Global Rigidity of Solvable Group Actions on S1

GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1

LIZZIE BURSLEM AND AMIE WILKINSON

Introduction In this paper we find all solvable groups that act effectively on the circle as real- analytic diffeomorphisms, and we find all of their actions. Our starting point is the procedure of ramified lifting of group actions with a global fixed point, which we develop in Section 1. To summarize the discussion there, we say that a real analytic map π : S1 → S1 is a ramified covering map over p ∈ S1 if the restriction π to S1 \ π−1({p}) is a regular covering map onto its image S1 \{p}. To any subgroup G < Diff ω(S1) with the property that γ(p) = p, for all γ ∈ G, and any ramified covering map π over p we then associate a subgroup Gˆπ < Diff ω(S1), the π-ramified lift of G, whose elements are the real-analytic lifts under π of elements of G. Each such ˆπ ω 1 G is abstractly isomorphic to an H-extension of G+, where G+ = Diff +(S ) ∩ G, and H is a subgroup of a dihedral group determined by π. Applying this procedure to the affine group 1 1 Aff(R) = {x 7→ cx + d : RP → RP | c, d ∈ R, c 6= 0}, which fixes ∞ ∈ RP 1, we obtain countably infinitely many different abstract isomor- ω 1 phism classes of subgroups of Diff (S ), each a finite extension of the group Aff+(R), and each class containing countably many distinct conjugacy classes in Diff (S1). These groups have strong rigidity properties. In the first part of the paper, we start with an abstract group, the solvable Baumslag- Solitar group BS(1, n), which has a unique faithful representation, up to conjugacy, into Aff(R). In Theorem 0.1, we show that any faithful representation of BS(1, n) ω into Diff (S1) is a ramified lift of this Aff(R) - action. Hence, up to conjugacy in Diff ω(S1), the group BS(1, n) has countably many faithful representations. In The- orem 0.3, we show that each representation belonging to this countable set is locally rigid in Diff r(S1), where r depends nontrivially on the representation. Due to the nature of ramified lifting, the same representation that is Cr locally rigid also fails to be Cr−1 locally rigid (Proposition 0.5). In the second part of the paper, we start with a solvable subgroup G < Diff ω(S1) (or more generally, an eventually solvable subgroup, see 0.2 for the definition) and prove in Theorem 0.9 that if G is not virtually abelian, then it must be conjugate to a subgroup of a ramified lift of Aff(R). We thus give a complete classification of all solvable subgroups of Diff ω(S1), both algebraically and up to conjugacy in Diff ω(S1). 1 2 LIZZIE BURSLEM AND AMIE WILKINSON

As a corollary we obtain that any subgroup of Diff ω(S1) is either virtually solvable or contains no nontrivial normal solvable subgroups. The same conclusions hold for any eventually solvable subgroup of Diff ∞(S1) without infinitely flat elements. Before describing these results in greater detail, we fix notation. Notation and preliminary definitions: At times we will use three different analytic coordinatations of the circle S1. To denote an element of the additive group, R/Z, we will use u, for a real coordinate, e.g. for the real projective line RP 1, we will use x, and to represent a complex coordinate, e.g., for the multiplicative group {z ∈ C | |z| = 1}, we will use z. These coordinate systems are identified by: u ∈ R/Z 7→ z = exp(2πiu) ∈ {z ∈ C | |z| = 1} 1 − x2 + 2xi x ∈ P 1 7→ z = ∈ {z ∈ | |z| = 1} R 1 + x2 C 1 u ∈ R/Z 7→ x = tan(πu) ∈ RP

When we are not specifying a coordinate system, we will use p or q to denote an element of S1. We fix an orientation on S1 and use “<” to denote the counterclockwise cyclic ordering on S1. r If G is a group, then we denote by R (G) the set of all representations ρ0 : G → r 1 r Diff (S ), and we denote by R+(G) the set of all orientation-preserving representa- r r r 1 tions in R (G). Two representations ρ1, ρ2 ∈ R (G) are conjugate (in Diff (S )) if r 1 −1 there exists h ∈ Diff (S ) such that, for every γ ∈ G, hρ1(γ)h = ρ2(γ) (equiva- lently, if this holds on a generating set for G). We use the standard Cr topology on representations of a finitely-generated group into Diff r(S1), r ∈ {1,..., ∞, ω}. Let Γ be a finitely-generated group with generat- r ing set {γ1, . . . , γk}. If ρ0 ∈ R (Γ) is a representation, then any collection U1,..., Uk r 1 r of neighborhoods of ρ0(γ1), . . . , ρ0(γk), respectively, in Diff (S ) defines a C neigh- r borhood U of ρ0 in R (Γ) by: r U = {ρ ∈ R (Γ) | ρ(γi) ∈ Ui, i = 1, . . . k}. These neighborhoods generate a topology; a different initial choice of finite generating set will produce the same topology. r r 1 We say that ρ0 ∈ R (Γ) is (C ) locally rigid if there exists a C neighborhood U r r 1 of ρ0 in R (Γ) such that every ρ ∈ U is conjugate in Diff (S ) to ρ0. Finally, we say that Γ is globally rigid in Diff r(S1) if there exists a countable set of locally rigid representations in Rr(Γ) such that every faithful representation in Rr(Γ) is conjugate to an element of this set.

