Diffeomorphism groups of critical regularity

Let M be a circle or a compact interval, and let α=k+τ≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =k+\tau \ge 1$$\end{document} be a real number such that k=⌊α⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=\lfloor \alpha \rfloor $$\end{document}. We write Diff+α(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Diff}\,}}_+^{\alpha }(M)$$\end{document} for the group of orientation preserving Ck\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G≤Diff+α(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\le {{\,\mathrm{Diff}\,}}_+^\alpha (M)$$\end{document} with the property that G admits no injective homomorphisms into ⋃β>αDiff+β(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcup _{\beta >\alpha }{{\,\mathrm{Diff}\,}}_+^\beta (M)$$\end{document}. We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of ⋂β<αDiff+β(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcap _{\beta <\alpha }{{\,\mathrm{Diff}\,}}_+^\beta (M)$$\end{document} with the property that G admits no injective homomorphisms into Diff+α(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Diff}\,}}_+^\alpha (M)$$\end{document}. 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≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 1$$\end{document} is a real number not equal to 2, then there is no nontrivial homomorphism Diff+α(S1)→⋃β>αDiff+β(S1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Diff}\,}}_+^\alpha (S^1)\rightarrow \bigcup _{\beta >\alpha }{{\,\mathrm{Diff}\,}}_+^{\beta }(S^1)$$\end{document}. Finally, we obtain an independent result that the class of finitely generated subgroups of Diff+1(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Diff}\,}}_+^1(M)$$\end{document} 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 → R is Hölder continuous with exponent τ if there is a constant C such that for all x, y ∈ 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 ≥ 1 and for a smooth manifold M, we write Diff k+τ + (M) for the group of orientation preserving C k diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent τ ∈ [0, 1). For compactness of notation, we will write Diff α + (M) for Diff k+τ + (M), where k = α and τ = α − k. By convention, we will write Diff 0 + (M) = Homeo + (M). The purpose of this paper is to study the algebraic structure of finitely generated groups in Diff α + (M), as α varies. We note that the isomorphism types of finitely generated subgroups in Diff α + (I ) coincide with those in Diff α c (R), the group of compactly supported C α diffeomorphisms on R; see Theorem A.3.
Let us denote by G α (M) the class of countable subgroups of Diff α + (M), considered up to isomorphism. It is clear from the definition that if α ≤ β then G β (M) ⊆ G α (M). In general, it is difficult to determine whether a given element G ∈ G α (M) also belongs to G β (M). A motivating question is the following: Question 1.1 Let k ≥ 0 be an integer.
The answer to the above question was previously known only for k ≤ 1 in part (1), and only for k = 0 in part (2). A first obstruction for the C 1regularity comes from the Thurston Stability [66], which asserts that every finitely generated subgroup of Diff 1 + (I ) is locally indicable. An affirmative answer to part (1) of Question 1.1 follows for k = 0 and M = I ; that is, G 0 (I )\G 1 (I ) contains a finitely generated group. Using Thurston Stability, Calegari proved that G 0 (S 1 )\G 1 (S 1 ) contains a finitely generated group; see [15] for the proof and also for a general strategy of "forcing" dynamics from group presentations. Navas [57] produced an example of a locally indicable group in G 0 (M)\G 1 (M); 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 / ∈ G 1 (S 1 ). This result was built on work of Witte [70]. More generally, Navas [55] showed that every countably infinite group G with property (T) satisfies G / ∈ G 1 (I ) and G / ∈ G 1.5+ (S 1 ) for all > 0; it turns out that G / ∈ G 1.5 (S 1 ) 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 [62] proved that if N is a nonabelian nilpotent group, then N / ∈ G 2 (M). By Farb-Franks [28] and Jorquera [36], every finitely generated residually torsion-free nilpotent group belongs to G 1 (M). For instance, the integral Heisenberg group belongs to G 1 (M)\G 2 (M). 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 (M) [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 (M) for all compact one-manifolds M [2]; see also [27,61]. Mapping class groups of once-punctured hyperbolic surfaces belong to G 0 (S 1 ); see [9,33,59].
Simplicity of subgroups often plays a crucial role in the study of group actions [13,25,38,65]. Examples of countable simple groups in G 0 (I )\G 1 (I ) 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 (I )\G 1 (I ) 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 ∈ {I, S 1 }. In this article, we give the first construction of finitely generated groups and simple groups in G α (M)\G β (M).
Main Theorem For all α ∈ [1, ∞), each of the sets 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 + (M) is not a group for β < 1. Using [24], we will prove a stronger fact that G Lip (M)\G 1

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 > 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 ≤ Diff α + (M) and if β > α, an injective homomorphism G → Diff β + (M) is called an algebraic smoothing of G. The Main Theorem implies that for each α ≥ 1, there exists a finitely generated subgroup G ≤ Diff α + (M) 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 Sect. 2.1 we give the definition of concave moduli (of continuity), a strict partial order between them, and the symbol k 0. For instance, ω τ (x) = x τ is a concave modulus satisfying ω τ k 0 for each τ ∈ (0, 1] and k ∈ N. For a concave modulus ω, we let Diff k,ω + (M) denote the group of C k -diffeomorphisms on M whose kth derivatives are ω-continuous. We also write Diff k,0 + (M) := Diff k + (M). We denote by Diff k,bv + (I ) the group of diffeomorphisms f ∈ Diff k + (I ) such that f has bounded total variation. Note that Diff k,bv + (I ) contains Diff k,Lip + (I ), the group of C k -diffeomorphisms whose kth derivatives are Lipschitz.
For a concave modulus ω or for ω ∈ {0, bv}, the set of all countable subgroups of Diff k,ω + (M) is denoted as G k,ω (M). We will deduce the Main Theorem from a stronger, unified result as can be found below. Theorem 1.4 For each k ∈ N, and for each concave modulus μ ω 1 , there exists a finitely generated group Q = Q(k, μ) ≤ Diff k,μ + (I ) such that the following hold.
(i) [Q, Q] is simple and every proper quotient of Q is abelian; (ii) if ω = bv, or if ω is a concave modulus satisfying μ ω k 0, then Theorem 1.4 will imply the Main Theorem after making suitable choices of μ above. See Sect. 6.4 for details.
We let F n denotes a rank-n free group. Let BS(1, 2) denote the solvable Baumslag-Solitar group of type (1, 2); see Sect. 3. In the case when M = I , our construction for Theorem 1.4 builds on a certain quotient of the group Let us describe our construction more precisely. (ii) Every diffeomorphism f ∈ φ k,μ (G † ) is C ∞ on I \∂ I .
We deduce that the group admits no injective homomorphisms into Diff k,ω + (I ). We will then bootstrap this construction to produce simple groups in Sect. 6.
We define the critical regularity on M of an arbitrary group G as Here, we adopt the convention sup ∅ = −∞. The critical regularity spectrum of M that is defined as Another consequence of the Main Theorem is the following. Theorem 1.5 gives the first examples of groups whose critical regularities are determined (and realizable) and belong to (1, ∞). 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 [56]. Second, Castro-Jorquera-Navas proved ( [22], combined with [62]) 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 4 × 4 integral lower triangular matrices with ones on the diagonal satisfies CritReg I (N 4 ) = 3/2.
It is not known whether or not the critical regularity 3/2 of N 4 is realizable.
The case G ∈ G 1 (M)\G 0 (M) 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 + (M) is topologically conjugate to a group of bi-Lipschitz homeomorphisms [24]. Thus, it is reasonable to say that [0, 1) is missing from from the critical regularity spectrum.
The authors proved in [40] that for each integer 2 ≤ k ≤ ∞, the class of finitely generated group in G k (M) is not closed under taking finite free products. From [8] and from the consideration of BS (1,2) 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 Sect. 3.4 for details.

