly the number of components of supports of generators of G : needed to cover the support of f . We prove a key technical result governing this complexity; this result is called the Slow Progress Lemma and applies to an action of an arbitrary finitely generated group on I. To have a starting diffeomorphism with finite complexity, we build an element 1 ‰ u P G : such that if ψ : G : Ñ Diff 1 pIq is an arbitrary representation then the support of ψpuq is compactly contained in the support of ψpG : q.
Next, we build an action φ of G : so that certain judiciously chosen conjugates w j uw´1 j of u, which depend strongly on the regularity pk, µq, result in a sequence of diffeomorphisms φpw j uw´1 j q whose complexity grows linearly in j. We show that under an arbitrary representation ψ : G : Ñ Diff k,ὼ pIq, the complexity of ψpw j uw´1 j q grows more slowly than that of φpw j uw´1 j q, a statement which follows from the Slow Progress Lemma. Thus for each ψ, we find an element g P G : which survives under φ but dies under ψ. In particular, φpG : q cannot be realized as a subgroup of Diff k,ὼ pIq.

1.4.
Outline of the paper. We strive to make this article as self-contained as possible. In Section 2, we build up the analytic tools we need. Section 3 summarizes the dynamical background used in the sequel, and proves Corollary 1.7. Section 4 establishes the Slow Progress Lemma for a general finitely generated group action on intervals. In Section 5, we fix a concave modulus µ, and construct a representation φ of the group G : into Diff k,μ pIq with desirable dynamical properties and prove Theorem 1.5. In Section 6, we complete the proof of the Main Theorem and gather the various consequences of the main results.

Probabilistic dynamical behavior
Throughout this section and for the rest of the paper, we will let I denote a nonempty compact subinterval of R. All homeomorphisms considered in this paper are assumed to be orientation preserving. We continue to let M " I or M " S 1 .
We wish to develop the concepts of fast and expansive homeomorphisms (Definition 2.8). These concepts establish a useful relationship between the dynamical behavior of a diffeomorphism supported on I and its analytic behavior, which is to say its regularity.
2.1. Moduli of continuity. We will use the following notion in order to guarantee the convergence of certain sequences of diffeomorphisms.
(2) Let ω be a concave modulus . For U Ď R or U Ď S 1 , we define the ω-norm of a map f : U Ñ R as We say f is ω-continuous if f has a bounded ω-norm.
The notion of ω-continuity depends only on the germs of ω for bounded functions, as can be seen from the following easy observation.

Lemma 2.2.
Let ω be a concave modulus , and let f : U Ñ R be a bounded function for some U Ď R. If there exist constants K, δ ą 0 such that | f pxq´f pyq| ď K¨ωp|x´y|q for all 0 ă |x´y| ď δ, then we have r f s ω ă 8. Remark 2.3. It is often assumed in the literature that a concave modulus ωpxq is defined only locally at x " 0, namely on r0, δs for some δ ą 0 [52,53]. This restriction does not alter the definition of ω-continuity for compactly supported functions. The reason goes as follows. Suppose ω : r0, δs Ñ r0, ωpδqs is a strictly increasing concave homeomorphism. By an argument in the proof of Lemma A.9, we can find a concave modulus µ : r0, 8q Ñ r0, 8q such that ωpsq ď µpsq ď p2`δ{ωpδqq ωpsq for all s P r0, δs. By Lemma 2.2, we conclude that the ω-continuity coincides with the µ-continuity for a compactly supported function.
The complex plane C has a natural lexicographic order ă C ; that is, we write z ă C w in C if Re z ă Re w, or if Re z " Re w and Im z ă Im w. For two complex numbers a, b P C, we let pa, bs C :" tz P C | a ă C z ď C bu.
In particular, we have that p0, 1s C :" ts We similarly define pa, bq C , together with the other types of intervals.
Then ω z is a small perturbation of ω τ pxq " x τ " expp´τ logp1{xqq. By simple computations of the derivatives, one sees that ω z is a concave modulus defined for all small x ě 0. See Figure 1 for the graphs of ω z . We will use the notation in Example 2.4 for the rest of the paper. The Hölder continuity of exponent τ P p0, 1q is equivalent to the ω τ -continuity.
(2) For two positive real sequences ta j u and tb j u, we will write ta j u À tb j u if ta j {b j u is bounded.
In particular, the expression ω ą k 0 is vacuously true for k ą 1. Compare this condition to Mather's Theorem (Definition 3.12 and Theorem 3.13).
Lemma 2.6. The following hold for k P N and for a concave modulus ω.
(3) Assume that we have positive sequences ta j u and tb j u such that ta k´1 j ωpa j qu À tb k´1 j ωpb j qu. If ω ą k 0, then we have ta j u À tb j u.
Proof. Proofs of (1) and (2) are obvious from monotonicity and concavity. Assume (3) does not hold. Passing to a subsequence, we may assume tt j :" b j {a j u converges to 0. Then we have a contradiction because b k´1 j ωpb j q a k´1 j ωpa j q " t k´1 j¨ω pt j a j q ωpa j q Ñ 0 as j Ñ 8.
Suppose ω and µ are concave moduli. We define a strict partial order ω ! µ if lim xÑ`0 ωpxq log K p1{xq µpxq " 0 for all K ą 0. Here, we use the notation log K t " plog tq K .
From z ă C w, we see that the above limit equals´8. This is as desired.
Let k P N and let ω be a concave modulus. A C k,ω -diffeomorphism on M is defined as a diffeomorphism f of M such that f pkq is ω-continuous. We say the pair pk, ωq is a regularity of f . If ω " ω τ for some τ P p0, 1q then a C k,ω -diffeomorphism means a C k`τ -diffeomorphism. We have C k,ω 1 " C k,Lip .
Let f : I " rp, qs Ñ R be a map. Recall that the (total) variation of f is given by where the supremum is taken over all possible finite partitions of I. A function has bounded variation on I if Varp f, Iq is finite on I. If M " S 1 , we use the same definition for Varp f, Iq with p " q. We say that a diffeomorphism f : M Ñ M is C k,bv if f is C k and if in addition we have f pkq has bounded variation.
Let ω be a concave modulus, or let ω " bv. We write for The set of all C k,ω diffeomorphisms of M is denoted as Diff k,ὼ pMq, which turns out to be a group for k P N (Proposition A.2). We define G k,ω pMq to be the set of the isomorphism classes of countable subgroups of Diff k,ὼ pMq.
If we have two concave of moduli ω ! µ, then we have Diff k,ὼ pMq ď Diff k,μ pMq.
In particular, if z, w P p0, 1s C satisfy z ă C w, then we see from Lemma 2.7 that Diff k,ω z pMq ě Diff k,ω ẁ pMq.

2.2.
Fast and expansive homeomorphisms. From now on until Section 6, we will be mostly concerned with the case M " I. For a measurable set J Ď R, we denote by |J| its Lebesgue measure. We write J 1 for the derived set of J, which is to say the set of the accumulation points of J. If X is a set, we let #X denote its cardinality.
Let f : X Ñ X be a map on a space X. We use the standard notations Fix f " tx P X | f pxq " xu, The set supp f is also called the (open) support of f . We note the identity map Id : R Ñ R satisfies Id p jq pxq " δ 1 j for j ě 1.
Definition 2.8. Let f : I Ñ I be a homeomorphism, and let J Ď I be a compact interval such that f pJq " J. We let k P N.
(1) We say f is k-fixed on J if one of the following holds: (3) We say f is λ-expansive on J for some λ ą 0 if sup yPJ | f pyq´y| dpty, f pyqu, BJq ě λ.
We note that f has one of the above three properties if and only if so does f´1. Note also that f is λ-expansive on J " rp, qs if and only if there exists some y P J satisfying one of the following (possibly overlapping) alternatives: (E1) p ă y ă f pyq ă q and f pyq´y ě λpy´pq; (E2) p ă y ă f pyq ă q and f pyq´y ě λpq´f pyqq; (E3) p ă f pyq ă y ă q and y´f pyq ě λp f pyq´pq; (E4) p ă f pyq ă y ă q and y´f pyq ě λpq´yq.
For a set A Ď N, we define its natural density as if the limit exists. A crucial analytic tool of this paper is the following probabilisitic description of fast and expansive homeomorphisms.
Theorem 2.9. Let k P N, and let ω ą k 0 be a concave modulus. Suppose we have (i) a diffeomorphism f P Diff k,ὼ pIq Y Diff k,bv pIq; (ii) a sequence tN i u Ď N such that sup iPN N i p1{iq k´1 ωp1{iq ă 8; (iii) a sequence of compact intervals tJ i u in I such that f is k-fixed on each J i and such that sup iPN #t j P N | J i X J j ‰ ∅u ă 8. Then for each δ ą 0 and λ ą 0, the following set has the natural density zero: The proof of the theorem is given in Section 2.3.

2.3.
Proof of Theorem 2.9. Let k and ω be as in Theorem 2.9. We first note a classical result in number theory.
Lemma 2.10. For sets A, B Ď N, the following hold.
(1) If d N pAq " 1 for some A Ď N and if i P N, then d N ppA´iq X Nq " 1.
(2) If d N pAq " d N pBq " 1 for some A, B Ď N, then d N pA X Bq " 1.
(3) ( [65,54]) If ř iPA 1{i is convergent, then d N pAq " 0. Fastness and expansiveness constants of "roots" of a diffeomorphism behave like arithmetic and geometric means, respectively: Lemma 2.11. Let f P Homeo`pJq for some compact interval J, and let N P N.
Proof. Let us write J " rp, qs.
(1) For some y P J we have Hence there exists some y 1 " f i pyq such that | f py 1 q´y 1 | ě δ N |J|. (2) Assume the alternative (E1) holds as described after Definition 2.8. That is, p ă y ă f pyq ă q for some y P J such that f N pyq´y ě λpy´pq. Note that So, for some y 1 " f i pyq, we have f py 1 q´y 1 y 1´p`1 . This is the desired inequality. The other alternatives are similar.
Lemma 2.12. For a C k -map f : I Ñ R, the following hold.
(2) If f is k-fixed on a compact interval J Ď I, then p f´Idq p jq has a root in J for each j " 0, 1, . . . , k.
Proof. For each j P t0, 1, . . . , ku, we define S j : " S j p f q " tx P I | f p jq pxq " Id p jq pxqu.
(1) We have S 1 j Ď S j . It now suffices for us to show the following: Let us assume x P S 1 j for some 0 ď j ă k. Then there exists a sequence tx i u Ď S j ztxu converging to x. There exists y i between x i and x such that Since y i P S j`1 converges to x, we see that x P S 1 j`1 . This proves S 1 j Ď S 1 j`1 . (2) By part (1), it suffices to consider the case that #pJ X Fix f q ě k`1. We inductively observe that p f´Idq p jq has at least pk`1´jq roots for each j " 0, 1, . . . , k by the Mean Value Theorem.