0.1. Rigidity of Baumslag-Solitar groups. Let BS(1, n) =< a, b | aba−1 = bn > . GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 3

We study here actions of BS(1, n) on the circle S1 by Cr diffeomorphisms, up to Cr conjugacy. We consider values of r ranging from 2 to ∞, plus r = ω, the analytic category. Below this range, the actions of BS(1, n) can be quite flexible; for instance, any representation ρ : BS(1, n) → Diff 1(S1) admits uncountably many deformations inside of Diff 1(S1) (Proposition 0.5). 1 In this work, we show that, as r increases, the set of rigid representations of BS(1, n) into Diffr(S1) grows incrementally larger until precisely the point r = ω, where all faithful representations of BS(1, n) into Diffω(S1) fall into countably many distinct real-analytic conjugacy classes. That is BS(1, n) is globally rigid in Diff ω(S1). Furthermore, up to finite index subgroups, every analytic representation of BS(1, n) that is not faithful must factor through a representation of Z, see Theorem 0.1 below. We remark that, by contrast, there are uncountably many topologically distinct ω faithful representations of BS(1, n) into Diff (R) (see [FF], Proposition 5.1). The proof of our results uses the existence of a global fixed point on S1 for a finite index subgroup of BS(1, n); such a point need not exist when BS(1, n) acts on R. Farb and Franks [FF] studied actions of Baumslag-Solitar groups on the line and circle. Among their results, they prove that if m > 1, the (nonsolvable) Baumslag-Solitar group: BS(m, n) =< a, b | abma−1 = bn >, has no faithful C2 actions on S1 if m does not divide n. They ask whether the actions of B(1, n) on the circle can be classified. This question inspired the the present paper. We now describe this collection of rigid representations in more detail. In real pro- 1 ω 1 jective coordinates on RP , the standard representation ρn of BS(1, n) into Diff +(S ) takes the generators a and b to the affine maps x 7→ nx, and x 7→ x + 1. Taking ramified lifts of this representation, a procedure which we describe in Sec- tion 1, gives a countable family V of representations of BS(1, n) into Diff ω(S1). Tak- ing orientation-preserving ramified lifts of ρn gives another countable family V+ ⊂ ω R+(BS(1, n)). Theorem 0.1. Let ρ : BS(1, n) → Diff ω(S1) be a representation. Then either: k (1) There exists k ∈ N>0 such that ρ(b ) = id, or (2) ρ is faithful and is conjugate in Diff ω(S1) to a unique element of V. ω 1 ω 1 Further, if ρ takes values in Diff +(S ), and (2) holds, then ρ is conjugate in Diff +(S ) to a unique element of V+ The conclusion of Theorem 0.1 does not hold when Cω is replaced by a lower differentiability class such as C∞, even when analytic conjugacy is replaced by topological

1In addition, it is not hard to see that any homeomorphism g : S1 → S1 with rotation number 0 is contained in a faithful representation of BS(1, n) into Homeo(S1). 4 LIZZIE BURSLEM AND AMIE WILKINSON

conjugacy in the statement. Nonetheless, as r increases, there is a sort of “quantum rigidity” phenomenon. Let ρ : BS(1, n) =< a, b | aba−1 = bn >→ Diﬀ 2(S1) be a representation, and let f = ρ(a). We make a preliminary observation: Lemma 0.2. If the rotation number of f is irrational, then gk = id, for some k ≤ n + 1, where g = ρ(b). (See the beginning of Section 5 for a proof). Hence, if ρ ∈ R2(BS(1, n)) is faithful, then f must have periodic points. For ρ ∈ R2(BS(1, n)) a faithful representation, we deﬁne the inner spectral radius σ(ρ) by:

k 0 1 k k 0 σ(ρ) = sup{|(f ) (p)| k | p ∈ Fix(f ) and |(f ) (p)| ≤ 1}. 1 For the standard representation, σ(ρn) = n , and in general, if ρ ∈ V, then σ(ρ) = 1 1 s n , for some s ∈ N≥1. Theorem 0.3. Let ρ : BS(1, n) → Diff r(S1) be a faithful representation, where 1 1 r−1 r ∈ [2, ∞]. If either r < ∞ and σ(ρ) ≤ n , or r = ∞ and σ(ρ) < 1, then ρ r 1 r 1 is conjugate in Diff (S ) to a unique element of V. If ρ takes values in Diff +(S ), r 1 then ρ is conjugate in Diff +(S ) to a unique element of V+. Theorem 0.3 has the following corollary:

1 ∞ 1 r−1 Corollary 0.4. Every ρ ∈ V is C locally rigid. Further, if σ(ρ) < n , then ρ is Cr locally rigid. This corollary implies that the standard representation is Cr locally rigid, for all r ≥ 3, and every representation in V is locally rigid in some finite differentiability classes. This local rigidity breaks down, however, if the differentiability class is lowered.