Corollary 1.7
The group (Z×BS(1, 2)) * Z does not embed into Diff 1 + (M). In particular, the class of finitely generated subgroups of Diff 1 + (M) 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 ≥ 1, we let Diff α c (X ) 0 denote the group of C α diffeomorphisms of X isotopic to the identity through a compactly supported C α isotopy. If 1 ≤ α < β, then there is a natural embedding Diff β c (X ) 0 → Diff α c (X ) 0 defined simply by the inclusion. The main result (and its proof) of [47] (1) and (2), and if α > 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 [69] and of Cantwell-Conlon [21], that is originally proved for integer regularities. Corollary 1.9 Let α ≥ 1 be a real number. Then for every closed orientable Here, a homeomorphism of foliations is a homeomorphism of the underlying foliated manifolds which respects the foliated structures.

Automatic continuity
K. Mann proved that if X is a compact manifold then the group Homeo 0 (X ) of homeomorphisms isotopic to the identity has automatic continuity, so that every homomorphism from Homeo 0 (X ) into a separable group is continuous [48]. She uses this fact to prove that Homeo 0 (X ) has critical regularity 0 and hence has no algebraic smoothings. For discussions of a similar ilk, the reader may consult [47] and [35]. The Main Theorem implies that the critical regularity of Diff α + (M) is α, for M ∈ {I, S 1 } and for α ≥ 1.

Superrigidity
Recall that Margulis Superrigidity says that under suitable hypotheses, a representation of a lattice in a higher rank Lie group G is actually given by the restriction of a representation of G to (see [50]). For the continuous groups Diff α + (M) 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.

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 is a topological conjugacy of φ into Diff β + (M) for some β > α; that is, the conjugation hφh −1 of φ by some homeomorphism h on M such that we have hφ(G)h −1 ≤ Diff β + (M). A topological smoothing of a subgroup is obviously an algebraic smoothing, but not conversely; compare [22] and [37]. By a result of Tsuboi [69], there exists a two-generator solvable group G and a faithful action ϕ k of G on the interval such that ϕ k (G) ≤ Diff k + (I ) but such that ϕ k (G) is not topologically conjugate into Diff k+1 + (I ). Since ϕ k is injective, these actions are algebraically smoothable. See Sect. 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 [G, G] is simple, and such that [G, G] 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 [G, G] is infinite and simple, [G, G] stabilizes each component of M. It follows that G has critical regularity α with respect to faithful actions on M.

Kernel structures
In Theorem 1.5, let us fix ∈ (0, 1) such that ω μ. It will be impossible to find a finite set S ⊆ G † \ ker φ such that for all ψ ∈ Hom(G † , Diff k+ + (I )) we have S ∩ ker ψ = ∅. Indeed, Lemma 3.5 implies that for all finite set S ⊆ G † \{1} there exists a C ∞ 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.

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 + (I ). For ∈ (0, 1] satisfying ω := ω μ, we also show that the group φ(G † ) admits no algebraic smoothing into Diff k,ω + (I ). We remark that Diff k+1 + (I ) ≤ Diff k,ω + (I ). To study maps into Diff k,ω + (I ), 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 ∈ G † such that if ψ : G † → Diff 1 + (I ) is an arbitrary representation then the support of ψ(u) is compactly contained in the support of ψ(G † ).
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 (k, μ), result in a sequence of diffeomorphisms φ(w j uw −1 j ) whose complexity grows linearly in j. We show that under an arbitrary representation ψ : G † → Diff k,ω + (I ), the complexity of ψ(w j uw −1 j ) grows more slowly than that of φ(w j uw −1 j ), a statement which follows from the Slow Progress Lemma. Thus for each ψ, we find an element g ∈ G † which survives under φ but dies under ψ. In particular, φ(G † ) cannot be realized as a subgroup of Diff k,ω + (I ).

Outline of the paper
We strive to make this article as self-contained as possible. In Sect. 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 Sect. 5, we fix a concave modulus μ, and construct a representation φ of the group G † into Diff k,μ + (I ) with desirable dynamical properties and prove Theorem 1.5. In Sect. 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.

Moduli of continuity
We will use the following notion in order to guarantee the convergence of certain sequences of diffeomorphisms. Definition 2.1 (1) A concave modulus of continuity (or concave modulus, for short) means a homeomorphism ω : 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 → R be a bounded function for some U ⊆ R. If there exist constants K , δ > 0 such that

Remark 2.3
It is often assumed in the literature that a concave modulus ω(x) is defined only locally at x = 0, namely on [0, δ] for some δ > 0 [51,52]. This restriction does not alter the definition of ω-continuity for compactly supported functions. The reason goes as follows. Suppose ω : [0, δ] → [0, ω(δ)] is a strictly increasing concave homeomorphism. By an argument in the proof of Lemma A.9, we can find a concave modulus μ : for all s ∈ [0, δ]. 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 < C ; that is, we write z < C w in C if Re z < Re w, or if Re z = Re w and Im z < Im w. For two complex numbers a, b ∈ C, we let In particular, we have that We similarly define (a, b) C , together with the other types of intervals.
Then ω z is a small perturbation of ω τ (x) = x τ = exp(−τ log(1/x)). By simple computations of the derivatives, one sees that ω z is a concave modulus defined for all small x ≥ 0. See Fig. 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 ∈ (0, 1) is equivalent to the ω τ -continuity. (2) For two positive real sequences {a j } and {b j }, we will write In particular, the expression ω k 0 is vacuously true for k > 1. Compare this condition to Mather's Theorem (Definition 3.12 and Theorem 3.13).

Lemma 2.6
The following hold for k ∈ N and for a concave modulus ω.
(3) Assume that we have positive sequences {a j } and {b j } such that If ω k 0, then we have {a j } {b j }.
Proof Proofs of (1) and (2) are obvious from monotonicity and concavity. Assume (3) does not hold. Passing to a subsequence, we may assume {t j := b j /a j } converges to 0. Then we have a contradiction because Suppose ω and μ are concave moduli. We define a strict partial order ω μ if for all K > 0. Here, we use the notation From z < C w, we see that the above limit equals −∞. This is as desired.
Let k ∈ N and let ω be a concave modulus.
Let f : I = [p, q] → 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 Var( f, I ) is finite on I . If M = S 1 , we use the same definition for Var( f, I ) with p = q. We say that a diffeomorphism f : 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 which turns out to be a group for k ∈ N (Proposition A.2). We define G k,ω (M) to be the set of the isomorphism classes of countable subgroups of Diff k,ω + (M). Note that We have that If we have two concave of moduli ω μ, then we have In particular, if z, w ∈ (0, 1] C satisfy z < C w, then we see from Lemma 2.7 that Diff k,ω z

Fast and expansive homeomorphisms
From now on until Sect. 6, we will be mostly concerned with the case M = I . For a measurable set J ⊆ R, we denote by |J | its Lebesgue measure. We write J 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 → X be a map on a space X . We use the standard notations The set supp f is also called the (open) support of f . We note the identity map Id : R → R satisfies Id ( j) (x) = δ 1 j for j ≥ 1.