"
p f´1q pkq ‰ ω˘¨| J| k´1 ωp|J|q ě λ. Proof. For each j ď k, Lemma 2.12 implies that there exists s j P J satisfying f p jq ps j q " Id p jq ps j q.
Let y 0 P J be arbitrary. We see (cf. Lemma A.4) that | f py 0 q´y 0 | "ˇˇˇˇż (1) Pick y 0 P J such that | f py 0 q´y 0 | ě δ|J|. We see If f is C k,ω , then we further deduce that f pkq ‰ ω¨| J| k ωp|J|q.
By applying the same estimate for s 0 " p, we see that If f is C k,ω , we further have The other alternatives can be handled in the same manner; in particular, we use the diffeomorphism g " f´1 for (E2) and (E3).
Proof of Theorem 2.9: C k,ω case. We assume f : I Ñ I is a C k,ω -diffeomorphism. Let δ, λ ą 0, and define We let K ą 0 be the larger value of the suprema in the conditions (ii) and (iii). The following claim is obvious from (iii) and from a maximality argument. Claim 1. The sequence of intervals tJ i u can be partitioned into at most K collections such that each collection consists of disjoint intervals. In particular, we have It now suffices for us to establish the two claims below.
By Lemmas 2.11 and 2.13, we have that Lemma 2.10 now implies the claim.
There is a constant K 0 ą 0 such that Hence, Lemmas 2.11 and 2.13 imply that As in Claim 2, we have ř B λ 1{i ă 8 and d N pB λ q " 0. Proof of Theorem 2.9: C k,bv case. We now assume f is a C k,bv -diffeomorphism. Let us closely follow the proof of C k,ω case, using the same notation. In particular, we define the same sets A δ and B λ .
and f pkq pz i q " Id pkq pz i q " δ 1k . Again, it suffices to prove the following two claims.
By Lemmas 2.11 and 2.13, we have that So, for some constant K 0 , K 1 ą 0 we deduce from Hölder's inequality that We conclude from Lemma 2.10 that d N pA δ q " 0.
We apply Lemma 2.13 and also the proof of Claim 3. For each i P N we put We have We again apply Hölder's inequality. For some constant K 0 , K 1 ą 0, we see We obtain d N pB λ q " 0.

2.4.
Diffeomorphisms of optimal regularity. Let us now describe a method of constructing a fast diffeomorphism of a specified regularity on a given support.
Theorem 2.14. We let k P N, let δ P p0, 1q and let µ be a concave modulus satisfying µ " ω 1 . Suppose that tJ i u iPN is a disjoint collection of compact intervals such that J i Ď IzBI, and that tN i u iPN Ď N is a sequence such that Then there exists f P Diff k,μ pRq satisfying the following: if an open neighborhood U of x P R intersects only finitely many J i 's, then f is C 8 at x.
Since I is compact, it is necessary that ř i |J i | ă 8. From the above theorem we will deduce that some C k,µ diffeomorphism is "faster" than all C k,ω diffeomorphisms for ω ! µ, in a precise sense as described in Corollary 2.20.
Throughout Section 2.4, we will fix the following constants.
Setting 2.15. Let k, δ, µ be as in Theorem 2.14. Pick a constant 0 P p0, 1q and put A priori, we will choose 0 so small that we have estimates We also pick 0 P p0, 0 s such that µp 0 q ď 0 .
We will prove Theorem 2.14 through a series of lemmas. Let us first note the following standard construction of a bump function Ψ; see Figure 2 (a). Lemma 2.16. There exists an even, C 8 map Ψ : R Ñ R such that the following hold: Let us introduce a constant The following technical lemma establishes the existence of a bump function with a long flat interval and with a controlled C k -norm. See Figure 2 (b).
Proof. There exists a unique C 8 map g satisfying the following conditions: Hence, we have (i). If t P p0, {2q, then It follows that g 1 ptq P r0, 1{2s. Since we have the symmetry gptq " gp ´tq, we obtain (ii). We see }g} 8 " C k µp q ď Cµp q ď K 0 µp q. If t ď {2 and i ě 1, then The condition (iii) follows. To verify (iv), let us estimate |g pkq pxq´g pkq pyq|. We have that g pkq " 0 on p´8, 0q Y pD , p1´Dq q Y p , 8q.
Using the symmetry |g pkq pxq| " |g pkq p ´xq|, we may only consider x P r0, D s.
So, we have an inequality We now have the following three possibilities for y. Case 1. y P p´8, 0s Y rD , p1´Dq s Y r , 8q.
Lemma 2.18. For each compact interval J Ď R with 0 ă :" |J| ď 0 , there exists a diffeomorphism f P Diff 8 pRq satisfying the following: Proof. We may assume J " r0, s. Let g be as in Lemma 2.17, and put f " Id`g. By symmetry and the condition (ii) on g, we have for all t. We have (B), and in particular, f is a C 8 diffeomorphism.
The claims (A), (C) and (E) are immediate from Lemma 2.17. Observe that For each N ě 1{p k´1 µp qq, we have that This establishes the claim (D), and hence the conclusion of the lemma.
Proof of Theorem 2.14. Put i " |J i |. As For each n ě i 0 , consider the composition For m ě n ě i 0 , we have that Hence tF n u uniformly converges to a C k map F : R Ñ R in the C k -norm [26].
Since F is the composition of infinitely many homeomorphisms with disjoint supports, we see F is also a homeomorphism. In particular, we see supp F " Claim. For all x, y P R we have In order to prove the claim, we may assume x P J i for some i ě i 0 . If y P J i , then the condition (E) implies the claim. If y R supp F, then we can find x 0 P BJ i such that |x´y| ě |x´x 0 |. So, Finally, if y P J j for some i ‰ j ě i 0 , then we can find x 0 P BJ i and y 0 P BJ j such that |x´y| ě |x´x 0 |`|y´y 0 |. As µ is increasing, we see that Hence, the claim is proved. We have that F P Diff k,μ pRq.
Finally, we can pick F˚P Diff 8 pRq such that: Then the diffeomorphism f " F˝F˚P Diff k,μ pIq satisfies the conclusions (i) and (ii). To see the conclusion (iii), observe from the hypothesis that either In all cases, f coincides with some f i locally at x, and hence, is locally C 8 .
Remark 2.19. In the above proof, the modulus of continuity was used to guarantee a uniform convergence of partially defined diffeomorphisms. This idea can be found in the construction of a Denjoy counterexample, which is a C 1` diffeomorphism f : S 1 Ñ S 1 such that f is not conjugate to a rotation and such that f has an irrational rotation number. Denjoy's Theorem implies that there are no such C 1`bv examples [23,59].
We note the following consequence of Theorem 2.14.
Corollary 2.20. Let K˚ą 0, and let tJ i u iPN be a collection of disjoint compact intervals contained in the interior of I satisfying Then for k P N and for a concave modulus µ " ω 1 , there exists Proof. Let us write i " |J i | and We have f P Diff k,μ pIq as given by Theorem 2.14 with respect to tJ i u and some δ P p0, 1q. Let us pick ω such that 0 ă k ω ! µ.
For all sufficiently large i, we have So we see that Note that BJ i are accumulated fixed points of f . Since f N i is δ-fast on J i for all i, Theorem 2.9 implies that f is not C k,ω . For C k,bv , we simply set ω " ω 1 and apply Theorem 2.9 again.
We will need the following properties of density-one sets.
(1) If A Ď N satisfies d N pAq " 1, then for each s P N we have d N ti P N : i`rss˚Ď Au " 1.
(2) Let β 0 P N, and let X, Y Ď N. Assume that d N pX Y ppY´βq X Nqq " 1 for each integer β ě β 0 . Then we have d N pX Y Yq " 1.
Proof of Lemma 2.21.
(1) We can rewrite the given set as The conclusion follows from the first two parts of Lemma 2.10.
(2) Pick an arbitrary integer N ě β 0 . For each β P N, define Hence, for each s P N and t P rNs˚we compute pN´β 0 q¨#`S N 2 X pt`Nrss˚q˘ď # tm P t`rN s´N`1s˚| m P X Y Yu . By summing up the above for t P rNs˚, we have pN´β 0 q#`S N 2 X rN ss˚˘ď N# tm P rN ss˚| m P X Y Yu . After dividing both sides by N 2 s and sending s Ñ 8, we see that Since N is arbitrary, we have d N pX Y Yq " 1.