1 1 r−1 Proposition 0.5. For every ρ ∈ V, if σ(ρ) = n , for some r ≥ 2 then there r exists a family of representations ρt ∈ R (BS(1, n)), t ∈ (−1, 1), with the following properties:

(1) ρ0 = ρ, r−1 r (2) t 7→ ρt is continuous in the C topology on R (BS(1, n)), 1 1 (3) for every t1, t2 ∈ (−1, 1), if ρt1 is conjugate to ρt2 in Diff (S ) then t1 = t2. It follows from our characterization of V in the next section that, for each value of 1 1 r r ∈ [1, ∞), there are infinitely many elements ρ ∈ V satisfying σ(ρ) = n . Hence, for each r ≥ 3 there are infinitely many representations in V that are Cr locally rigid, but not Cr−1 locally rigid. A. Navas has given a complete classification of C2 solvable group actions, up to topological semiconjugacy. As in [FF], a main tool in this classification is Kopell’s lemma (Lemma 3.6), which classifies the centralizer of a C2 contraction on [0, ∞). GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 5

One corollary of his result is that every faithful C2 representation ρ of BS(1, n) into Diff 2(S1) is is topologically semiconjugate to the standard representation: Theorem 0.6. [Nav1] Let ρ : BS(1, n) → Diff r(S1) be a representation, where r ≥ 2. Then either ρ is unfaithful, in which case ρ(b)m = id, for some m, or there exists an integer m ≥ 1, a finite collection of closed, connected sets I1,...,Ik, and a surjective continuous map ϕ : S1 → RP 1 with the following properties: m (1) ρ(b) is the identity on each set Ik. (2) ϕ sends each set Ik to ∞; 1 Sk r (3) the restriction of ϕ to S \ i=1 Ii is a C covering map of R; (4) For every γ ∈ BS(1, n), the following diagram commutes: ρ(γ) S1 - S1 ϕ ϕ ? ? ρn(γ) S1 - S1 ω 1 where ρn : BS(1, n) → Diff (S ) is the standard representation. The map ϕ in Theorem 0.6 is a sort of “broken Cr ramified cover.” The regularity of ϕ at the preimages of the point ∞ can be poor, and the map can be infinite- to-one on the sets I1,...,Ik, but a map with these features is nothing more than a deformation of a ramified covering map. Combining Theorem 0.6 with Theorem 0.3 and the proof Proposition 0.5, we obtain: Corollary 0.7. Let ρ : BS(1, n) → Diff r(S1), be any representation, with r ≥ 2. Then either: (1) ρ is not faithful, and there exists an m ≥ 1 such that ρ(b)m = id; (2) ρ admits Cr−1 deformations as in Proposition 0.5; or (3) ρ is Cr conjugate to an element of V. Since the statement of Theorem 0.6 does not appear explicitly in Navas’s paper, and we don’t use this result elsewhere in the paper, we sketch the proof at the end of Section 5. Finally, note that the trivial representation ρ0(a) = ρ0(b) = id is not rigid in any topology; it can be approximated by the representation ρ(b) = id, ρ(a) = f, where f is an any diffeomorphism close to the identity. Another nice consequence of Navas’s theorem is that this is the only way to C2 deform the trivial representation. Corollary 0.8. There is a C2 neighborhood U ⊂ R2(BS(1, n)) of the trivial representation such that, for all ρ ∈ U, ρ(b) = id.

Proof: Let ρ be a C2 representation. Since ρ(b) is conjugate by ρ(a) to ρ(b)n, it k will have rotation number of the form n−1 if ρ(a) is orientation-preserving, and of 6 LIZZIE BURSLEM AND AMIE WILKINSON

k 0 the form n+1 if ρ(a) is orientation-reversing. Therefore, if ρ is sufficiently C -close to m ρ0 and if ρ(b) = id, for some m ≥ 1, then m = 1. So we may assume that there 1 S exists a map ϕ as in Theorem 0.6 and that m = 1. On a component of S \ Ii, ϕ is a diffeomorphism conjugating the action of ρ to the restriction of the standard representation ρn to R (in general ϕ fails to extend to a diffeomorphism at either endpoint of R). But in the standard action, the element ρn(a) has a fixed point in R 1 of derivative n. If ρ is sufficiently C close to ρ0, this can’t happen.

0.2. General solvable groups. Several works address the properties of solvable subgroups of Diff r(S1); we mention a few here. Building on work of Kopell [Ko], Plante and Thurston [PT] showed that any nilpotent subgroup of Diff 2(S1) is in fact abelian. Ghys [Gh] proved that any solvable subgroup of Diff ω(S1) is metabelian, i.e., two-step solvable. In the same work, he remarks that there are solvable subgroups of Diff ∞(S1) that are not metabelian. The subgroups he constructs contain infinitely flat elements – nontrivial diffeomorphisms g ∈ Diff ∞(S1) with the property that g(p) = p, for some p ∈ S1, and:

g0(p) = 1; g(k)(p) = 0, ∀k > 1. Navas [Nav1] constructed further examples of solvable subgroups of Diff ∞(S1) with arbitrary degree of solvability, again using infinitely flat elements. Our main result in this part of the paper, Theorem 0.9, implies that any solvable subgroup of Diff ∞(S1) that does not contain infinitely flat elements is either virtually abelian or conjugate to a ramified lift of the affine group Aff(R). We describe ramified ω lifting in the next section. Each ramified lift of Aff(R) is a subgroup of Diff (S1), isomorphic (as a group) to a product Aff+(R)×G, where G is a subgroup of the dihedral group Dk, for some k ∈ N. For each such G, there are countably infinitely many ramified lifts, no two of which are conjugate in Diff ω(S1) (or indeed, in Diff 1(S1)). As mentioned in the previous section, Navas’s work also contains a topological classification of solvable subgroups of Diff 2(S1). As part of a study of ergodicity of S1-actions, Rebelo and Silva [RS] also study the solvable subgroups of Diff ω(S1). We say that a group G is eventually solvable if there exists a sequence of subgroups:

G = G0 B G1 B G2 B ··· B Gk 6= {1},

(with Gi+1 normal in Gi), such that Gk is solvable. We remark that G is eventually solvable if and only if there is such a subnormal series with Gk 6= {1} abelian.