Definition 2.8
Let f : I → I be a homeomorphism, and let J ⊆ I be a compact interval such that f (J ) = J . We let k ∈ N.
(1) We say f is k-fixed on J if one of the following holds: 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 = [p, q] if and only if there exists some y ∈ J satisfying one of the following (possibly overlapping) alternatives: For a set A ⊆ 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 ∈ N, and let ω k 0 be a concave modulus. Suppose we have Then for each δ > 0 and λ > 0, the following set has the natural density zero: The proof of the theorem is given in Sect. 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 ⊆ N, the following hold.
Fastness and expansiveness constants of "roots" of a diffeomorphism behave like arithmetic and geometric means, respectively: Lemma 2.11 Let f ∈ Homeo + (J ) for some compact interval J , and let N ∈ N.
Proof Let us write J = [p, q].
(1) For some y ∈ J we have Hence there exists some (2) Assume the alternative (E1) holds as described after Definition 2.8. That is, So, for some y = f i (y), we have This is the desired inequality. The other alternatives are similar.

Lemma 2.12
For a C k -map f : I → R, the following hold.
Proof For each j ∈ {0, 1, . . . , k}, we define (1) We have S j ⊆ S j . It now suffices for us to show the following: Let us assume x ∈ S j for some 0 ≤ j < k. Then there exists a sequence {x i } ⊆ S j \{x} converging to x. There exists y i between x i and x such that Since y i ∈ S j+1 converges to x, we see that x ∈ S j+1 . This proves S j ⊆ S j+1 .

Lemma 2.13
Let J ⊆ I be a compact interval, and let δ, If, furthermore, f is C k,ω then we have (2) If f is λ-expansive on J , then If, furthermore, f is C k,ω then we have Proof For each j ≤ k, Lemma 2.12 implies that there exists s j ∈ J satisfying Let y 0 ∈ J be arbitrary. We see (cf. Lemma A.4) that If f is C k,ω , then we further deduce that (2) Write J = [p, q]. Assume the alternative (E1) holds for y 0 ∈ J ; that is, 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 → I is a C k,ωdiffeomorphism. Let δ, λ > 0, and define We let K > 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 {J i } 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 > 0 such that Hence, Lemmas 2.11 and 2.13 imply that As in Claim 2, we have B λ 1/i < ∞ and d N (B λ ) = 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 .
So, for some constant K 0 , K 1 > 0 we deduce from Hölder's inequality that We conclude from Lemma 2.10 that d N (A δ ) = 0.
We apply Lemma 2.13 and also the proof of Claim 3. For each i ∈ N we put We have We again apply Hölder's inequality. For some constant We obtain d N (B λ ) = 0.

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 ∈ N, let δ ∈ (0, 1) and let μ be a concave modulus satisfying μ Then there exists f ∈ Diff k,μ + (R) satisfying the following: Since I is compact, it is necessary that i |J i | < ∞. 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 Sect. 2.4, we will fix the following constants.
We will prove Theorem 2.14 through a series of lemmas. Let us first note the following standard construction of a bump function ; see Fig. 2a.

Lemma 2.16
There exists an even, C ∞ map : R → 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 Fig. 2b.
Proof There exists a unique C ∞ map g satisfying the following conditions: Hence, we have (i).
To verify (iv), let us estimate Using the symmetry So, we have an inequality We now have the following three possibilities for y.
We see that Proof We may assume J = [0, ]. 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 ∞ diffeomorphism.
The claims (A), (C) and (E) are immediate from Lemma 2.17. Observe that .
This establishes the claim (D), and hence the conclusion of the lemma.
Proof of Theorem 2.14 Put i = |J i |. As i i < ∞, there exists i 0 such that i ≤ * for all i ≥ i 0 . For each i ≥ i 0 , we apply Lemma 2.18 to obtain Hence {F n } uniformly converges to a C k map F : R → 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 Claim For all x, y ∈ R we have In order to prove the claim, we may assume x ∈ J i for some i ≥ i 0 . If y ∈ J i , then the condition (E) implies the claim. If y / ∈ supp F, then we can Hence, the claim is proved. We have that F ∈ Diff k,μ Finally, we can pick F * ∈ Diff ∞ + (R) such that: Then the diffeomorphism f = F • F * ∈ Diff k,μ + (I ) 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 ∞ .

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 → 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 [23,58].
We note the following consequence of Theorem 2.14.

Corollary 2.20
Let K * > 0, and let {J i } i∈N be a collection of disjoint compact intervals contained in the interior of I satisfying Then for k ∈ N and for a concave modulus μ ω 1 , there exists Proof Let us write i = |J i | and .
We have f ∈ Diff k,μ + (I ) as given by Theorem 2.14 with respect to {J i } and some δ ∈ (0, 1). Let us pick ω such that 0 ≺ k ω μ.
Claim lim For all sufficiently large i, we have So we see that For C k,bv , we simply set ω = ω 1 and apply Theorem 2.9 again.

More on natural density
For N ∈ N, let us use the notation We will need the following properties of density-one sets.
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.
Hence, for each s ∈ N and t ∈ [N ] * we compute By summing up the above for t ∈ [N ] * , we have After dividing both sides by N 2 s and sending s → ∞, we see that

Background from one-dimensional dynamics
In this section, we gather the relevant facts regarding one-dimensional dynamics that we require in the sequel.

Covering distance and covering length
Throughout Sect. 3.1, we let G be a group with a finite generating set V , and let ψ : G → Homeo + (I ) be an action. We develop some notions of complexity of an element in ψ(G) 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 ⊆ I , we define its V -covering length as Here, we use the convention inf ∅ = ∞. We also let CovLen V (∅) = 0. We define the V -covering distance of x, y ∈ I as That is to say, once a generating set for G has been fixed, 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 ψ(G), then the covering distance between them is necessarily infinite.
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 CovLen(A) and CovDist(x, y). We will also write gx := ψ(g).x for g ∈ G and x ∈ I .
Covering distance behaves well in the sense that it satisfies the triangle inequality:   If 1 = w ∈ G, we define the syllable length of w, written ||w||, to be where v i ∈ V and n i ∈ Z for each 1 ≤ i ≤ . The following lemma relates the algebraic structure of the given group G = V with the dynamical behavior of actions of G: 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 ∈ 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 ∈ V and n ∈ Z. Then either whenever |i − j| ≥ 2, and such that U i ∩ U i+1 is a nonempty proper subset of both U i and U i+1 for 1 ≤ i ≤ n − 1. Then we say (U 1 , . . . , U n ) 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.

Lemma 3.3 If U is a collection of open intervals such that I ⊆ U , then a minimal subcover V ⊆ U of I is a chain of finitely many open intervals.
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 ψ(G) efficiently in the following sense, which provides a converse to Lemma 3.2: Then there exists an element g ∈ G such that gx > y and such that ||g|| = N .
We remark that ideas in a very similar spirit to Lemma 3.4 were used extensively in [41].
for each i, and such that these intervals witness the fact that CovDist(x, y) = N . Lemma 3.3 implies {U i } is a chain. Renumbering these intervals if necessary, we may assume that x ∈ U 1 \U 2 , that y ∈ U N \U N −1 , and that for each i (cf. Fig. 3). Note that we allow sup For a suitable choice of n 1 , we have v n clearly has syllable length at most N and satisfies gx > y. Lemma 3.2 implies that ||g|| = N .