Background from one-dimensional dynamics
In this section, we gather the relevant facts regarding one-dimensional dynamics that we require in the sequel.
3.1. Covering distance and covering length. Throughout Section 3.1, we let G be a group with a finite generating set V, and let ψ : G Ñ Homeo`pIq be an action. We develop some notions of complexity of an element in ψpGq which will be useful for our purposes.
We use the notation Note that supp ψ may have multiple components. Define Then V is an open cover of supp ψ consisting of intervals. For a nonempty subset A Ď I, we define its V -covering length as Here, we use the convention inf ∅ " 8. We also let CovLen V p∅q " 0. We define the V -covering distance of x, y P I as That is to say, once a generating set for G has been fixed, CovDist V px, yq is the least number of components of supports of generators of G needed to traverse the interval from x to y. Also, if x and y lie in different components of supp ψpGq, then the covering distance between them is necessarily infinite. We let CovDist V px, xq " 0.
Both covering distance and covering length depend not just on G and ψ but also on a generating set V. When the meaning is clear, we will often omit V , and write CovLenpAq and CovDistpx, yq. We will also write gx :" ψpgq.x for g P G and x P I.
Covering distance behaves well in the sense that it satisfies the triangle inequality: Lemma 3.1. For x, y, z P I and for A, B Ď I, the following hold.
(1) CovDistpx, yq ă 8 if and only if x and y are contained in the same component of supp ψ.
are open covers of A and B which witness the fact that CovLenpAq " n and CovDistpBq " m respectively. Then (2).
where v i P V and n i P Z for each 1 ď i ď . The following lemma relates the algebraic structure of the given group G " xVy with the dynamical behavior of actions of G: For each x P I and w P G, we have CovDistpx, wxq ď ||w||.
Here we are implicitly measuring the covering distance with respect to the generating set V of G.
Proof of Lemma 3.2. Clearly we may assume that x P supp ψ, since otherwise there is nothing to prove. We proceed by induction on ||w||. If ||w|| " 1 then w " v n for some v P V and n P Z. Then either x " v n x or x P J P π 0 supp ψpvq. It follows that CovDistpx, v n xq is 0 or 1. Now assume ||w|| " ě 2. We can write w " v n¨w1 , where ||w 1 || " ´1. By induction, CovDistpx, w 1 xq ď ´1. As CovDistpw 1 x, v n¨w1 xq ď 1, the estimate follows from Lemma 3.1.
Let pU 1 , . . . , U n q be a sequence of open intervals in R such that U i X U j " ∅ whenever |i´j| ě 2, and such that U i X U i`1 is a nonempty proper subset of both U i and U i`1 for 1 ď i ď n´1. Then we say pU 1 , . . . , U n q is a chain of intervals in R. Figure 3 gives an example of a chain of four intervals.
A finite set F of intervals is also called a chain of intervals if F becomes so after a suitable reordering. Chains of intervals arise naturally when we consider an open cover of a compact interval. The proof of the following lemma is straightforward. When we discuss a chain of intervals, we assume those intervals are open. It will be useful for us to be able to move points inside a connected component of supp ψpGq efficiently in the following sense, which provides a converse to Lemma 3.2: Then there exists an element g P G such that gx ą y and such that ||g|| " N.
We remark that ideas in a very similar spirit to Lemma 3.4 were used extensively in [41].
Proof of Lemma 3.4. Let tU 1 , . . . , U N u be intervals such that U i P π 0 supp ψpv i q for some v i P V for each i, and such that these intervals witness the fact that CovDistpx, yq " N. Lemma 3.3 implies tU i u is a chain. Renumbering these intervals if necessary, we may assume that x P U 1 zU 2 , that y P U N zU N´1 , and that Figure 3). Note that we allow sup U i´1 " inf U i`1 .
For a suitable choice of n 1 , we have v n 1 1 x " x 2 P U 2 . By induction, we have that v n i i x i " x i`1 P U i`1 for a suitable choice of n i . Once v n N´1 N´1¨¨¨v clearly has syllable length at most N and satisfies gx ą y. Lemma 3.2 implies that ||g|| " N.

3.2.
A residual property of free products. For a compact interval J Ď R, we let Diff 8 0 pJq denote the group of C 8 -diffeomorphisms of R supported in J. One can identify Diff 8 0 pJq with the group of C 8 -diffeomorphisms on J which are C 8tangent to the identity at BJ. For a group G and a subset S Ď G, we let xxS yy denote the normal closure of S . Lemma 3.5. Suppose G ď Diff 8 0 pIq has a connected support, and suppose 1 ‰ g P pGˆxsyq˚xty -pGˆZq˚Z.
Then there exists a representation φ g : pGˆxsyq˚xty Ñ Diff 8 0 pIq with a connected support such that φ g pgq ‰ 1 and such that supp φ g pGqXsupp φ g psq " ∅. Furthermore, we can require that φ g pGq -G.
After a suitable conjugation, we may assume g " t p pg s q q¨¨¨t p 1 pg 1 s q 1 q for some P N, g i P G and p i , q i P Z. For each i, we can further require that p i ‰ 0, and that either g i ‰ 1 or q i ‰ 0. There exists a representation Here, ρ i is C 8 -conjugate to ρ˘if g i ‰ 1, and to ρ 1˘o therwise. In particular, we require supp ρ i pGˆxsyq " p2i´1, 2iq.
We pick x 2 `1 and z i so that We can find a C 8 -action The nontriviality of φ g pgq comes from a Ping-Pong argument for free products (cf. [42,3]); that is, φ g pgqpx 1 q " x 2 `1 ą x 1 . The first conclusion follows from We may assume g i ‰ 1 for at least one i. This is because, the above construction also works for a finite set A Ď Gzt1u after setting g as a suitable concatenation of the elements in A. In particular, ρ i ae G and φ g ae G are faithful. Here, the symbol ae denotes the restriction of a representation.
3.3. Centralizers of diffeomorphisms. We recall the following standard result. It was proved for C 2 maps by Kopell [44] and generalized later to C 1`bv maps by Navas [58] in his thesis.
We continue to let M P tI, S 1 u. We say f P Homeo`pMq is grounded if Fix f ‰ ∅. In particular, every homeomorphism of I is grounded. An important and relatively straightforward corollary of Kopell's Lemma is the following fact: Lemma 3.7 (Disjointness Condition; see [2]). Let f, g P Diff 1`bv pMq be commuting grounded diffeomorphisms, where M P tI, S 1 u, and let U and V be components of supp f and supp g respectively. Then either U X V " ∅ or U " V.
If ω is a concave modulus or if ω P t0, bv, Lipu, then we define the C k,ωcentralizer group of G ď Homeo`pMq as Z k,ω pGq :" th P Diff k,ὼ pMq : rg, hs " 1 for all g P Gu.
Let Z k,ω pgq :" Z k,ω pxgyq for g P Homeo`pMq. We write Fix G " X gPG Fix g.
Proof. (1) Suppose g P ker ρzt1u. We may write g " y p x q for some p, q P Z so that xgx´1 " pxyx´1q p x q " y 2p x q P ker ρ.
(2) We may assume ρ is faithful by part (1). The case m " 2 precisely coincides with [8, Proposition 1.8]. The proof for the case m ą 2 is essentially identical.
If g P Diff 1`bv pS 1 q is an infinite order element having a finite orbit, then every element in Z 1`bv pgq has a finite orbit and every element in rZ 1`bv pgq, Z 1`bv pgqs is grounded; see [27] and [2]. This is a dynamical consequence of classical theorems of Hölder [34] and of Denjoy [23], combined with Kopell's Lemma. In this paper, we will need a C 1 -analogue of this consequence, as described below. The role of xgy is now played by the group BSp1, 2q. Lemma 3.9. Suppose we have an isomorphic copy of BSp1, 2q given as B " xx, y | xyx´1 " y 2 y ď Diff 1 pS 1 q.
Then the following hold.
(1) The C 1 -centralizer group Z 1 pBq of B has a finite orbit.
(2) For some finite index subgroup Z 0 of Z 1 pBq, we have supp Z 0 X supp y " ∅.
For g P Homeo`pS 1 q, we consider an arbitrary liftg : R Ñ R and define the rotation number of g as rotpgq :" lim A " tJ P π 0 supp B 0 : the restriction of B 0 on J is nonabelianu.
We may regard B 0 ď Diff 1 r0, 1s. It follows from [8, Theorem 1.7] that A is a finite set. Since Z 1 pBq ď Z 1 pB 0 q, the group Z 1 pBq permutes A and has a finite orbit inside X " Ť JPA BJ Ď S 1 . This proves part (1). Let Z 0 be the kernel of the above homomorphism Z 1 pBq Ñ Homeo`pXq.
Since every element of Z 0 fixes BJ for J P A , we can regard xZ 0 , B 0 y ď Diff 1 r0, 1s. Lemma 3.8 implies part (2).
Part (3) is not essential for the content of this paper, but we include it here for completeness and for its independent interest. To see the proof, note first that the finite cyclic group action ρ 0 : Z 1 pBq{Z 0 Ñ Homeo`pXq is free. By a variation of Hölder's Theorem given in [40,Corollary 2.3], there exists a free action ρ : Z 1 pBq{Z 0 Ñ Homeo`pS 1 q extending ρ 0 such that rot˝ρ is a monomorphism; see also [27]. We have a commutative diagram as below: Let g P rZ 1 pBq, Z 1 pBqs. The commutativity of the lower square implies that rot restricts to a homomorphism on Z 1 pBq. In particular, we have that rotpgq " 0 and that g is grounded. Since g centralizes B, and since Fix B 0 ‰ ∅, we see that FixxB 0 , gy ‰ ∅. So, we may regard xB 0 , gy ď Diff 1 pIq. Lemma 3.8 implies that supp g X supp y " ∅, as desired.