Examples: Solvable groups are of course eventually solvable; other basic examples of eventually solvable groups can be constructed using central extensions and semidirect products. Here are a few examples. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 7

(1) If S is a surface of negative Euler characteristic, and T 1S is its unit tangent 1 bundle, then π1(T S) is not solvable, but it is eventually solvable, since it has a normal Z subgroup: 1 1 → Z → π1(T S) → π1(S) → 1. 1 ˜ (2) Let H < Homeo+(S ) be any subgroup, and let H < Homeo(R) be the set of all lifts of H to the universal cover R of S1. Then H˜ is eventually solvable. 2 (3) Let G = Z ×α SL(2, Z) (the semidirect product), where α is the linear action of SL(2, Z) on Z2. This group is not solvable but contains a normal Z2 subgroup, and so is eventually solvable. (4) Let Fk be a free group on k generators and let ρ : Fk → SL(n, Z) be a faithful n n representation. Then Z ×ρ Fk is eventually solvable. Note that both Z and 1 Fk act faithfully on S by real-analytic diffeomorphisms. In [FS] it is shown that the centralizer of any nontrivial diffeomorphism g ∈ Diff ω(S1) is virtually abelian. From this it follows that the groups in (1) and (2) cannot act faithfully as analytic diffeomorphisms of S1. The groups in (3) do not act faithfully as C2, or even C1+α, diffeomorphisms of S1 (the C2 case was remarked by Tsuboi, see [Ca], and the C1+α was proved by Navas [Nav2]). As far as we know, there was no previously known restriction on the actions of the groups in (4) on the circle. In Section 1, we construct a collection: s ω 1 RAFF := {Affc (R) < Diff (S ) | s ∈ S}, where S is a countably infinite index set, with the following properties: s s 1 2 1 1 (1) if s1, s2 ∈ S and Affc (R) is conjugate to Affc (R) in Diff (S ), then s1 = s2; (2) for each s ∈ S, there exists a subgroup H of a dihedral group such that s Affc (R) ' Aff+(R) × H, (3) for each finite dihedral or cyclic group H, there exist infinitely many s ∈ S so s that Affc (R) ' Aff+(R) × H, We also construct a collection s ω 1 RAFF + := {Affc +(R) < Diff +(S ) | s ∈ S+}, 1 1 with the same properties, except that in (1), the conjugacy is taken in Diff +(S ), and in (2) and (3), H is cyclic. Theorem 0.9. Let G < Diff r(S1) be eventually solvable, where r ∈ {∞, ω}. Then either: (1) G is virtually abelian, (2) G contains infinitely flat elements (which can’t happen if r = ω), or (3) G is conjugate in Diff r(S1) to a subgroup of an element of RAFF. r 1 r 1 Further, if G < Diff +(S ) and (3) holds, then G is conjugate in Diff +(S ) to a subgroup of an element of RAFF +. 8 LIZZIE BURSLEM AND AMIE WILKINSON

As a corollary to Theorem 0.9, we obtain that any nonsolvable eventually solvable group, such as those presented in (1) -(4) above, cannot act faithfully as a group of analytic diffeomorphisms of S1. Corollary 0.10. Let G < Diff ω(S1) be any subgroup. Then either: (1) G is virtually solvable, or (2) G has no normal solvable subgroups. We end this section with a remark and a conjecture. The proof of Theorem 0.9 uses Takens’ result [Ta] that every C∞ diffeomorphism of [0, 1) without fixed points in its interior is either infinitely flat or embeds in a C∞ flow. If it is true that every Cr diffeomorphism of [0, 1) without fixed points in its interior is either r−1-flat (meaning tangent to the identity to order r − 1 at its fixed point) or embeds in a Cr flow, then modification of the proof of Theorem 0.9 shows that the following conjecture is true:

Conjecture: Let G < Diff r(S1) be eventually solvable, with r ≥ 1. Then either: (1) G is virtually abelian, (2) G contains r-flat elements, or (3) G is conjugate in Diff r(S1) to a subgroup of an element of RAFF.

1. Introduction to ramified lifts and the definitions of V, V+, RAFF, and RAFF + Let G be a group and let ρ : G → Diff ω(S1) be a representation with a global fixed point p. Restricting each element of this representation to a suitably small neighborhood of p, we obtain a representationρ ˜ : G → Gω, where Gω is the group of analytic germs of diffeomorphisms. A theorem of Nakai [Nak] implies that if G is solvable, then for some k ≥ 1,ρ ˜ is conjugate in Gω to a representation taking values k in the ramified affine group Aff (R): k x Aff (R) = { 1 | a, b ∈ R, b > 0} (axk + b) k (see [Gh] for a proof in the context of circle diffeomorphisms). The name “ramified k affine group” is explained by the fact that the elements of Aff (R) are lifts of the elements of the affine group under the branched (or ramified) cover z 7→ zk. More precisely, the affine group, regarded as a subgroup of the conformal group PSL(2, C), fixes the point ∞ ∈ CP 1. In projective coordinates about infinity, each z −1 element takes the form z 7→ az+b . In the disk |z| < ba , this element can be lifted k 1 under z 7→ z to an analytic map; fixing a branch of z 7→ z k , this lift takes the form: z z 7→ 1 . (azk + b) k Choosing a branch that preserves the real locus in |z| < ba−1, we obtain the elements k of the real ramified affine group Aff (R). Note that no nontrivial element of the GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 9 affine group admits a global lift under z 7→ zk to an analytic diffeomorphism of the Riemann sphere; this is consistent with the fact that there is only one conjugacy class 2 of representations of the affine group into PSL(2, C). Despite the fact that the standard representation of Aff(R) as a group of conformal diffeomorphisms of CP 1 does not admit a global ramified lift, it is still possible to create new actions on RP 1 from the standard action. The method is to consider rational maps with real coefficients that act as “ramified covering maps” on RP 1. In contrast to a ramified cover of CP 1, which must have at least 2 critical values, a ramified cover of RP 1 can have exactly one critical value, which can be chosen to coincide with the global fixed point of Aff(R). Any subgroup G of Aff(R) will have at least one lift (possibly several) under such a ramified cover to a group of real analytic diffeomorphisms of RP 1. Each such ramified covering map is specified by a vector in Nd × {±1}d, for some d, called its signature, and, up to real analytic conjugacy, there are finitely many representations of G for each signature (Proposition 1.2 and Lemma 1.3 below).