A residual property of free products
For a compact interval J ⊆ R, we let Diff ∞ 0 (J ) denote the group of C ∞diffeomorphisms of R supported in J . One can identify Diff ∞ 0 (J ) with the group of C ∞ -diffeomorphisms on J which are C ∞ -tangent to the identity at ∂ J . For a group G and a subset S ⊆ G, we let S denote the normal closure of S.
has a connected support, and suppose Then there exists a representation with a connected support such that φ g (g) = 1 and such that supp Proof of Lemma 3. 5 We have embeddings with full supports. Let ρ − and ρ − denote the "opposite" representations of ρ + and ρ + , respectively. That is, we let ρ − (g)(x) = 1 − ρ + (g)(1 − x) and similarly for ρ − .
After a suitable conjugation, we may assume for some ∈ N, g i ∈ G and p i , q i ∈ 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 and a point x 2i−1 such that We pick x 2 +1 and z i so that We can find a C ∞ -action The nontriviality of φ g (g) comes from a Ping-Pong argument for free products (cf. [3,42]); that is, φ g (g)(x 1 ) = x 2 +1 > 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 ⊆ G\{1} after setting g as a suitable concatenation of the elements in A. In particular, ρ i G and φ g G are faithful. Here, the symbol denotes the restriction of a representation.

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 [57] in his thesis. Theorem 3.6 (Kopell's Lemma; see [44]) Let f, g ∈ Diff 1+bv We continue to let M ∈ {I, S 1 }. We say f ∈ Homeo + (M) 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: If ω is a concave modulus or if ω ∈ {0, bv, Lip}, then we define the C k,ωcentralizer group of G ≤ Homeo + (M) as Let Z k,ω (g) := Z k,ω ( g ) for g ∈ Homeo + (M). We write Fix G = ∩ g∈G Fix g. Let BS(1, m) denote the Baumslag-Solitar group of type (1, m), given as below. (1) If ρ(y) = 1, then ρ is faithful.
(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 > 2 is essentially identical.
If g ∈ Diff 1+bv + (S 1 ) is an infinite order element having a finite orbit, then every element in Z 1+bv (g) has a finite orbit and every element in [Z 1+bv (g), Z 1+bv (g)] 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 g is now played by the group BS(1, 2). Lemma 3.9 Suppose we have an isomorphic copy of BS(1, 2) given as Then the following hold.
(1) The C 1 -centralizer group Z 1 (B) of B has a finite orbit. Let Z 0 be the kernel of the above homomorphism Since every element of Z 0 fixes ∂ J for J ∈ A , we can regard Z 0 , B 0 ≤ Diff 1 + [0, 1]. 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 (B)/Z 0 → Homeo + (X ) is free. By a variation of Hölder's Theorem given in [40,Corollary 2.3], there exists a Fig. 4 The relators of G † . The horizontal double edge denotes the relator aea −1 = e 2 and the other two edges denote commutators free action ρ : Z 1 (B)/Z 0 → Homeo + (S 1 ) extending ρ 0 such that rot •ρ is a monomorphism; see also [27]. We have a commutative diagram as below: The commutativity of the lower square implies that rot restricts to a homomorphism on Z 1 (B). In particular, we have that rot(g) = 0 and that g is grounded. Since g centralizes B, and since Fix B 0 = ∅, we see that Fix B 0 , g = ∅. So, we may regard B 0 , g ≤ Diff 1 + (I ). Lemma 3.8 implies that supp g ∩ supp y = ∅, as desired.

A universal compactly-supported diffeomorphism
Throughout this paper, we will fix a finite presentation: 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 ψ(G † ): If ψ : G † → Homeo + (I ) is a representation and f ∈ ψ(G † ), there is little reason to believe that CovLen(supp f ) < ∞, even if we restrict to a component of supp ψ(G † ). 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 ∈ G † for which CovLen(supp ψ(u 0 )) < ∞.
We will build such an element u 0 ∈ G † . We say a set A ⊆ R is compactly contained in a set B ⊆ R if there exists a compact set C such that A ⊆ C ⊆ B. (1) Then α, β, t is not isomorphic to Z 2 * Z.
(2) If M = I , then the support of is compactly contained in supp α, β, t .
Proof Suppose we have a faithful representation Consider first the case when M = I . By Lemma 3.8, we see that supp ψ(c) ∩ supp ψ(e) = ∅. It follows from Lemma 3.10 that This is a contradiction, for ψ is faithful. Assume M = S 1 . By Lemma 3.9 (2), we have some p ∈ N such that supp ψ(c p ) ∩ supp ψ(e) = ∅.
We again deduce a contradiction from Lemma 3.10, for we have We will apply abt-lemma to the triple (c, e, d). For this, we let Proof Since ψ(c) ∈ Z 1 ( a, e ), we see from Lemma 3.8 (2) that supp ψ(c) ∩ supp ψ(e) = ∅. Lemma 3.10 implies the desired conclusion.

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 (X ) 0 denote the set of C k,ω diffeomorphisms isotopic to the identity through compactly supported isotopies; this set is indeed a group [51]. Note that

Definition 3.12
Let ω be a concave modulus.
Mather [51,52] proved the simplicity of Diff k + (X ), 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 [51,52]) Suppose X is a smooth nmanifold without boundary. Let k ∈ 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 (X ) 0 is simple.

Corollary 3.15 Let X be a smooth n-manifold without boundary, and let k ∈
N. If some z ∈ (0, 1] C satisfies Re(k + z) = n +1, then the group Diff k,ω z c (X ) 0 is simple.
Proof We use Lemma 3.14 and Mather's Theorem. If Re z ∈ (0, 1), then ω z is sup-and sub-tame, and so, Diff k,ω z c (X ) 0 for all k ∈ N. If z = s √ −1 for some s < 0, then ω z is sup-tame; in this case, Diff k,ω z c (X ) 0 is simple for all integer k = n + 1. If z = 1 + s √ −1 for some s ≥ 0, then ω z is sub-tame and Diff k,ω z c (X ) 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 ∈ {S 1 , R}, the following hold.

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 [0, 1] 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 [14,19]).
We will denote the standard piecewise linear representation of F as A typical choice of a generating set for F is {x 0 , x 1 }, which are determined by the breakpoints data: Recall that a group action on a topological space is minimal if every orbit is dense. The action ρ F is minimal on (0, 1), but it has an even stronger property: the diagonal action of ρ F on X = {(x, y) ∈ (0, 1) × (0, 1) | x < y} 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 (0, 1) is locally dense [10]. The general definition of local density is not important for our purposes. For a chain group G ≤ Homeo + [0, 1] (see Remark 3.19 below for a definition), the local density of the action of G on (0, 1) 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 (0, 1); 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: The original construction of Ghys-Sergiescu is a C ∞ 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 Note ρ GS (F) acts minimally on (0, 1). There exists a homeomorphism h GS : [0, 1] → [0, 1] such that for all g ∈ F we have It will be convenient for us to denote a i = ρ GS (x i ) for i = 0, 1.

Corollary 3.18
There exists a chain of two intervals (U 1 , U 2 ) and C ∞ diffeomorphisms f 1 and f 2 supported on U 1 and U 2 respectively such that 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. . . , f n is called a pre-chain group (cf. [41]). The group f 1 , . . . , f n is called a chain group if moreover we have f i , f i+1 ∼ = F for each 1 ≤ i < n. If f 1 , . . . , f n is a pre-chain group then for all sufficiently large N , we have f N 1 , . . . , f N n is a chain group [41].