3.4.
A universal compactly-supported diffeomorphism. Throughout this paper, we will fix a finite presentation:  Figure 4. The relators of G : . The horizontal double edge denotes the relator aea´1 " e 2 and the other two edges denote commutators.
Whenever we have an action ψ of G : on I, we will define the covering length and the covering distance by the following open cover of supp ψpG : q: V " ď vPV : π 0 supp ψpvq.
If ψ : G : Ñ Homeo`pIq is a representation and f P ψpG : q, there is little reason to believe that CovLenpsupp f q ă 8, even if we restrict to a component of supp ψpG : q. In order to use the covering length of a support as a meaningful notion of complexity of a diffeomorphism, we need to find an element 1 ‰ u 0 P G : for which CovLenpsupp ψpu 0 qq ă 8.
We will build such an element u 0 P G : . We say a set A Ď R is compactly contained in a set B Ď R if there exists a compact set C such that A Ď C Ď B. (1) Then xα, β, ty is not isomorphic to Z 2˚Z .
We have that supp γ X supp δ " ∅. By Lemma 3.10 of [40], we have supp φz supp β is compactly contained in supp γ Y supp δ. Since u " rφ, αs, we see that We can now deduce Corollary 1.7 in Section 1. The authors were told by A. Navas of the following proof for M " I.
Consider first the case when M " I. By Lemma 3.8, we see that supp ψpcq X supp ψpeq " ∅. It follows from Lemma 3.10 that ψxc, e, dy fl Z 2˚Z -xc, e, dy. This is a contradiction, for ψ is faithful.
Assume M " S 1 . By Lemma 3.9 (2), we have some p P N such that supp ψpc p q X supp ψpeq " ∅.
We again deduce a contradiction from Lemma 3.10, for we have ψxc p , e, dy fl Z 2˚Z -xc p , e, dy.
3.5. Simplicity and diffeomorphism groups. We will require some classical results about the simplicity of certain groups of diffeomorphisms of manifolds. For a manifold X, we let Diff k,ω c pXq 0 denote the set of C k,ω diffeomorphisms isotopic to the identity through compactly supported isotopies; this set is indeed a group [52]. Note that Diff k,ω c pS 1 q 0 " Diff k,ὼ pS 1 q, Diff k,ω c pRq 0 " Diff k,ω c pRq. Definition 3.12. Let ω be a concave modulus.
Mather [52,53] proved the simplicity of Diff k pXq, where X is an n-manifold and k ‰ n`1. The following is a straightforward generalization from his argument.
Theorem 3.13 (Mather's Theorem [52,53]). Suppose X is a smooth n-manifold without boundary. Let k P N, and let ω be a concave modulus satisfying the following: ‚ if k " n, then we further assume ω is sup-tame; ‚ if k " n`1, then we further assume ω is sub-tame. Then the group Diff k,ω c pXq 0 is simple. In Example 2.4, we have defined a concave modulus ω z for each z P p0, 1s C . Lemma 3.14. We have the following.
Corollary 3.15. Let X be a smooth n-manifold without boundary, and let k P N. If some z P p0, 1s C satisfies Repk`zq ‰ n`1, then the group Diff k,ω z c pXq 0 is simple. Proof. We use Lemma 3.14 and Mather's Theorem. If Re z P p0, 1q, then ω z is sup-and sub-tame, and so, Diff k,ω z c pXq 0 for all k P N. If z " s ?´1 for some s ă 0, then ω z is sup-tame; in this case, Diff k,ω z c pXq 0 is simple for all integer k ‰ n`1. If z " 1`s ?´1 for some s ě 0, then ω z is sub-tame and Diff k,ω z c pXq 0 for all integer k ‰ n. The conclusion follows.
We will later use the following form of simplicity results. The proof is given in Appendix (Theorem A.10).
Theorem 3.16. For each X P tS 1 , Ru, the following hold.
(1) If α ě 1 is a real number, then every proper quotient of Diff α c pXq 0 is abelian. If, furthermore, α ‰ 2, then Diff α c pXq 0 is simple. (2) If α ą 1 is a real number, then every proper quotient of Ş βăα Diff β c pXq 0 is abelian. If, furthermore, α ą 3, then Ş βăα Diff β c pXq 0 is simple. 3.6. Locally dense copies of Thompson's group F. Recall that Thompson's group F is defined to be the group of piecewise linear homeomorphisms of the unit interval r0, 1s such that the discontinuities of the first derivatives lie at dyadic rational points, and so that all first derivatives are powers of two. It is well-known that Thompson's group F is generated by two elements (see [19,14]).
We will denote the standard piecewise linear representation of F as ρ F : F Ñ Homeo`r0, 1s.
Recall that a group action on a topological space is minimal if every orbit is dense. The action ρ F is minimal on p0, 1q, but it has an even stronger property: the diagonal action of ρ F on X " tpx, yq P p0, 1qˆp0, 1q | x ă yu is minimal. This follows from the transitivity of F on a pair of dyadic rationals in X; see [19] and [14].
Alternatively, the action ρ F on p0, 1q is locally dense [10]. The general definition of local density is not important for our purposes. For a chain group G ď Homeo`r0, 1s (see Remark 3.19 below for a definition), the local density of the action of G on p0, 1q is equivalent to the minimality of the action of G on X, which in turn is equivalent to the minimality of the action of G on p0, 1q; this is proved in [41, Lemma 6.3]. Thompson's group F is an example of a chain group (Corollary 3.18).
We will require the following result: Theorem 3.17 (Ghys-Sergiescu, [30]). The standard piecewise-linear realization ρ F of Thompson's group F is topologically conjugate to a C 8 action on r0, 1s such that each element is C 8 tangent to the identity at t0, 1u.
The original construction of Ghys-Sergiescu is a C 8 action of Thompson's group T for a circle; the above theorem is an easy consequence by restricting their action on an interval. Let us denote this action as ρ GS : F Ñ Diff 8 0 r0, 1s. Note ρ GS pFq acts minimally on p0, 1q. There exists a homeomorphism h GS : r0, 1s Ñ r0, 1s such that for all g P F we have ρ GS pgq " h GS˝ρF pgq˝h´1 GS .
It will be convenient for us to denote a i " ρ GS px i q for i " 0, 1.
Corollary 3.18. There exists a chain of two intervals pU 1 , U 2 q and C 8 diffeomorphisms f 1 and f 2 supported on U 1 and U 2 respectively such that x f 1 , f 2 y " ρ GS pFq.
Proof. It is routine to check that f 1 " a´1 1 a 0 and f 2 " a 1 satisfy the conclusion. See [41] for details.
Remark 3.19. More generally, if pU 1 , . . . , U n q is a chain of intervals and if f 1 , . . . , f n P Homeo`pRq satisfy that supp f i " U i for each i, then the group x f 1 , . . . , f n y is called a pre-chain group (cf. [41]). The group x f 1 , . . . , f n y is called a chain group if moreover we have x f i , f i`1 y -F for each 1 ď i ă n. If x f 1 , . . . , f n y is a pre-chain group then for all sufficiently large N, we have x f N 1 , . . . , f N n y is a chain group [41].

The Slow Progress Lemma
Throughout this section, we assume the following. Let k P N, and let G be a group with a finite generating set V. We will consider an arbitrary representation ψ of G given in one of the following two types: ‚ ψ : G Ñ Diff k,ὼ pIq, where ω ą k 0 is some concave modulus; ‚ ψ : G Ñ Diff k,bv pIq, in which case we will put ω " ω 1 . We denote by }h} the syllable length of h P G with respect to V as in Section 3.1. We also use the notation V " Y vPV π 0 supp ψpvq.
Suppose we have sequences tN i u iPN Ď N and tv i u iPN Ď V such that the following two conditions hold. First, for some K ą 0 we assume Second, for each v P V we assume the following set has a well-defined natural density: (A2) N v :" ti P N | v i " vu. Let us define a sequence of words tw i u iě0 Ď G by w 0 " 1 and The main content of this section is the following: The proof of the lemma occupies most of this section. As a consequence of this lemma, we will then describe a group theoretic obstruction for algebraic smoothing.
Remark 4.2. The statement of the Slow Progress Lemma is topological. In other words, even after ψ is replaced by an arbitrary topologically conjugate representation, the same conclusion holds. 4.1. Reduction to limit superior. For brevity, we simply write CovLen and CovDist for CovLen V and CovDist V . We write gx " ψpgqx for g P G and x P I. (i) lim sup iÑ8 pi´CovDistpx, w i xqq " 8; (ii) lim iÑ8 pi´CovDistpx, w i xqq " 8.
Proof. Assume (ii) does not hold. There exists M 0 ą 0 and an infinite set A Ď N such that for all a P A we have a´CovDistpx, w a xq ă M 0 .
If (i) is true, then we have an increasing sequence t jpsqu sPN such that lim sÑ8 p jpsq´CovDistpx, w jpsq xqq " 8.
This is a contradiction, and (i)ñ(ii) is proved. The converse is immediate.

4.2.
Markers of covering lengths. In order to prove Lemma 4.1 by contradiction, let us make the following standing assumption of this section: there exists a point x P U P π 0 supp ψpGq and a real number M 0 ą 0 such that the sequence tx i :" w i xu iě0 satisfies (A3) for all i ě 0, we have i´M 0 ď CovLenrx, x i q ď i.

By Lemma 4.3, it suffices for us to deduce a contradiction from (A3).
The sequence tx i u accumulates at BU. Since the sequence cannot accumulate simultaneously at the both endpoints of U by assumption (A3), we may make an additional assumption: For each i P N, we define zi " suptz P rx, sup Uq | CovLenrx, zq ď iuu.
The point zi is the "length-i marker" of covering lengths in the following sense.
(3) There exists M 1 , M 2 ą 0 such that for all i ě M 1 we have that zi´M 2 ă x i ă zi´M 2`1 .

Proof. (1) Monotonicity of h is clear. For the left-continuity and surjectivity, it
suffices to show CovLenrx, zi q " i. Let us define Since each point in I belongs to at most |V| intervals in V , each z 1 i is realized as sup J for some J P V .
We claim that zi " z 1 i and that CovLenrx, zi q " i for each i P N. The case i " 1 is trivial. Let us assume the claim for i´1. Then we have CovLenrx, z 1 i q " i and z 1 i ď zi . If z 1 i ă zi then there exists t P pz 1 i , zi q such that CovLenrx, tq " i. But whenever t P J P V we have z 1 i´1 R J, by the choice of z 1 i . This shows CovLenrx, tq ą i, a contradiction. Hence the claim is proved.
The opposite inequality is immediate from the definition of z 1 i . For the second equation, it suffices to further note that CovLenrx, zi s " i`1.
(3) By (A3), the following holds for all but finitely many i: For such an i, we have that x i P pzj´1, zj s and x i`1 P pzj , zj`1s for j " CovLenrx, x i q. If x i " zj , then x i`1 ă zj`1 and moreover, x i` ă zj` for all P N.
Let us write z i " zi´M 2 . After increasing M 0 if necessary, we have the following for all i ě M 0 and j ą 0: We may also assume: Consider the set of "significant generators" and their minimum density: By further increasing M 0 , we may require: for all v P V 1 and N ě M 0 . We note the following.
Lemma 4.5. Let v P V 1 , and let N v " t j 1 ă j 2 ă j 3 ă¨¨¨u. Then there exists a constant K 1 ě K such that whenever m P N satisfies j m ě M 0 , we have Proof. Note that m " #pN v X r1, j m sq ě δ 1 j m . Hence, we have j m ď m{δ 1 . Lemma 2.6 implies that ωp1{ j m q ě ωpδ 1 {mq ě δ 1 ωp1{mq.
The desired inequality is now immediate.