Deﬁnition: A real analytic surjection π : S1 → S1 is called a ramiﬁed covering map over p ∈ S1 if the restriction of π to π−1(S1 \{p}) is a regular analytic covering map onto S1 \{p} of degree d ≥ 1.

Examples: The map π1 : R/Z → R/Z given by 2 π1(u) = sin (πu) −1 1 is a ramified covering map over 0, with critical points of order 2 at π1 (0) = {0, 2 }. 1 1 The rational map π2 : RP → RP given by: (x + 1)2(x − 1)2 π (x) = 2 x(x2 + 1) is also a ramified covering map over 0, with critical points of order 2 at ±1. It is clear that ±1 are critical points of π , and one verifies directly that the other critical 2 √ 1 1 p points of π2 in CP occur off of RP , at ±i 3 ± 8. We will define an equivalence relation on ramified covering maps in which π1 and π2 are equivalent. In Section 2 we show that, under this notion of equivalence, all possible ramified covering maps occur as rational maps. If π : S1 → S1 is a ramified covering map over p then for each q ∈ π−1(p), there exists an integer s(q) ≥ 1 such that the leading (nonconstant) term in Taylor expansion of π at q is of order s(q). A regular covering map is a ramified covering map; in this case, d is the topological degree of the map, and s(q) = 1, for each q ∈ π−1(p). As the examples show, a ramified covering map need not be a regular covering map (even topologically), as it is possible to have s(q) > 1.

2However, though we do not discuss them here, there are other representations of the affine group as real-analytic diffeomorphisms of the sphere that do admit ramified lifts. 10 LIZZIE BURSLEM AND AMIE WILKINSON

Let π be a ramiﬁed covering map over p, and let q1, . . . , qd be the elements of −1 π (p), ordered so that p ≤ q1 < q2 < ··· < qd < p. For each i ∈ {1, . . . , d} we deﬁne oi ∈ {±1} by: ( 1 if π|(qi,qi+1) is orientation-preserving, oi = −1 if π|(qi,qi+1) is orientation-reversing.

d d We call the vector s(π) = (s(q1), s(q2), . . . , s(qd), o1, . . . od) ∈ N × {±1} the signature of π. Geometrically, we think of a signature as a regular d-gon in R2 with vertices labelled by s1, . . . , sd and edges labelled by o1, . . . , od. Every signature vector s = (s1, . . . , sd, o1, . . . , od) has the following two properties: (1) The number of vertices with an even label is always even:

#{1 ≤ i ≤ d | si ∈ 2N} ∈ 2N. (2) If a vertex has an odd label, then both edges connected to that vertex have the same label, and if a vertex has an even label, then the edges have opposite labels:

si+1 (−1) = oi−1 oi, where addition is mod d. We will call any vector s ∈ Nd × {±1}d with these properties a signature vector. Note that a signature vector of length 2d is determined by its ﬁrst d + 1 entries. Let Sd be the set of all signature vectors with length 2d, and let S be the set of all signature vectors. We prove the following proposition in Section 2.

Proposition 1.1. Given any s ∈ S and p ∈ S1, there is a ramiﬁed covering map π : S1 → S1 over p with signature s.

If π is a ramiﬁed covering map, then the cyclic and dihedral groups:

d Cd = hb : b = idi, and d 2 −1 −1 Dd = ha, b : b = id, a = id, aba = b i, respectively, act on π−1(p) and on the set E(π) of oriented components of S1 \ π−1(p) in a natural way. By an orientation-preserving homeomorphism, we identify the circle with a regular oriented d-gon, sending the elements of π−1(p) to the vertices and the elements of E(π) to the edges. The groups Cd C Dd act by symmetries of the d- gon, inducing actions on π−1(p) and E(π) that are clearly independent of choice of −1 homeomorphism. For q ∈ π (p), e ∈ E(π), and ζ ∈ Dd, we write ζ(q) and ζ(e) for their images under this action. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 11

These symmetry groups also act on the signature vectors in Sd in the natural way, permuting both vertex labels and edge labels. For ζ ∈ Dd, we will write ζ(s) for the image of s ∈ Sd under this action. In this notation, the action is generated by:

b(s1, . . . , sd, o1, . . . , od) = (s2, s3, . . . , sd, s1, o2, o3, . . . , od, o1), and

a(s1, . . . , sd, o1, . . . , od) = (s1, sd, sd−1 . . . , s3, s2, −od, −od−1,..., −o2, −o1).