The Slow Progress Lemma
Throughout this section, we assume the following. Let k ∈ 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: where ω k 0 is some concave modulus; • ψ : G → Diff k,bv + (I ), in which case we will put ω = ω 1 . We denote by h the syllable length of h ∈ G with respect to V as in Sect. 3.1. We also use the notation V = ∪ v∈V π 0 supp ψ(v).
Suppose we have sequences {N i } i∈N ⊆ N and {v i } i∈N ⊆ V such that the following two conditions hold. First, for some K > 0 we assume Second, for each v ∈ V we assume the following set has a well-defined natural density: Let us define a sequence of words {w i } i≥0 ⊆ G by w 0 = 1 and The main content of this section is the following:

Lemma 4.1 (Slow Progress Lemma)
For each x ∈ I , we have 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.

Reduction to limit superior
For brevity, we simply write CovLen and CovDist for CovLen V and CovDist V . We write gx = ψ(g)x for g ∈ G and x ∈ I .
Proof Assume (ii) does not hold. There exists M 0 > 0 and an infinite set A ⊆ N such that for all a ∈ A we have For each s ∈ N, let us choose a(s) ∈ A such that j (s) < a(s). We see that This is a contradiction, and (i) ⇒ (ii) is proved. The converse is immediate.

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 ∈ U ∈ π 0 supp ψ(G) and a real number M 0 > 0 such that the sequence {x i := w i x} i≥0 satisfies By Lemma 4.3, it suffices for us to deduce a contradiction from (A3). The sequence {x i } accumulates at ∂U . Since the sequence cannot accumulate simultaneously at the both endpoints of U by assumption (A3), we may make an additional assumption: For each i ∈ N, we define The point z * i is the "length-i marker" of covering lengths in the following sense.

Lemma 4.4 (1) Define h : (x, sup U ) → N by h(z) := CovLen[x, z). Then h is a surjective, monotone increasing, left-continuous function. (2) For all 1 ≤ i < i + j, we have
Proof (1) Since each point in I belongs to at most |V | intervals in V , each z i is realized as sup J for some J ∈ V .
We claim that z * i = z i and that CovLen[x, z * i ) = i for each i ∈ N. The case i = 1 is trivial. Let us assume the claim for i − 1. Then we have CovLen[x, z i ) = i and z i ≤ z * i . If z i < z * i then there exists t ∈ (z i , z * i ) such that CovLen[x, t) = i. But whenever t ∈ J ∈ V we have z i−1 / ∈ J , by the choice of z i . This shows CovLen[x, t) > i, a contradiction. Hence the claim is proved.
(2) Note that The opposite inequality is immediate from the definition of z i . For the second equation, it suffices to further note that CovLen[x, z * i ] = i + 1. (3) By (A3), the following holds for all but finitely many i: For such an i, we have that x i ∈ (z * j−1 , z * j ] and x i+1 ∈ (z * j , z * j+1 ] for j = CovLen[x, x i ). If x i = z * j , then x i+1 < z * j+1 and moreover, x i+ < z * j+ for all ∈ N.
Let us write z i = z * i−M 2 . After increasing M 0 if necessary, we have the following for all i ≥ M 0 and j > 0: (A5) We may also assume: CovLen Consider the set of "significant generators" and their minimum density: By further increasing M 0 , we may require: for all v ∈ V 1 and N ≥ M 0 . We note the following.

Lemma 4.5 Let
Proof Note that Hence, we have j m ≤ m/δ 1 . Lemma 2.6 implies that The desired inequality is now immediate.

Estimating gaps
Let i ≥ M 0 . Since we can find J i ∈ π 0 supp ψ(v i ) such that {x i−1 , x i } ⊆ J i . We define As illustrated in Fig. 5, we will write Roughly speaking, L i is obtained from J i by successively attaching adjacent components of supp ψ(v i ) on the left until we have included at least k +1 fixed points of ψ(v i ) or an accumulated fixed point of ψ(v i ). By (A4) and (A6), the intervals L i and R i are compactly contained in U . (1) The map ψ(v i ) is k-fixed on L i and also on R i . (1) and (2) are obvious from the definition and from the fact that For part (3), suppose x ∈ A ∈ π 0 supp ψ(v) for some v ∈ V . There exist at most 2k indices i ≥ M 0 such that v j = v and such that A ⊆ L j ∪ 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 ≥ 8k. We call each x i as a ball, and the interval  For each δ > 0 and v ∈ 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 Sect. 2:  we see that either L i ⊆ gap(m) or R i ⊆ gap(m). As m ∈ Bag δ , we have i ∈ Ball δ and hence, By a similar argument, This is a contradiction.

Lemma 4.10
For all λ ≥ 1, the following set has the natural density one.

Completing the proof of the Slow Progress Lemma
We see from Lemma 4.5 and Theorem 2.9 that for each v ∈ V 1 . This implies d N (E λ ) = 0, contradicting Lemma 4.10. Hence the assumption (A3) is false and the proof is complete.

Consequences of the Slow Progress Lemma
The following is the main obstruction of algebraic smoothing in the Main Theorem.

Lemma 4.11
Let u ∈ G and let U ∈ π 0 supp ψ(G). If supp ψ(u) ∩ U is compactly contained in U , then for each real number T 0 > 0 and for all sufficiently large i ∈ N, there exists h i ∈ G such that the following hold: Proof Let u, U and T 0 be given as in the hypothesis. We write Put T = CovDist(x, y). By the Slow Progress Lemma, whenever i 0 we have This gives the desired relations.

A dynamically fast subgroup of Diff k,μ + (I)
Recall we have defined G † in Sect. 3.4. We will now build a representation φ : G † → Diff k,μ 0 (I ) such that supp φ(G † ) is connected and φ(G † ) admits no injective homomorphisms into Diff k,ω + (I ) for all 0 ≺ 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 ∈ G † be given such that supp φ(u 0 ) is compactly contained in supp φ(G † ). We build a sequence a elements {w i } i≥0 ⊆ G † which depend on k, μ such that, after replacing u 0 by a suitable conjugate u in G † if necessary, we have We build the representation φ in several steps.

A configuration of intervals in I
Let us now build an infinite chain of bounded open intervals in R as shown in Fig. 7. The union of F will be also bounded. We will simultaneously define representations As in Lemma 3.11, we put The standard affine action of BS(1, 2) is conjugate to a C ∞ -action on R supported in [0, 1]; see [68] or [58,Section 4.3], for instance. Applying Lemma 3.5 to we have an action such that ρ 0 (b) = 1, ρ 0 (u † ) = 1 and moreover, I 0 := (−1, 1) = supp ρ 0 . By the same lemma, we can also require that ρ 0 a, e ∼ = a, e ∼ = BS(1, 2).
We will include six more open intervals to the chain F as shown in the configuration (Fig 7). We will require that B − = −B + and so forth, where we use the notation for 0 ≤ r < s ≤ ∞. Also, we set sup C + = 2 and sup D + = 3. By Corollary 3.18, there exists a C ∞ diffeomorphisms c + 1 , d + 1 supported on C + , D + respectively such that c + 1 , d + 1 ∼ = F and c + 1 , d + 1 acts locally densely on C + ∪ D + . We may require c Note that 1 / 2 < 2 and that the sequence { i / i+1 } decreases to 1. Hence,

Let us inductively define
Note that |L + i ∩ L + i+1 | = κ i ; see Fig. 8. Since Fig. 8 The bounded open intervals L i 's we see that In other words, the collection {L + i } has no triple intersections. Then we define symmetrically L − i = −L + i and add L ± i to F . This completes the definition of the infinite chain F . As i i < ∞, there exists some compact interval I such that By applying Theorem 2.14 to the parameter Note that we are invoking the hypothesis that . We also define a ± 2 completely analogously with respect to the parameter Then we define We see from the construction that Hence, the map φ extends to a group action Let us summarize the properties of φ below. The proofs are obvious from construction and from Theorem 2.14. We continue to use the notation from Sect. 5.1.