Estimating gaps. Let
As illustrated in Figure 5, we will write Roughly speaking, L i is obtained from J i by successively attaching adjacent components of supp ψpv i q on the left until we have included at least k`1 fixed points of ψpv i q or an accumulated fixed point of ψpv i q. By (A4) and (A6), the intervals L i and R i are compactly contained in U. Lemma 4.6. For each i P N X rM 0 , 8q, the following hold.
(1) The map ψpv i q is k-fixed on L i and also on R i .
Proof. Parts (1) and (2) are obvious from the definition and from the fact that CovLenrz i , z i`1 s " 2. For part (3), suppose x P A P π 0 supp ψpvq for some v P V. There exist at most 2k indices i ě M 0 such that v j " v and such that A Ď L j Y R j . Hence, the total number of L i 's and R i 's containing a given arbitrary point x is at most 2k|V|. Part (4) follows similarly.
Let us pick an integer C ě 8k. We call each x i as a ball, and the interval rz i , z i`C q as a bag (of size C). For each m ě M 0 , we define bagpmq " rz m , z m`C q, gappmq " rx m , x m`C´1 s. See Figure 6.
For each δ ą 0 and v P V, we let Intuitively speaking, Ball δ is the collection of balls which are δ-fast neither on L i nor on R i . Also, Bag δ is the set of bags which "involve" only balls from Ball δ . We now use the analytic estimate from Section 2: Lemma 4.7. For each δ ą 0, the sets Ball δ and Bag δ have the natural density one.
Proof. Let v P V 1 . By Lemmas 4.5 and 4.6, we can apply Theorem 2.9 to f " ψpvq. We see that It follows that d N pBall δ q " 1. By Lemma 2.21 (1), we have d N pBag δ q " 1.
x m´1 z m x m z m`1 x m`1 z m`C´1 x m`C´1 z m`C x m`C gappmq bagpmq Figure 6. The gap in a bag.
Proof. We may assume λ ą 8k. For δ ą 0, we define Then we see D C,λ " X λ Y ppY λ´C q X rM 0 , 8qq . Lemmas 4.7 and 4.9 imply that d N pD C,λ q " 1, Hence by Lemma 2.21, we obtain that d N pX λ Y Y λ q " 1. This implies d N pE λ q " 1.
Completing the proof of the Slow Progress Lemma. We see from Lemma 4.5 and Theorem 2.9 that for each v P V 1 . This implies d N pE λ q " 0, contradicting Lemma 4.10. Hence the assumption (A3) is false and the proof is complete.

4.4.
Consequences of the Slow Progress Lemma. The following is the main obstruction of algebraic smoothing in the Main Theorem.
Lemma 4.11. Let u P G and let U P π 0 supp ψpGq. If supp ψpuq X U is compactly contained in U, then for each real number T 0 ą 0 and for all sufficiently large i P N, there exists h i P G such that the following hold: (iii) For each v P V and for at least one h 1 P tv¨h i , v´1¨h i u, we have Proof. Let u, U and T 0 be given as in the hypothesis. We write x " infpsupp ψpuq X Uq, y " suppsupp ψpuq X Uq.
Put T " CovDistpx, yq. By the Slow Progress Lemma, whenever i " 0 we have CovDistpx, w i xq ă i´pT 0`T q, CovDistpy, w i yq ă i´pT 0`T q.
Put u i " w i uw´1 i . Since supp ψpu i q X U Ď pw i x, w i yq, we see from Lemma 3.4 that there exists h i P G with }h i } ă 2i´T 0 satisfying h i w i x ą w i y. Furthermore, for each v P V there is a spvq P t1,´1u such that v spvq h i w i x ě h i w i x ą w i y. We see that This gives the desired relations.

A dynamically fast subgroup of Diff k,μ pIq
Recall we have defined G : in Section 3.4. We will now build a representation φ : G : Ñ Diff k,µ 0 pIq such that supp φpG : q is connected and φpG : q admits no injective homomorphisms into Diff k,ὼ pIq for all 0 ă k ω ! µ.
The criticality of the regularity will be encoded in a dynamically fast condition described as follows. As in Lemma 3.11, we let 1 ‰ u 0 P G : be given such that supp φpu 0 q is compactly contained in supp φpG : q. We build a sequence a elements tw i u iě0 Ď G : which depend on k, µ such that, after replacing u 0 by a suitable conjugate u in G : if necessary, we have CovDistpinf supp φpw i uw´1 i q, sup supp φpw i uw´1 i qq ě 2i.
We build the representation φ in several steps. 5.1. Setting up notation. Let us prepare some notation which we will use throughout this section. We fix k P N and µ " ω 1 . We put δ " 9{10 and recall the notation t 0 , δ 0 , 0 , i , N i u from Setting 2.15 and from Corollary 2.20. Namely, we pick a universal constant 0 P p0, 1q, and define δ 0 ě 9{10 from 0 . For instance, we can set 0 " 1{1000. We have defined a constant 0 depending on µ, so that 0 , µp 0 q P p0, 0 s. We will choose K˚P N, and let i " 1{`pi`K˚q log 2 pi`K˚q˘. We have defined another sequence Possibly after increasing K˚ą 0, we may assume that 1 ď 0 and that κ :" 2 {p2 2` 1 q ą 1{4.
Recall we have a generating set V : " ta, b, c, d, eu Ď G : as in Section 3.4. For i P N, we let v 2i´1 " b and v 2i " a. Define a sequence tw i u iPN Ď G : by w 0 " 1 and w i " v N i i¨w i´1 .  Figure 7. The union of F will be also bounded. We will simultaneously define representations ρ 0 , ρ 1 , ρ 2 : G : Ñ Diff k,μ pRq.
We will include six more open intervals B˘, C˘, Dt o the chain F as shown in the configuration (Fig 7). We will require that B´" B`and so forth, where we use the notatioń pr, sq " p´s,´tq for 0 ď r ă s ď 8. Also, we set sup C`" 2 and sup D`" 3. By Corollary 3.18, there exists a C 8 diffeomorphisms c1 , d1 supported on C`, Dr espectively such that xc1 , d1 y -F and xc1 , d1 y acts locally densely on C`Y D`. We may require c1 pxq ą x for x P C`and d1 pxq ą x for x P D`. We define c1 , d1 symmetrically so that c1 p´xq "´c1 pxq and d1 p´xq "´d1 pxq. In particular, supp d1 " D˘, supp c1 " C˘.
(2) For each g P G : , the restriction φpgqae IzBI is a C 8 diffeomorphism.
(3) For each i ě 1, the map φpv N i i q is δ 0 -fast on Lȋ . (4) Every orbit of φxa, c, d, ey in I 0 is accumulated at BI 0 .

5.3.
The behavior of tw i u iě0 under φ. Whereas we have good control over the compactly supported diffeomorphism φpuq, we will need to have good control over commutators of conjugates of φpuq.

Lemma 5.2.
For each nonempty open interval U 0 Ď supp φpG : q, there exists a suitably chosen f P φpG : q such that f pU 0 qXL1 ‰ ∅ and such that f pU 0 qXL1 ‰ ∅.
Intuitively, Lemma 5.2 says that no matter how small an interval we choose inside supp φpG : q, we may find an element of f P φpG : q so that f pU 0 q stretches across Of course, f pU 0 q might be much larger than this union, though this is unimportant.
Proof of Lemma 5.2. Let U 0 " pz 1 , z 2 q be given as in the hypotheses of the lemma. By Lemmas 5.1 (4) and 3.4, there exists an f P φpG : q such that f pz 2 q P D`X L1 . So, we may assume z 2 P D`X L1 . We may then assume that z 1 ě sup L1 ; for, otherwise there is nothing to show. There are four (overlapping) cases to consider.
Case 1: z 1 P B´Y C´Y D´. For sufficiently large n 1 , n 2 , n 3 P N and for f 1 " φpd n 3 c n 2 b n 1 q, we have f 1 pz 2 q P L1 zD`and f 1 pz 1 q P D´X L1 . This is the desired configuration.
Case 2: z 1 P I 0 . By Lemma 5.1 (4), there is f 1 P φxa, c, d, ey such that f 1 pz 1 q P B´X I 0 . Note that is a nonempty set which is disjoint from supp φxa, c, d, ey. Hence, f 1 pz 2 q R Q; see Figure 9. We have f 1 pz 2 q P C`Y D`. As in Case 1, we can find sufficiently large n 1 , n 2 , n 3 P N such that for f 2 " φpd n 3 c n 2 b n 1 q we have f 2 f 1 pz 1 q P L1 . and f 2 f 1 pz 2 q P L1 . This is the desired. Case 3: z 1 P B`.
There exist sufficiently large n 1 P N such that for f 1 " φpb´n 1 q, we have f 1 pz 2 q P D`X L1 and f 1 pz 1 q P I 0 X B`. So, we again have Case 2.
We use the fact that the restriction of φxc, dy to C`YD`generates a locally dense copy of Thompson's group F. As we have seen in Section 3.6, for some suitable f 1 P φxc, dy we may arrange f 1 pz 1 q P B`X C`and f 1 pz 2 q P D`X L1 , thus reducing to the previous case. Figure 9. The point f 1 pz 2 q stays in C`Y D`.
We retain the elements tw i u iPN as defined in Section 5.1. The following lemma measures the complexity of certain diffeomorphisms in φpG : q and shows that the complexities grow linearly. Lemma 5.3. Let u P G : z ker φ be an element such that supp φpuq is compactly contained in supp φ. Then for some conjugate u 1 P G : of u, and for some component U 1 of supp φpu 1 q, we have that whenever i P N the bounded open interval φpw i qU 1 intersects both Lì`1 and Lí`1. In particular, we have that CovLenpφpw i qU 1 q ą 2i, and that Bpφpw i qU 1 q Ď supp φpaq Y supp φpbq.

Proof.
Choose an open interval U 0 P π 0 supp φpuq compactly contained in I. By Lemma 5.2, there is a conjugate u 1 P G : of u such that the image U 1 of U 0 under this conjugation intersects L1 . Conjugating by a further power of b if necessary, we may assume ps´, s`q Ď U 1 for some s˘satisfying the following.
Assume by induction that Figure 10. Replacing u by a suitable conjugate u 1 .
In order to cover φpw i qU 1 by intervals in F , we need at least tI 0 , B˘, C˘, D˘, L1 , . . . , Lȋ u.
The conclusion is now obvious.