Denote by StabCd (s) and StabDd (s) the stabilizer of s in Cd and Dd, respectively, under this action:

StabH (s) = {ζ ∈ G | ζ(s) = s},

for H = Cd or Dd.

2 Examples: The signature vector of π1(u) = sin (πu) is s1 = (2, 2, 1, −1). The

stabilizer of s1 in Dd is StabDd (s1) =< a >, and the stabilizer of s1 in Cd is trivial. 2 2 2 The signature vector of π2(x) = ((x − 1) (x + 1) )/(x(x + 1)) is s2 = (2, 2, −1, 1).

Note that s2 lies in the Cd-orbit of s1, and so StabCd (s2) and StabDd (s2) must be

conjugate to StabCd (s1) and StabDd (s1), respectively, by an element of Cd. In this simple case, the stabilizers are equal. For another example, consider the signature vector (2, 3, 1, 2, 3, 1, −1, −1, −1, 1, 1, 1), which geometrically is represented by the following labelled graph: 2 + Z − = ZZ~ 1 3 + − ? ? 3 1 Z + ZZ~ = − 2 This labelling has no symmetries, despite the fact that the edge labels have a ﬂip symmetry and the vertex labels have a rotational symmetry. By contrast, the signature/labelling: 2 − > Z}Z + Z 1 4 − 6 − ? 4 1 Z + ZZ~ = − 2 12 LIZZIE BURSLEM AND AMIE WILKINSON

3 has a 180 degree rotational symmetry corresponding to the element b ∈ C6, and so both stabilizer subgroups are < b3 >.

1.1. Definitions of V, and V+. To define V, and V+, we will take ramified lifts of ω 1 the standard representation ρn : BS(1, n) → Diff +(S ). The next proposition gives the key tool for lifting representations under ramified covering maps.

ω 1 Proposition 1.2. Let G be a group, and let ρ : G → Diff +(S ) be a representation with global fixed point p. Let π : S1 → S1 be a ramified covering map over p with signature s ∈ Sd, for some d ≥ 1.

Then for every homomorphism h : G → StabDd (s), there is a unique representation ρˆ =ρ ˆ(π, h): G → Diﬀ ω(S1) such that, for all γ ∈ G, (1)ρ ˆ(γ)(q) = h(γ)(q), for each q ∈ π−1(p); (2)ρ ˆ(γ)(e) = h(γ)(e), for each oriented component e ∈ E(π); (3) the following diagram commutes:

ρˆ(γ) S1 - S1 π π ? ? ρ(γ) S1 - S1 ω 1 Furthermore, if h takes values in StabCd (s), then ρˆ takes values in Diﬀ +(S ).

Proposition 1.2 is a special case of Proposition 4.5, which is proved in Section 4. We callρ ˆ a ramified lift of ρ. Some form of ramified lifting has been used before in dynamical systems, although we have not found a systematic treatment of it in the literature. For example, the ramified cover u 7→ sin2(πu) mentioned above may also be regarded as a ramified cover over the interval [0, 1]. It was observed by Kolmogorov that, under this covering map, the quadratic map x 7→ 4x(1 − x) lifts to the doubling map u 7→ 2u on R/Z. We thank Michal Misiurewicz for pointing this out to us. Note that the representation ρ in Proposition 1.2 must be orientation preserving, although the liftρ ˆ might not be, depending on where the image of h lies. There is also a criterion for lifting representations into Diff ω(S1) that are not necessarily orientation-preserving. Since we don’t need such lifts to define V and V+, we postpone discussion of this issue to the next subsection.

Lemma 1.3. Suppose that π1 and π2 are two ramiﬁed covering maps over p such that s(π2) lies in the Dd-orbit of s(π2); that is, suppose there exists ζ ∈ Dd such that:

s(π2) = ζ(s(π1)). GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 13

ω 1 Then given any representation ρ : G → Diff +(S ) with global fixed point p and any −1 homomorphism h : G → StabDd (s1), the representations ρˆ(π1, h) and ρˆ(π2, ζhζ ) are conjugate in Diff ω(S1), where (ζhζ−1)(γ) := ζh(γ)ζ−1.

−1 Furthermore, if ζ ∈ Cd and h takes values in StabCd (s), then ρˆ(π1, h) and ρˆ(π2, ζhζ ) ω 1 are conjugate in Diﬀ +(S ).

Lemma 1.3 follows from Lemma 4.6, which is proved in Section 4. Lemma 1.3 allows us to deﬁne V and V+. Note that the elements of Sd are totally ordered by the n lexicographical order on R . Hence we can write Sd as a disjoint union of Cd-orbits: G Sd = Cd(sα),

α∈A+

where for each α ∈ A+, sα is the smallest element in its Cd-orbit. Similarly, there is an index set A ⊃ A+ such that: G Sd = Dd(sα). α∈A + S Let Sd = {sα | α ∈ A}, and let Sd = {sα | α ∈ A+}. Finally, let S = d Sd and let S + S = d Sd .