Lemma 5.1 The following hold for
Every orbit of φ a, c, d, e in I 0 is accumulated at ∂ I 0 .

The behavior of {w i } i≥0 under φ
Whereas we have good control over the compactly supported diffeomorphism φ(u), we will need to have good control over commutators of conjugates of φ(u).

Lemma 5.2 For each nonempty open interval U
Intuitively, Lemma 5.2 says that no matter how small an interval we choose inside supp φ(G † ), we may find an element of f ∈ φ(G † ) so that f (U 0 ) stretches across Of course, f (U 0 ) might be much larger than this union, though this is unimportant.
Proof of Lemma 5.2 Let U 0 = (z 1 , z 2 ) be given as in the hypotheses of the lemma. By Lemmas 5.1 (4) and 3.4, there exists an f ∈ φ(G † ) such that f (z 2 ) ∈ D + ∩ L + 1 . So, we may assume z 2 ∈ D + ∩ L + 1 . We may then assume that z 1 ≥ sup L − 1 ; for, otherwise there is nothing to show. There are four (overlapping) cases to consider.
We use the fact that the restriction of φ c, d to C + ∪ D + generates a locally dense copy of Thompson's group F. As we have seen in Sect. 3.6, for some suitable f 1 ∈ φ c, d we may arrange f 1 (z 1 ) ∈ B + ∩ C + and f 1 (z 2 ) ∈ D + ∩ L + 1 , thus reducing to the previous case.
We retain the elements {w i } i∈N as defined in Sect. 5.1. The following lemma measures the complexity of certain diffeomorphisms in φ(G † ) and shows that the complexities grow linearly.

Lemma 5.3
Let u ∈ G † \ ker φ be an element such that supp φ(u) is compactly contained in supp φ. Then for some conjugate u ∈ G † of u, and for some component U 1 of supp φ(u ), we have that whenever i ∈ N the bounded open interval φ(w i )U 1 intersects both L + i+1 and L − i+1 . In particular, we have that Proof Choose an open interval U 0 ∈ π 0 supp φ(u) compactly contained in I . By Lemma 5.2, there is a conjugate u ∈ G † of u such that the image U 1 of U 0 under this conjugation intersects L ± 1 . Conjugating by a further power inf Fig. 10. We now apply φ to the conjugates w i u w −1 i .

Assume by induction that
Here, we used κ > 1/4 > 2(1 − δ 0 ). By induction, we see that φ(w i )s ± ∈ L ± i+1 . In order to cover φ(w i )U 1 by intervals in F , we need at least The conclusion is now obvious.

Certificates of non-commutativity
The following fact will be used in order to show that φ(G † ) cannot be smoothed algebraically.
Proof Write U = (z 1 , z 2 ) and CovDist(z 1 , But this would imply that one of the following holds: This then violates Lemma 3.2. Let f = φ(u) and g = φ(huh −1 ). Since supp f is compactly contained in I , there exists a compact interval J such that Since φ(G † ) is C ∞ at each point x ∈ I \∂ I , we may regard f, g ∈ Diff ∞ + (J ). A corollary to Kopell's Lemma (Corollary 3.7) implies that if f and g commute, then U and φ(h)U 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 ∈ G † satisfies that supp φ(u) is a nonempty set compactly contained in supp φ(G † ). Then there exists a conjugate u of u in G † such that for all i ∈ N, for all s, t ∈ {−1, 1} and for all h ∈ G † satisfying h < 2i, we have Proof Using Lemma 5.3, we obtain a conjugate u of u such that for each i ∈ N, the set supp φ(w i u w −1 i ) has a component U i whose covering length is larger than 2i.
Note that for at least one h ∈ {h, a s · h, b t · h}, we have that and that ||h || ≤ 2i. The nontriviality of follows immediately from Lemma 5.4.

Finishing the proof of Theorem 1.5
So far, we have constructed Theorem 5. 6 Suppose ω is a concave modulus satisfying 0 ≺ k ω μ, or suppose ω = bv. If we have a representation then we have that Proof Let u 1 := u † ∈ [G † , G † ] be the element considered in Lemma 3.11 and Sect. 5.2. By the same lemma, supp ψ(u 1 ) is compactly contained in supp ψ(G † ). We see from the construction that φ(u 1 ) = 1. So, we may assume ψ(u 1 ) = 1. Let us choose a minimal collection There exists a conjugate u 1 of u 1 satisfying the conclusion of Lemma 5.5. Recall from Sect. 5.1 that we have Hence, we can apply Lemma 4.11 to u 1 and U 1 . We obtain some i ∈ N, some h 1 ∈ G † with h 1 < 2i, and some s, t ∈ {1, −1} such that As u 1 has been chosen to satisfy Lemma 5.5, there exists a choice of h 1 such that Note that supp φ(u 2 ) is still compactly contained in supp φ(G † ). We now have supp ψ(u 2 ) ⊆ U 2 ∪ · · · ∪ U n .
Inductively, we use u 2 to obtain u 2 satisfying Lemma 5.5. The same argument as above yields Continuing this way, we obtain an element u m ∈ [G † , G † ] ∩ ker ψ\ ker φ for some m ≤ 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 PL[0, 1] 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 [63]. 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 (G † ) 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.

The Rank Trick
If φ : G → Homeo + [0, 1] be a representation, then a priori, it is possible that the rank of the abelianization H 1 (φ(G), Z) is less than that of H 1 (G, Z). Let us now describe a systematic way of producing another representation φ 0 such that the rank of H 1 (φ 0 (G), Z) is maximal. such that supp ρ is bounded, then there exists another representation satisfying the following: Proof Let H 1 (G, Z) ∼ = Z m for some m ≥ 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 → Homeo + (R) by the recipe for each g ∈ G. It is clear that ρ 0 satisfies parts (i) and (ii). Since α decomposes as we see that ρ 0 (G) surjects onto Z m . This proves part (iii).
Remark 6.2 Algebraically, the group ρ 0 (G) is a subdirect product of ρ(G) and Z m .

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 + (R). We will need the following result of the authors with Lodha: 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 {a 0 , a 1 } as defined in Sect. 3.6. By an n-generator group, we mean a group generated by at most n elements. (0, 1). We put G = G, ρ GS (F) .