5.4.
Certificates of non-commutativity. The following fact will be used in order to show that φpG : q cannot be smoothed algebraically.
Let f " φpuq and g " φphuh´1q. Since supp f is compactly contained in I, there exists a compact interval J such that Since φpG : q is C 8 at each point x P IzBI, we may regard f, g P Diff 8 pJq. A corollary to Kopell's Lemma (Corollary 3.7) implies that if f and g commute, then U and φphqU must either be equal or disjoint. They are not disjoint by the previous paragraph and they are not equal by the hypothesis.
We remark that the above fact can be generalized to arbitrary compactly supported representations which are C 2 in the interior. The following lemma extracts the main content of this section which will be necessary in the sequel.
Lemma 5.5. Suppose u P G : satisfies that supp φpuq is a nonempty set compactly contained in supp φpG : q. Then there exists a conjugate u 1 of u in G : such that for all i P N, for all s, t P t´1, 1u and for all h P G : satisfying }h} ă 2i, we have for at least one h 1 P th, a s¨h , b t¨h u.
Proof. Using Lemma 5.3, we obtain a conjugate u 1 of u such that for each i P N, the set supp φpw i u 1 w´1 i q has a component U i whose covering length is larger than 2i.
Note that for at least one h 1 P th, a s¨h , b t¨h u, we have that and that ||h 1 || ď 2i. The nontriviality of φrw i u 1 w´1 i , h 1 w i u 1 w´1 i ph 1 q´1s follows immediately from Lemma 5.4.
Proof. Let u 1 :" u : P rG : , G : s be the element considered in Lemma 3.11 and Section 5.2. By the same lemma, supp ψpu 1 q is compactly contained in supp ψpG : q. We see from the construction that φpu 1 q ‰ 1. So, we may assume ψpu 1 q ‰ 1. Let us choose a minimal collection tU 1 , . . . , U n u Ď π 0 supp ψpG : q such that There exists a conjugate u 1 1 of u 1 satisfying the conclusion of Lemma 5.5. Recall from Section 5.1 that we have lim iÑ8 N i p1{iq k´1 ωp1{iq " 0.
Hence, we can apply Lemma 4.11 to u 1 1 and U 1 . We obtain some i P N, some h 1 P G : with }h 1 } ă 2i, and some s, t P t1,´1u such that for all choice of h 1 1 P th 1 , a s¨h 1 , b t¨h 1 u. As u 1 1 has been chosen to satisfy Lemma 5.5, there exists a choice of h 1 1 such that Note that supp φpu 2 q is still compactly contained in supp φpG : q. We now have supp ψpu 2 q Ď U 2 Y¨¨¨Y U n .
Inductively, we use u 2 to obtain u 1 2 satisfying Lemma 5.5. The same argument as above yields u 3 P rG : , G : sz ker φ such that supp ψpu 3 q Ď U 3 Y¨¨¨Y U n .
Continuing this way, we obtain an element u m P rG : , G : s X ker ψz ker φ for some m ď n`1.
Remark 5.7. The idea of finding a nontrivial kernel element of an interval action by successively taking commutators appeared in [11], where Brin and Squier proved that PLr0, 1s does not contain a nonabelian free group. One can trace this idea back to the proof of the Zassenhaus Lemma on Zassenhaus neighborhoods of semisimple Lie groups [64]. This idea was also used in [2,40].
Proof of Theorem 1.5. Let φ k,µ " φ be the representation constructed in this section. Theorem 5.6 implies the conclusion (i). We have already verified (ii).
Remark 5.8. The group φ k,µ pG : q we constructed is never a subgroup of a rightangled Artin group, or even a subgroup of a braid group; see [40,Theorem 3.12] and [39, Corollary 1.2].