ω 1 Definition: Let ρn : BS(1, n) → Diff +(S ) denote the standard projective action, with global fixed point at ∞ ∈ RP 1. Then we define:

V = {ρˆn(πs, h) | s ∈ Sd, h ∈ Hom (BS(1, n), StabDd (s)), d ∈ N, d ≥ 1}, and let + V+ = {ρˆn(πs, h) | s ∈ Sd , h ∈ Hom (BS(1, n), StabCd (s)), d ∈ N, d ≥ 1}, 1 1 where, for s ∈ Sd, πs : S → S is the ramiﬁed cover over ∞ with signature s given by Proposition 1.1.

Proposition 1.4. Each element of V and V+ represents a distinct conjugacy class of faithful representation. 1 1 That is, if ρˆn(πs1 , h1), ρˆn(πs2 , h2) ∈ V (resp. ∈ V+) are conjugate in Diﬀ (S ) 1 1 (resp. in Diﬀ +(S )), then s1 = s2 and h1 = h2.

Proposition 1.4 is proved at the end of Section 4. Our main result, Theorem 0.1, states that the elements of V and V+ are the only faithful representations of BS(1, n), ω 1 ω 1 up to conjugacy in Diﬀ (S ) and Diﬀ +(S ), respectively. 14 LIZZIE BURSLEM AND AMIE WILKINSON

1.2. Definitions of RAFF, and RAFF +. We next define the collections of rami- s fied affine groups RAFF and RAFF +. In basic terms, each Affc (R) ∈ RAFF is the set of all possible ramified lifts of elements of Aff(R) under some fixed ramified cover s πs of signature s, and each Affc +(R) ∈ RAFF + is the set of orientation-preserving s elements of Affc (R). To describe these groups this more explicitly, we need to define ramified lifts of orientation-reversing diffeomorphisms. In the end, our description is complicated by the following fact. In contrast to lifts by regular covering maps, ramified lifts of orientation-preserving diffeomorphisms can be orientation-reversing, and vice versa. s Thus to define Affc +(R) it is not sufficient to take ramified lifts of elements of Aff+(R) alone. There is another action of the dihedral group 2 d −1 −1 Dd =< a, b, | a = 1, b = 1, aba = b > on Sd that ignores the edge labels completely. To distinguish from the action of Dd # on Sd already defined, we will write ζ : Sd → Sd for the action of an element ζ ∈ Dd. In this notation, the action is generated by: # b (s1, . . . , sd, o1, . . . , od) = (s2, s3, . . . , sd, s1, o1, o2, . . . , od), and # a (s1, . . . , sd, o1, . . . , od) = (s1, sd, sd−1 . . . , s3, s2, o1, o2, . . . , od). For s ∈ S , we denote by Stab# (s) and Stab# (s) the stabilizers of s in D and C , d Dd Cd d d respectively under this action.

Lemma 1.5. For each s ∈ Sd, there exists a homomorphism ∆ : Stab# (s) → /2 . s Dd Z Z

so that StabDd (s) = ker(∆s).

Proof: Let s ∈ S be given. Clearly Stab (s) is a subgroup of Stab# (s). Let d Dd Dd I : Sd → Sd be the involution:

I(s1, . . . , sd, o1, . . . , od) = (s1, . . . , sd, −o1,..., −od). We show that for every ζ ∈ Stab# (s), either ζ(s) = s (so that ζ ∈ Stab (s)) or Dd Dd ζ(s) = I(s). This follows from the property (2) in the definition of signature vector, which implies that every element of Sd is determined by its first d + 1 entries. Hence we may define ∆s(ζ) to be 0 if ζ(s) = s and 1 otherwise. Since I is an involution, ∆s is a homomorphism.

Example: Consider the signature s = (2, 1, 2, 1, 2, 1, 2, 1, 1, 1, −1, −1, 1, 1, −1, −1) corresponding to the following labelling: GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 15 − 2∗ - 1 + @ − @ © @@R 1 2 6 + + ? 2 1 @I@ −@ + @ − 21

(The ∗ indicates the ﬁrst element in the signature vector, which is ﬁxed by a ∈ D8). For this example we have Stab (s) =< b4 >, Stab (s) =< a, b4 >, Stab# (s) =< C8 D8 C8 b2 >, and Stab# (s) =< a, b2 >. In this example the homomorphism ∆ is surjective, D8 s with nontrivial kernel. For s = (2, 1, 4, 1, 2, 1, 4, 1, 1, 1, −1, −1, 1, 1, −1, −1), on the other hand, the image of ∆ is trivial, and Stab (s) = Stab# (s) =< a, b4 >, s D8 D8 Stab (s) = Stab# (s) =< b4 >. C8 C8

For a third example, recall that the stabilizer StabD6 (s) of the signature vector s = (2, 3, 1, 2, 3, 1, −1, −1, −1, 1, 1, 1) is trivial. Because of the rotational symmetry of the vertex labels, however, Stab# (s) = Stab# (s) =< b3 >' /2 . In this C6 D6 Z Z example, ∆s is an isomorphism. Let G < Diﬀ ω(S1) be a subgroup with global ﬁxed point p ∈ S1: f(p) = p, ∀f ∈ G.

We now show how to assign, to each s ∈ S, a subgroup Gˆs consisting of ramified lifts ω 1 of elements of G. We first write G = G+ t G−, where G+ = G ∩ Diff +(S ) is the kernel of the homomorphism O : G → Z/2Z given by: ( 0 if f is orientation-preserving, O(f) = 1 otherwise.