Lemma 6.4 (Chain Group Trick) Let G be an n-generator subgroup of Homeo + (R) such that supp G is compactly contained in
(1) Then G is an (n +2)-chain group acting minimally on (0, 1). In particular, [ G, G] is simple and every proper quotient of G is abelian. Since s i ∈ Q GS , we can find f 1 ∈ ρ GS (F) such that supp f 1 = (s 2 , s 3 ) and such that f 1 (t) ≥ t for all t ∈ [0, 1]. We fix t 0 ∈ (s 2 , s 3 ) ∩ Q GS , so that After conjugating G by a suitable element of ρ GS (F) if necessary, we may assume that the closure of supp G is contained in (t 0 , f 1 (t 0 )).
Claim If g = g i for some 1 ≤ i ≤ n, then we have that If t / ∈ (s 2 , s 3 ), then a 1 •g(t) = a 1 (t) and the claim is obvious. If t ∈ (s 2 , s 3 ), then This proves the claim. We define u 0 = a 1 , and u i = a 1 g i for i = 1, . . . , n. We also let u * 0 = u −1 0 a 0 , u * n+1 = a n 0 u n a −n 0 and Then we have The group G acts minimally on (0, 1) since so does ρ GS (F). It now suffices to show that the collection {u * 0 , u * 1 , . . . , u * n+1 } is a generating set for an (n + 2)-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 ∈ ρ GS (F) in part (1). We put For all distinct i, j ∈ Z we have Let we have an embedding G → [G 1 , G 1 ] defined by The proof is complete since Remark 6.5 In the above lemma, put V := {g 1 , . . . , g n }\ρ GS (F).
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 [G, G].
Proof Let G 0 ≤ 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 [G, G] acts trivially on G/G 0 . This implies [G, G] ≤ G 0 .

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 the representation φ constructed in the previous section. We put T 1 := φ(G † ).
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 such that the conclusions of Lemma 6.1 hold. We put T 2 := φ 0 (G † ) so that We may assume supp T 2 ⊆ I ⊆ (0, 1).

Claim 1
We have that T 1 , T 2 ≤ Diff k,μ 0 (I ) and that This claim for T 1 follows from Theorem 5.6. In order to prove the claim for T 2 , we let 0 ≺ k ω μ or let ω = bv. Suppose ψ : T 2 → Diff k,ω + (I ) is a representation. By applying Theorem 5.6 again to the composition Since φ 0 (g) = φ(g) = 1 by Lemma 6.1 (ii), we have φ 0 (g) ∈ ker ψ\{1}. This proves the claim. We can apply the Chain Group Trick (Lemma 6.4) to T 2 , and obtain acting minimally on (0, 1) as a seven-generator chain group. From Claim 1 and from the fact T 2 → [T 3 , T 3 ], we obtain the following and complete the proof of Theorem 1.4 for M = I .

Claim 2 The countable simple group
Let us now consider the case M = S 1 . After a conjugation, we may assume supp T 3 ⊆ I ⊆ (0, 1). As BS(1, 2) embeds into Diff ∞ 0 (I ), we may regard

Claim 3
We have the following: is an injective homomorphism. By Lemma 3.9 (2), a proper compact subset of S We can apply the Rank Trick to ρ, since Then we obtain a representation Let T 4 be the image of ρ 0 . We may require that supp T 4 ⊆ I ⊆ (0, 1) and that Claim 3 now implies the following.

Claim 4
The group T 5 is a nine-generator group such that Since H 1 (T 5 , Z) ∼ = H 1 (T 4 , Z) ⊕ Z is free abelian, we can finally apply the Chain Group Trick to obtain a minimally acting eleven-chain group Q = Q(k, μ) with Summarizing, we have the following. Proposition 6.7 Let k ∈ N, and let μ ω 1 be a concave modulus. Then there exists an eleven-generator group Q = Q(k, μ) such that the following hold.
(1) [Q, Q] 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 ∈ {I, S 1 }.
Proof Part (1) follows from that Q is a minimally acting chain group (Theorem 6.3). Part (2) is established above. We deduce part (3) from Part (4) is a consequence of parts (1) and (3) along with Lemma 6.6.
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 isomorphisms ∼ = come from the Rank Trick and the embeddings → come from the Chain Group Trick.

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 ≥ 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 ∈ X α , we have that A ≤ Diff α 0 (I ) and that (ii) For each B ∈ Y α , we have that B ≤ β<α Diff β 0 (I ) and that (iii) No two groups in X α ∪ Y α have isomorphic commutator subgroups.
In order to prove Theorem 6.8, we set up some notations. For a complex number z ∈ C, we let z denote the largest integer m such that m < C z. For instance, we have k = k − √ −1 = k − 1 and k + 1/2 = k + √ i = k for an integer k. Let z > C 1, written as z = k + τ + s √ −1 for k = Re z and τ, s ∈ R. We put κ(z) := ( z , ω z− z ).
If α > 1 is a real number and if k = α , then we see that Using the notation Q(k, μ) from Proposition 6.7, we observe the following.

Lemma 6.9
The following hold for all complex numbers 1 < C z < C w.
For part (4), we first assume Re z = 1. There exists a real number t > s such that st ≥ 0 and such that z < C w := k + τ + t √ −1 < C w. Using part (1), we may assume w = w . Part (3) implies ω w− w k 0. We have that The conclusion of (4) follows from Lemma 2.7 and Proposition 6.7.
Let us assume Re z = 1, so that z = s √ −1 for some s > 0. We can pick w = 1 + τ < C w for some τ ∈ (0, 1). Again, we may set w = w so that ω w− w is sub-tame. The desired conclusion follows from the comparison Remark 6.10 In the case when z = 1 + s √ −1 and w = 1 + t √ −1 for some 0 < s < t, we cannot conclude that part (4) above holds. This is because ω w− w = ω t √ −1 may not be sub-tame. Let us now prove Theorem 6.8 for the case α > 1. We define Pick a real number s > 0 and put Let β > α be a non-integer real number. By Lemma 6.9, we have that [A, A] / ∈ G κ(β) (S 1 ) = G β (S 1 ). The conclusion (i) of the Theorem is satisfied.
Let us now pick a real number s < 0 and put B = Q • κ α + s √ −1 ∈ Y α . Let β < α be non-integer real number larger than 1. We have that Since α > C α + s √ −1 > C 1, we see from Lemma 6.9 that This proves the conclusion (ii). It is obvious from the conclusions (i) and (ii) that whenever A ∈ X α and B ∈ Y α , we have [A, A] [B, B]. Suppose we have real numbers 0 < s 1 < s 2 , and put A i = Q • κ(α + s i √ −1). Using α > 1 we deduce from Lemma 6.9 that In particular, 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 > 1, we pick G β ∈ X β . We put for each γ > 1. By the Rank Trick for the natural surjection from a free group onto G β for β ≥ 1, we obtain another groupḠ β ≤ Diff β 0 (I ) whose abelianization is free For each β > 1, we can apply the Chain Group Trick toḠ 1 ×Ḡ β to obtain a minimally acting chain group (β) such that It follows that [ (β), (β)] / ∈ G γ (S 1 ) for all γ > 1. From the consideration of critical regularities, we note thatḠ β Ḡ γ whenever 1 ≤ β < γ . Note also thatḠ β ≤ [ (β), (β)] and that a countable group contains at most countably many finitely generated subgroups. So, there exists a continuum X * ⊆ (1, ∞) such that for all distinct β, γ in X * , we have Then X 1 = { (β) | β ∈ X * } is the desired continuum of the theorem. Finally, let us construct a continuum Y 1 . To be consistent with the notations in Sect. 6.3, let us set As we noted in Remark 1.2, we have that G 0 (M) = G Lip (M). 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 + (I ) has a trivial image. In particular, H 1 (T 2 , Z) is trivial and T 2 ∈ G 0 (I )\G 1 (I ). We continue as in Sect. 6.3, after substituting (k, μ) = (0, 0) and (k, ω) = (1, 0) (and forgetting k, bv). We obtain groups T 3 , T 4 , T 5 and a minimally acting chain group Q ≤ Homeo + (I ) such that Let us put H 1 := Q. The construction of Y 1 is very similar to that of X 1 . For each β > 1, we can find a finitely generated groupH β ≤ γ <β Diff γ 0 (I ) such that H 1 (H β , Z) is free abelian, and such thatH β / ∈ G β (S 1 ). For each β > 1, we apply the Chain Group Trick toH 1 ×H β and obtain a minimal chain group (β) such that As before, there exists a continuum Y * ⊆ (1, ∞) such that Y 1 = { (β) | β ∈ Y * } is the desired collection. Note that no two groups in the collection X 1 ∪ Y 1 have isomorphic commutator subgroups.
Remark 6.11 Calegari [15] exhibited a finitely generated group in G 0 (S 1 )\G 1 (S 1 ). Lodha and the authors [41] gave (continuum many distinct) finitely generated groups inside G 0 (I )\G 1 (I ) having simple commutator groups, building on [46]. The last part of the above proof strengthens both of these results. 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 ∈ {I, S 1 }. Let B be a closed manifold with a universal coverB → B. Suppose we have a representation