Proof of the Main Theorem
Let us now complete the proofs of all the results in the introduction.
6.1. The Rank Trick. If φ : G Ñ Homeo`r0, 1s be a representation, then a priori, it is possible that the rank of the abelianization H 1 pφpGq, Zq is less than that of H 1 pG, Zq. Let us now describe a systematic way of producing another representation φ 0 such that the rank of H 1 pφ 0 pGq, Zq is maximal. Lemma 6.1 (Rank Trick). Let G be a group such that H 1 pG, Zq is finitely generated free abelian. If we have a representation ρ : G Ñ Homeo`pRq such that supp ρ is bounded, then there exists another representation ρ 0 : G Ñ xρpGq, Diff 8 pRqy ď Homeo`pRq satisfying the following: (i) supp ρ 0 is bounded; (ii) ρ 0 pgq " ρpgq for each g P rG, Gs; (iii) H 1 pρ 0 pGq, Zq -H 1 pG, Zq.
Proof. Let H 1 pG, Zq -Z m for some m ě 0. We can pick compactly supported for each i. The abelianization of G can be realized as some surjection We define a representation ρ 0 : G Ñ Homeo`pRq by the recipe ρ 0 pgq " ρpgqαpgq for each g P G. It is clear that ρ 0 satisfies parts (i) and (ii). Since α decomposes as we see that ρ 0 pGq surjects onto Z m . This proves part (iii).
Remark 6.2. Algebraically, the group ρ 0 pGq is a subdirect product of ρpGq and Z m .
6.2. The Chain Group Trick. Let us describe a general technique of embedding a finitely generated orderable group into a countable simple group. In Remark 3.19, we defined the notion of a chain group, which is a certain finitely generated subgroup of Homeo`pRq. We will need the following result of the authors with Lodha: . If H ď Homeo`pIq is a chain group acting minimally on IzBI, then rH, Hs is simple and every proper quotient of H is abelian.
In [41], it is shown that every finitely generated orderable group embeds into some minimally acting chain group. We will need a variation of this result for diffeomorphisms. Let us use notations ρ GS , h GS and ta 0 , a 1 u as defined in Section 3.6. By an n-generator group, we mean a group generated by at most n elements. Lemma 6.4 (Chain Group Trick). Let G be an n-generator subgroup of Homeo`pRq such that supp G is compactly contained in p0, 1q. We put r G " xG, ρ GS pFqy.
(1) Then r G is an pn`2q-chain group acting minimally on p0, 1q. In particular, r r G, r Gs is simple and every proper quotient of r G is abelian. (2) If H 1 pG, Zq is free abelian, then there is an embedding from G into r r G, r Gs.
Proof. We will follow the proof of [41,Theorem 1.3], taking extra care with elements of ρ GS pFq. Let us fix a generating set tg 1 , . . . , g n u of G.
Since s i P Q GS , we can find f 1 P ρ GS pFq such that supp f 1 " ps 2 , s 3 q and such that f 1 ptq ě t for all t P r0, 1s. We fix t 0 P ps 2 , s 3 q X Q GS , so that After conjugating G by a suitable element of ρ GS pFq if necessary, we may assume that the closure of supp G is contained in pt 0 , f 1 pt 0 qq.
Claim. If g " g i for some 1 ď i ď n, then we have that If t R ps 2 , s 3 q, then a 1˝g ptq " a 1 ptq and the claim is obvious. If t P ps 2 , s 3 q, then a´1 1 ptq ă a´1 1 ps 3 q " s 2 ă gptq ă s 3 " a´1 1 ps 4 q ă a´1 1˝a 0 ptq.
This proves the claim. We define u 0 " a 1 , and u i " a 1 g i for i " 1, . . . , n. We also let u0 " u´1 0 a 0 , un`1 " a n 0 u n a´n 0 and ui " pa i 0 u´1 i a´i 0 q¨pa i´1 0 u i´1 a 1´i 0 q, i " 1, . . . , n. Then we have r G " xG, a 0 , a 1 y " xu0 , . . . , un`1y.
The group r G acts minimally on p0, 1q since so does ρ GS pFq. It now suffices to show that the collection tu0 , u1 , . . . , un`1u is a generating set for an pn`2q-chain group; this is a routine computation of the supports using the above claim, and worked out in [41,Lemma 4.2].
(2) Recall we have defined f 1 P ρ GS pFq in part (1). We put For all distinct i, j P Z we have Let H 1 pG, Zq -Z m for some m ď n. Possibly after increasing the value of n if necessary, we may require that tg 1 , . . . , g m u generates H 1 pG, Zq, and that tg m`1 , . . . , g n u Ď rG, Gs.
we have an embedding G ãÑ rG 1 , G 1 s defined by The proof is complete since rG 1 , G 1 s ď r r G, r Gs.
Let us make a general observation.
Lemma 6.6. Let G be an infinite group such that every proper quotient of G is abelian. Then every finite index subgroup of G contains rG, Gs.
Proof. Let G 0 ď G be a finite index subgroup. Then G acts on the coset space G{G 0 by multiplication and hence there is a representation from G to the symmetric group of G{G 0 . Since every proper quotient is abelian, we see that rG, Gs acts trivially on G{G 0 . This implies rG, Gs ď G 0 .
6.3. Proof of Theorem 1.4. We will prove the theorem by establishing several claims. Let k and µ be as given in the hypothesis of the theorem. We denote by φ " φ k,µ : G : Ñ Diff k,µ 0 pIq the representation φ constructed in the previous section. We put T 1 :" φpG : q. From now on, we will assume supp T 1 is sufficiently smaller than I whenever necessary.
By the Rank Trick (Lemma 6.1), we can find φ 0 : G : Ñ Diff k,µ 0 pIq such that the conclusions of Lemma 6.1 hold. We put T 2 :" φ 0 pG : q so that We may assume supp T 2 Ď I Ď p0, 1q.
Claim 1. We have that T 1 , T 2 ď Diff k,µ 0 pIq and that This claim for T 1 follows from Theorem 5.6. In order to prove the claim for T 2 , we let 0 ă k ω ! µ or let ω " bv. Suppose ψ : T 2 Ñ Diff k,ὼ pIq is a representation. By applying Theorem 5.6 again to the composition we see that there exists g P rG : , G : sz ker φ such that ψ˝φ 0 pgq " 1. Since φ 0 pgq " φpgq ‰ 1 by Lemma 6.1 (ii), we have φ 0 pgq P ker ψzt1u. This proves the claim. We can apply the Chain Group Trick (Lemma 6.4) to T 2 , and obtain T 3 :" xT 2 , ρ GS pFqy ď Diff k,µ 0 r0, 1s acting minimally on p0, 1q as a seven-generator chain group. From Claim 1 and from the fact T 2 ãÑ rT 3 , T 3 s, we obtain the following and complete the proof of Theorem 1.4 for M " I. Claim 2. The countable simple group rT 3 , T 3 s ď Diff k,µ 0 r0, 1s satisfies that Let us now consider the case M " S 1 . After a conjugation, we may assume supp T 3 Ď I Ď p0, 1q. As BSp1, 2q embeds into Diff 8 0 pIq, we may regard T 3ˆB Sp1, 2q ď Diff k,μ pS 1 q.
Recall F denotes the Thompson's group acting on r0, 1s. We have a natural map ρ : T 2˚F Ñ T 3 ď Diff k,µ 0 pIq. We can apply the Rank Trick to ρ, since Then we obtain a representation ρ 0 : T 2˚F Ñ xT 3 , Diff 8 pRqy ď Diff k,μ pRq.
Let T 4 be the image of ρ 0 . We may require that supp T 4 Ď I Ď p0, 1q and that H 1 pT 4 , Zq is free abelian. Moreover, we have rT 3 , T 3 s -rT 4 , T 4 s.
Claim 3 now implies the following.
Claim 4. The group T 5 is a nine-generator group such that Since H 1 pT 5 , Zq -H 1 pT 4 , Zq ' Z is free abelian, we can finally apply the Chain Group Trick to obtain a minimally acting eleven-chain group Q " Qpk, µq with T 5 ãÑ rQ, Qs ď Q ď Diff k,µ 0 pIq ãÑ Diff k,μ pS 1 q. Summarizing, we have the following. Proposition 6.7. Let k P N, and let µ " ω 1 be a concave modulus. Then there exists an eleven-generator group Q " Qpk, µq such that the following hold.
(1) rQ, Qs is simple and every proper quotient of Q is abelian.
Then for an arbitrary finite index subgroup A of Q, and for all homomorphism the image is abelian, whenever M P tI, S 1 u.
We have now proved Theorem 1.4. For a later use, we record the inclusion relations between the groups appearing above: In the above diagram, the isomorphismscome from the Rank Trick and the embeddings ãÑ come from the Chain Group Trick. 6.4. Continua of groups of the same critical regularity. Recall a continuum means a set that has the cardinality of R. The Main Theorem is an immediate consequence of the following stronger result, combined with Theorem 6.3. Theorem 6.8. For each real number α ě 1, there exist continua X α , Y α of minimal chain groups acting on I such that the following conditions hold.
(i) For each A P X α , we have that A ď Diff α 0 pIq and that rA, As R ď βąα G β pIq Y G β pS 1 q.
(ii) For each B P Y α , we have that B ď Ş βăα Diff β 0 pIq and that rB, Bs R G α pIq Y G α pS 1 q.
(iii) No two groups in X α Y Y α have isomorphic commutator subgroups.
In order to prove Theorem 6.8, we set up some notations. For a complex number z P C, we let xzy denote the largest integer m such that m ă C z. For instance, we have xky " xk´?´1y " k´1 and xk`1{2y " xk`?iy " k for an integer k. Let z ą C 1, written as z " k`τ`s ?´1 for k " tRe zu and τ, s P R. We put κpzq :" pxzy, ω z´xzy q.
If α ą 1 is a real number and if k " tαu, then we see that , ω 1 q, if α " k.
Using the notation Qpk, µq from Proposition 6.7, we observe the following.
Lemma 6.9. The following hold for all complex numbers 1 ă C z ă C w.
(4) If z R N and Re w ą 1, then we have that rQ˝κpzq, Q˝κpzqs R G κpwq pS 1 q.
Note that G κpwq pIq Ď G κpwq pS 1 q by Theorem A.3.
The conclusion of (4) follows from Lemma 2.7 and Proposition 6.7.
Remark 6.10. In the case when z " 1`s ?´1 and w " 1`t ?´1 for some 0 ă s ă t, we cannot conclude that part (4) above holds. This is because ω w´xwy " ω t ?´1 may not be sub-tame.
Let us now prove Theorem 6.8 for the case α ą 1. We define X α :" tQ˝κpα`s ?´1 q : s ą 0u, Pick a real number s ą 0 and put A " Q˝κ`α`s ?´1˘P X α . Note that q 0 pIq ď Diff α 0 pIq. Let β ą α be a non-integer real number. By Lemma 6.9, we have that rA, As R G κpβq pS 1 q " G β pS 1 q. The conclusion (i) of the Theorem is satisfied.
This proves the conclusion (ii). It is obvious from the conclusions (i) and (ii) that whenever A P X α and B P Y α , we have rA, As fl rB, Bs. Suppose we have real numbers 0 ă s 1 ă s 2 , and put A i " Q˝κpα`s i ?´1 q. Using α ą 1 we deduce from Lemma 6.9 that q pS 1 q.
In particular, rA 1 , A 1 s fl rA 2 , A 2 s. Similarly, no two groups in Y α have isomorphic commutator subgroups. This proves the conclusion (iii).
Let us now construct a continuum X 1 . For each β ą 1, we pick G β P X β . We put G 1 :" Q˝κp1`?´1q so that rG 1 , G 1 s R G γ pS 1 q for each γ ą 1. By the Rank Trick for the natural surjection from a free group onto G β for β ě 1, we obtain another groupḠ β ď Diff β 0 pIq whose abelianization is free abelian such that rG β , G β s -rḠ β ,Ḡ β s. It follows thatḠ β R G γ pS 1 q for all γ ą β ě 1.
For each β ą 1, we can apply the Chain Group Trick toḠ 1ˆḠβ to obtain a minimally acting chain group Γpβq such that G 1ˆḠβ ãÑ rΓpβq, Γpβqs ď Γpβq ď Diff 1 0 pIq. It follows that rΓpβq, Γpβqs R G γ pS 1 q for all γ ą 1. From the consideration of critical regularities, we note thatḠ β flḠ γ whenever 1 ď β ă γ. Note also that G β ď rΓpβq, Γpβqs and that a countable group contains at most countably many finitely generated subgroups. So, there exists a continuum X˚Ď p1, 8q such that for all distinct β, γ in X˚, we have rΓpβq, Γpβqs fl rΓpγq, Γpγqs.
Then X 1 " tΓpβq | β P X˚u is the desired continuum of the theorem.
Finally, let us construct a continuum Y 1 . To be consistent with the notations in Section 6.3, let us set PSLp2, Rq ď Homeo`pRq.
As we noted in Remark 1.2, we have that G 0 pMq " G Lip pMq. So, it suffices to compare the regularities C 0 and C 1 . Kropholler and Thurston (see [6]) observed that the group T 2 is a finitely generated perfect group, and by Thurston Stability, that every homomorphism from T 2 to Diff 1 pIq has a trivial image. In particular, H 1 pT 2 , Zq is trivial and T 2 P G 0 pIqzG 1 pIq. We continue as in Section 6.3, after substituting pk, µq " p0, 0q and pk, ωq " p1, 0q (and forgetting k, bv). We obtain groups T 3 , T 4 , T 5 and a minimally acting chain group Q ď Homeo`pIq such that T 2 ãÑ rQ, Qs R G 1 pS 1 q.
Let us put H 1 :" Q. The construction of Y 1 is very similar to that of X 1 . For each β ą 1, we can find a finitely generated groupH β ď Ş γăβ Diff γ 0 pIq such that H 1 pH β , Zq is free abelian, and such thatH β R G β pS 1 q. For each β ą 1, we apply the Chain Group Trick toH 1ˆHβ and obtain a minimal chain group Λpβq such that H 1ˆHβ ãÑ rΛpβq, Λpβqs ď Λpβq ď Homeo`pIq.
As before, there exists a continuum Y˚Ď p1, 8q such that Y 1 " tΛpβq | β P Y˚u is the desired collection. Note that no two groups in the collection X 1 Y Y 1 have isomorphic commutator subgroups.
Remark 6.11. Calegari [15] exhibited a finitely generated group in G 0 pS 1 qzG 1 pS 1 q. Lodha and the authors [41] gave (continuum many distinct) finitely generated groups inside G 0 pIqzG 1 pIq having simple commutator groups, building on [46]. The last part of the above proof strengthens both of these results. 6.5. Algebraic and topological smoothability. Theorem 1.5 also implies that if α ě 1 is a real number, then there are very few homomorphism Diff ὰ pS 1 q Ñ Diff β pS 1 q and Diff α c pRq Ñ Diff β c pRq for all β ą α. Proof of Corollary 1.8. By the Main Theorem, none of the maps in (1) through (3) are injective. The desired conclusion now follows from Theorem 3.16.
Group actions of various regularities on manifolds are closely related to foliation theory (see [18], for instance). One of the canonical constructions in foliation theory is the suspension of a group action, a version of which we recall here for the convenience of the reader. Recall our hypothesis that M P tI, S 1 u. Let B be a closed manifold with a universal coverB Ñ B. Suppose we have a representation ψ : π 1 pBq Ñ Diff ὰ pMq.
The manifoldBˆM has a natural product foliation so that each copy ofB is a leaf. The group π 1 pBq has a diagonal action onBˆM, given by the deck transformation π 1 pBq Ñ HomeopBq and by the map ψ. The quotient space Epψq "`BˆM˘{π 1 pBq is a C α -foliated bundle. This construction is called the suspension of ψ; see [18] for instance. Two representations ψ, ψ 1 P Hompπ 1 pBq, Diff ὰ pMqq yield homeomorphic suspensions Epψq, Epψ 1 q as foliated bundles if and only if ψ and ψ 1 are topologically conjugate [17,Theorem2].
Let us now consider the case M " I and B " S g , a closed surface of genus g ě 2. Let k ě 0 be an integer. Cantwell-Conlon [21] and Tsuboi [70] independently proved the existence of a representation ψ k P Hompπ 1 pS g q, Diff k pIqq such that ψ k is not topologically conjugate to a representation in Hompπ 1 pS g q, Diff k`1 pMqq. So, they concluded: Theorem 6.12 (See [21] and [70]). For each integer k ě 0, there exists a C kfoliated bundle structure on S 2ˆI which is not homeomorphic to a C k`1 -foliated bundle.
We will now prove Corollary 1.9, which is the only remaining result in the introduction that needs to be shown. Assume α ě 1 is a real number and g ě 5. Theorem 1.5 implies that there exists a representation ψ α P Hompπ 1 pS g q, Diff α 0 pIqq such that ψ α is not topologically conjugate to a representation in Hence, we may replace the hypotheses C k and C k`1 in Theorem 6.12 by C α and Ť βąα C β , respectively. We can further extend this result to more general 3-manifolds, using the techniques in [20] described as follows. Every closed 3-manifold Y with H 2 pY, Zq ‰ 0 contains an embedded 2-sided closed surface S g for all sufficiently large g ą 0. Goodman used this observation to prove that Yz IntpS gˆI q admits a smooth foliation structure, based on Thurston's result; see [31, Corollary 3.1] and [68]. By adding in the aforementioned foliated bundle structure of S gˆI inside Y, we complete the proof of Corollary 1.9.