Algebraic and topological smoothability
The manifoldB × M has a natural product foliation so that each copy ofB is a leaf. The group π 1 (B) has a diagonal action onB × M, given by the deck transformation π 1 (B) → Homeo(B) and by the map ψ. The quotient space is a C α -foliated bundle. This construction is called the suspension of ψ; see [18] for instance. Two representations ψ, ψ ∈ Hom(π 1 (B), Diff α + (M)) yield homeomorphic suspensions E(ψ), E(ψ ) as foliated bundles if and only if ψ and ψ are topologically conjugate [17,Theorem2].
Let us now consider the case M = I and B = S g , a closed surface of genus g ≥ 2. Let k ≥ 0 be an integer. Cantwell-Conlon [21] and Tsuboi [69] independently proved the existence of a representation ψ k ∈ Hom(π 1 (S g ), Diff k + (I )) such that ψ k is not topologically conjugate to a representation in Hom(π 1 (S g ), Diff k+1 + (M)). So, they concluded: Theorem 6.12 (See [21] and [69]) For each integer k ≥ 0, there exists a C k -foliated bundle structure on S 2 × I which is not homeomorphic to a C k+1foliated bundle.
We will now prove Corollary 1.9, which is the only remaining result in the introduction that needs to be shown. Assume α ≥ 1 is a real number and g ≥ 5. Theorem 1.5 implies that there exists a representation ψ α ∈ Hom(π 1 (S g ), Diff α 0 (I )) 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 β>α 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 (Y, Z) = 0 contains an embedded 2-sided closed surface S g for all sufficiently large g > 0. Goodman used this observation to prove that Y \ Int(S g ×I ) admits a smooth foliation structure, based on Thurston's result; see [31, Corollary 3.1] and [67]. By adding in the aforementioned foliated bundle structure of S g × I inside Y , we complete the proof of Corollary 1.9.

Further questions
Let M ∈ {I, S 1 }. 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. 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: 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 (M).

Question 7.3 Let G be a finitely generated group. Does there exist an easily verifiable algebraic criterion which precludes G ∈ G β (M)?
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Appendix A. Diffeomorphism groups of intermediate regularities
Let M ∈ {I, S 1 }. We will record some basic properties of Diff k,ω + (M). 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 ∈ N, and let ω be a concave modulus. In [51], it is proved that for a smooth manifold X , the set Diff k,ω c (X ) 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 ∈ {0, bv}. Let k ∈ N, and let F, G : M → 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 [51] when ω = 0 or when ω is a concave modulus. So we assume ω = bv. We let {x i } be a partition of M.
(1) First consider the case k = 1. We note Hence, if F · G is C bv . If k > 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 ∈ M. Let us first assume k = 1, so that F ∈ C bv . Since G is bijective, we see that The induction step follows from Proposition A. 2 Let ω be a concave modulus, or let ω = {0, bv}. Then for each k ∈ N, the following is a group where the binary operation is the group composition: .
Proof Let f, g ∈ Diff k,ω + (M). It is well-known that Diff k + (M) is a group. So, we have f −1 , f • g ∈ Diff k + (M). It suffices to show that both are C k,ω . Note that ( f • g) = ( f • g)· g . Since f is C k−1,ω and g is C k , Lemma A.1 implies that f •g is C k−1,ω . By the same lemma, we see that ( f •g) is C k−1,ω . This proves f • g is C k,ω .
We can write where r : (0, ∞) → (0, ∞) is the C ∞ diffeomorphism r (x) = 1/x. Note that f stays away from 0. As f 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 ∈ N, the group Diff k,ω + (I ) is topologically conjugate to a subgroup of Diff k,ω c (R).
Muller [54] and Tsuboi [68] 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 ≥ 0, we implicitly assume to have a small number A > 0 so that f is defined as We let k and ω be as in Theorem A.3.
Then the following hold.
The rest of the proof for Theorem A.3 closely follows the argument in [68], as we summarize below. Let us fix a map that is defined near x = 0: φ(x) = e −1/x . Lemma A.5 For a C k,ω map g defined for x ≥ 0 satisfying g(0) = 0 and g (0) > 0, the following hold.
(1) The map h = g/x is a C k−1,ω map defined for x ≥ 0.
Proof (1) If T k g(x) 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/x − f /x = T k g/x is a polynomial.
It follows that G is C k−1,ω for x ≥ 0. Moreover, G is C k,ω for x > 0. We compute the following: From xh = g −h, we see that φ·(h •φ) is C k−1,ω and that lim x→0 G (x) = 1. We conclude that G exists for x ≥ 0 (even when k = 1), and is C k−1,ω . It follows that G is C k,ω .
It is a simple exercise on L'Hospital's Rule to see that − log y (log(− log y)) 2 log y log g − 1 = 0.
For all 0 ≤ i ≤ k, we have that By L'Hospital's Rule again, we have ( (g) − Id) (i) = 0 for all 0 ≤ i ≤ k.

A.3. Simplicity
Let us use the following terminology from [41]. Let X be a topological space, and let H ≤ Homeo(X Let X be a topological space. We say H ≤ Homeo(X ) has the fragmentation property for an open cover U of X , if each element h ∈ H can be written as 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 ≤ Homeo(X ). Assume the following. The following lemma is known for ω = 0 [60], detailed proofs of which can be found in [4,49]. The proof for a concave modulus ω is the same almost in verbatim.
Lemma A.8 Let k ∈ N, and let ω be a concave modulus. Then for a smooth manifold X without boundary, the group Diff k,ω c (X ) 0 has the fragmentation property for an arbitrary open cover of X .
From now on, we let X ∈ {S 1 , R}. We let C ω c (X, R) denote the set of real-valued compactly supported ω-continuous maps X → R. For each f ∈ C c (X, R) = C 0 c (X, R), we define the optimal modulus function of f as μ f (t) := sup{| f x − f y| : x, y ∈ X and |x − y| ≤ t}.
It is trivial that for all x, y ∈ X we have | f x − f y| ≤ μ f (|x − y|).
Lemma A.9 For X ∈ {S 1 , R} and for f ∈ C c (X, R), the following hold. Proof Part (1) is a consequence of the convexity of X and the uniform continuity of f . Part (2) is obvious when t ≤ s. If t > s, then part (2) follows from For part (3), we will use the idea described in [5, p.194]. Let F be the family of continuous, monotone increasing, concave functions h : [0, ∞) → [0, ∞) such that μ f (t) ≤ h(t) for all t ≥ 0. For instance, part (2) implies that the line h s (t) = (1 + t/s)μ f (s) belongs to F for each s > 0. Define