Further questions
Let M P tI, S 1 u. One can ask for a finer distinction at integer regularities. A difficulty with part (1) below is that there does not exist a concave modulus below ω 1 , by definition.
(1) Let k ě 1. Does there exist a finitely generated subgroup G ď Diff k,Lip pMq that does not admit an injective homomorphism into Diff k`1 pMq? (2) Does there exist a finitely generated group in the set č βPN G β pMqzG 8 pMq?
Many questions also persist about algebraic smoothability of groups. For instance, finite presentability as well as all other higher finiteness properties of the groups we produce are completely opaque at this time. We ask the following, in light of Theorem 6.8: Question 7.2. For which choices of α and β do there exist finitely presented groups G P G α pMqzG β pMq? What if α, β P N?
Moreover, the constructions we carry out in this paper are rather involved. It is still quite difficult to prove that a give group does not lie in G β pMq. Question 7.3. Let G be a finitely generated group. Does there exist an easily verifiable algebraic criterion which precludes G P G β pMq?

Appendix A. Diffeomorphism groups of intermediate regularities
Let M P tI, S 1 u. We will record some basic properties of Diff k,ὼ pMq. Most of these properties are well-known for the case ω " 0, but not explicitly stated in the literature for a general concave modulus ω. We will also include brief proofs.
A.1. Group structure. Let k P N, and let ω be a concave modulus. In [52], it is proved that for a smooth manifold X, the set Diff k,ω c pXq 0 is actually a group. We sketch a proof of this fact for one-manifolds, and also include the case ω " bv.
The following lemma is useful for inductive arguments on the regularities.
Lemma A.1. Suppose ω is a concave modulus, or ω P t0, bvu. Let k P N, and let F, G : M Ñ R be maps such that F is C k´1,ω and such that G is C k . Then the following hold.
Proof. This lemma is proved in [52] when ω " 0 or when ω is a concave modulus. So we assume ω " bv. We let tx i u be a partition of M.
(1) First consider the case k " 1. We note Hence, if F¨G is C bv . If k ą 1, then we use an induction to see that is C k´2,bv . This proves part (1).
(2) The map F˝G is well-defined for all x P M. Let us first assume k " 1, so that F P C bv . Since G is bijective, we see that ÿ i |F˝Gpx i q´F˝Gpx i´1 q| ď VarpF, Mq ă 8.
The induction step follows from pF˝Gq 1 " pF 1˝G q¨G 1 .
Proposition A.2. Let ω be a concave modulus, or let ω " t0, bvu. Then for each k P N, the following is a group where the binary operation is the group composition: Diff k,ὼ pMq.
Proof. Let f, g P Diff k,ὼ pMq. It is well-known that Diff k pMq is a group. So, we have f´1, f˝g P Diff k pMq. It suffices to show that both are C k,ω . Note that p f˝gq 1 " p f 1˝g q¨g 1 . Since f 1 is C k´1,ω and g is C k , Lemma A.1 implies that f 1˝g is C k´1,ω . By the same lemma, we see that p f˝gq 1 is C k´1,ω . This proves f˝g is C k,ω .
We can write p f´1q 1 " r˝f 1˝f´1 where r : p0, 8q Ñ p0, 8q is the C 8 diffeomorphism rpxq " 1{x. Note that f 1 stays away from 0. As f 1 is C k´1,ω and f´1 is C k , we again see that f´1 is C k,ω .
A.2. Groups of compactly supported diffeomorphisms. We now establish a topological conjugacy between certain diffeomorphism groups.
Theorem A.3. Let ω be a concave modulus. Then for each k P N, the group Diff k,ὼ pIq is topologically conjugate to a subgroup of Diff k,ω c pRq. Muller [55] and Tsuboi [69] established the above result for the case ω " 0. Our proof follows the same line, but an extra care is needed for a general concave modulus ω as described in the lemmas below.
When we say a function f is defined for x ě 0, we implicitly assume to have a small number A ą 0 so that f is defined as f : r0, As Ñ R.
We let k and ω be as in Theorem A.3.
Then the following hold.
For (3), we have some a i P Z such that Since f pk´iq is C i,ω , we see from part (2) that p f {xq pk´1q is C ω .
The rest of the proof for Theorem A.3 closely follows the argument in [69], as we summarize below. Let us fix a map that is defined near x " 0: φpxq " e´1 {x .
Lemma A.5. For a C k,ω map g defined for x ě 0 satisfying gp0q " 0 and g 1 p0q ą 0, the following hold.
(1) The map h " g{x is a C k´1,ω map defined for x ě 0.
(1) If T k gpxq denotes the k-th degree Taylor polynomial for g, then f " g´T k g satisfies the condition of Lemma A.4. The conclusion follows since g{xf {x " T k g{x is a polynomial.
By part (1), the map h is C k´1,ω for x ě 0. As x approaches to 0, the denominator of the above expression for G stays away from 0 because lim xÑ0 1´x log h˝φ " 1´0¨log g 1 p0q " 1.
From xh 1 " g 1´h , we see that φ¨ph 1˝φ q is C k´1,ω and that lim xÑ0 G 1 pxq " 1. We conclude that G 1 exists for x ě 0 (even when k " 1), and is C k´1,ω . It follows that G is C k,ω .
For all 0 ď i ď k, we have that Φpgqpxq´x φpxq¨φ pxq x i " 0.
By L'Hospital's Rule again, we have pΦpgq´Idq piq " 0 for all 0 ď i ď k.
For each g P Diff k,ὼ pIq, we define Φpgq " ψ 2˝g˝φ2 . Then Lemma A.5 (3) (after using the symmetry at x " 0 and x " 1) implies that Φpgq P Diff k,ω c pRq. A.3. Simplicity. Let us use the following terminology from [41]. Let X be a topological space, and let H ď HomeopXq. We say H acts CO-transitively (or, compactopen-transitively) if for each proper compact subset A Ď X and for each nonempty open subset B Ď X, there is u P H such that upAq Ď B. Lemma A.6 is a variation of a result commonly known as Higman's Theorem.
Lemma A.6 ([41, Lemma 2.5]). Let X be a non-compact Hausdorff space, and let Homeo`pXq denote the group of compactly supported homeomorphisms of X. If H ď Homeo`pXq is CO-transitive, then rH, Hs is simple.
Let X be a topological space. We say H ď HomeopXq has the fragmentation property for an open cover U of X, if each element h P H can be written as h " h 1¨¨¨h such that the support of h i is contained in some element of U . The following lemma is very useful when proving simplicity of homeomorphism groups. This lemma is originally due to Epstein [25]; let us state a generalization by Ling [45].
Lemma A.7 ( [25,45]). Let X be a paracompact Hausdorff space with a basis B, and let H ď HomeopXq. Assume the following.
(i) H has the fragmentation property for each subcover U of B; (ii) for each U, V P B there exists some h P H such that hpUq Ď V.
Then rH, Hs is simple.
The following lemma is known for ω " 0 [61], detailed proofs of which can be found in [4,50]. The proof for a concave modulus ω is the same almost in verbatim.
Lemma A.8. Let k P N, and let ω be a concave modulus. Then for a smooth manifold X without boundary, the group Diff k,ω c pXq 0 has the fragmentation property for an arbitrary open cover of X.
From now on, we let X P tS 1 , Ru. We let C ω c pX, Rq denote the set of realvalued compactly supported ω-continuous maps X Ñ R. For each f P C c pX, Rq " C 0 c pX, Rq, we define the optimal modulus function of f as µ f ptq :" supt| f x´f y| : x, y P X and |x´y| ď tu.
It is trivial that for all x, y P X we have | f x´f y| ď µ f p|x´y|q.
Lemma A.9. For X P tS 1 , Ru and for f P C c pX, Rq, the following hold.
(3) There exists a concave modulus µ such that f P C µ c pX, Rq and such that C µ c pX, Rq " č tC ω c pX, Rq | ω is a concave modulus and f P C ω c pX, Rqu. Proof. Part (1) is a consequence of the convexity of X and the uniform continuity of f . Part (2) is obvious when t ď s. If t ą s, then part (2) follows from µ f ptq ď µ f pt´stt{suq`tt{suµ f psq ď p1`t{sqµ f psq.
Then µ 1 is continuous, monotone increasing and concave. Put µ :" µ 1`I d, so that We see that µ is a concave modulus such that f P C µ c pX, Rq. Put T :" diam supp f ě 0. Suppose f P C ω c pX, Rq for some concave modulus ω. It only remains to show that C µ c pX, Rq Ď C ω c pX, Rq. For each t ą 0, we have µ f ptq " sup |x´y|ďt | f x´f y| ď r f s ω¨ω ptq.
It follows that g P C ω c pX, Rq and the lemma is proved. We are now ready to prove the simplicity of certain diffeomorphism groups.
Let µ be a concave modulus as in Lemma A.9 for the map f pkq P C c pX, Rq. Whenever k ă β ă α, we have f pkq P C β´k pX, Rq. The same lemma implies that C µ c pX, Rq Ď C β´k c pX, Rq. So, we have f P Diff k,µ c pXq 0 Ď Ş βăα Diff β c pXq 0 and completes the proof when α ‰ k. The proof of the case that α " k " xαy`1 is almost identical.
Claim 5. For each α ą 1, every proper quotient of Ş βăα Diff β c pXq 0 is abelian. The case X " R follows from Claim 1, so we may only consider the group G " č βăα Diff β pS 1 q.
By Lemma A.8 and Claim 4, the group G has the fragmentation property for an arbitrary cover. Since Diff 8 pS 1 q ď G, we can deduce Claim 5 from Lemma A.7.
Coming back to the proof of the theorem, we only need to prove the latter parts of (1) and (2). The latter part of (1) is a special case of Corollary 3.15. For the latter part of (2), assume α ą 3. We see from Mather's Theorem and from Claim 4 that the group Ş βăα Diff β c pXq 0 is a union of perfect groups. The conclusion follows.