Suppose that π : S1 → S1 is a ramiﬁed covering map over p. Then, for every

f ∈ G+, Proposition 1.2 implies that for every ζ ∈ StabDd (s(π)), there is a unique lift fˆ(π, ζ) ∈ Diﬀ ω(S1) satisfying: (1) fˆ(π, ζ)(q) = ζ(q), for all q ∈ π−1(p), (2) fˆ(π, ζ)(e) = ζ(e), for all e ∈ E(p), and (3) the following diagram commutes: 16 LIZZIE BURSLEM AND AMIE WILKINSON

fˆ(π, ζ) S1 - S1 π π ? ? f S1 - S1

(Further, this lift is orientation-preserving if ζ ∈ StabCd (s).) Suppose, on the other hand, that f ∈ G−. In Section 4, we prove Corollary 4.3, which implies that if ζ ∈ Stab# (s) satisfies: Dd (1) ζ(s) = I(s), then there exists a unique lift fˆ(π, ζ) ∈ Diff ω(S1) satisfying (1)-(3) (and, further, ˆ ω 1 f(π, ζ) ∈ Diff +(S ) if ζ ∈ StabCd (s).) We can rephrase condition (1) as:

∆s(ζ) = 1. To summarize this discussion, we have proved the following: Lemma 1.6. If f ∈ G, and ζ ∈ Stab# (s), then there exists a lift fˆ(π, ζ) satisfying Dd (1)-(3) if and only if: O(f) = ∆s(ζ).

For s ∈ S, let πs be the ramified covering map over p with signature s given by Proposition 1.1. If G < Diff ω(S1) has global fixed point p, we define Gˆs ⊂ Diff ω(S1) to be the fibered product of G and Stab# (s) with respect to O and ∆ : Dd s Gˆs := {fˆ(π , ζ) | (f, ζ) ∈ G × Stab# (s),O(f) = ∆ (ζ)}. s Dd s Similarly, we define: Gˆs := {fˆ(π , ζ) | (f, ζ) ∈ G × Stab# (s),O(f) = ∆ (ζ)}. + s Cd s ˆs ˆs ω 1 ω 1 It follows from Lemma 4.4 that G and G+ are subgroups of Diff (S ) and Diff +(S ), respectively, with: ˆ ˆ f1(πs, ζ1) ◦ f2(πs, ζ2) = f\1 ◦ f2(πs, ζ1ζ2). Further, we have: ˆs ˆs Proposition 1.7. Assume that G− is nonempty. Then G and G+ are both finite extensions of G+; there exist exact sequences: (2) 1 → G → Gˆs → Stab# (s) → 1, + Dd and (3) 1 → G → Gˆs → Stab# (s) → 1. + + Cd Furthermore, if the sequence O 1 → G+ → G → O(G) → 1 GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 17 splits, where O : G → Z/2Z is the orientation homomorphism, then the sequences: (4) 1 → G → Gˆs → Stab# (s) → 1, + Dd

(5) 1 → G → Gˆs → Stab# (s) → 1, + + Cd and ˆs ˆs ˆs 1 → G+ → G → O(G ) → 1(6) are also split.

Proof: The maps in the first sequence (2) are given by: ˆs ˆ ι : G+ → G f 7→ f(π, id) σ : Gˆs → Stab# (s) fˆ(π, ζ) 7→ ζ. Dd It is easy to see that ι is injective and σ is surjective. Moreover, fˆ(π, ζ) is in the kernel of σ if and only if ζ = id, if and only if f is orientation-preserving, if and only if fˆ(π, id) is in the image of ι. Hence the first sequence is exact. Similarly, the second sequence (3) is exact. Now suppose that 1 → G+ → G → O(G) → 1 is split exact. If O(G) is trivial then G+ = G, and there is nothing to prove. If O(G) = Z/2Z, then G contains an 2 involution g ∈ G− with g = id, namely the image of 1 under the homomorphism O(G) → G. We use g to define a homomorphism j : Stab# (s) → Gˆs as follows: Dd ( idˆ (π , ζ) if ∆ (ζ) = 0 j(ζ) = s s . gˆ(πs, ζ) if ∆s(ζ) = 1 Hence the sequence (4) is split. The restriction of j to Stab# (s) splits the sequence Cd (5). If O(Gˆs) is trivial, then the last sequence (6) is trivially split. If O(Gˆs) = Z/2Z, then there exists a ζ ∈ Stab# (s) such that ∆ (ζ) = 1. We then define k : O(Gˆs) → Gs Dd s by k(0) = id, k(1) =g ˆ(π, ζ), which implies that (6) is split.

Setting G = Aff(R), which has the global fixed point ∞ ∈ RP 1, we thereby define s s Affc (R) and Affc +(R), for s ∈ S. Since Aff(R) contains the involution x 7→ −x, the sequence O 1 → Aff+(R) → Aff(R) → Z/2Z → 1 splits. Proposition 1.7 implies that s s Aff ( ) ' Aff ( ) × Stab# (s), Aff ( ) ' Aff ( ) × Stab# (s), c R + R Dd c + R + R Cd s s s s and either Affc (R) = Affc +(R) or Affc (R) ' Affc +(R) × Z/2Z, depending on whether ∆s is surjective. 18 LIZZIE BURSLEM AND AMIE WILKINSON

s1 s2 Finally, Corollary 4.8 implies that Affc (R) and Affc (R) are conjugate subgroups in s s 1 1 1 2 Diff (S ), only if s1 ∈ Dds2. Similarly, Affc + (R) and Affc + (R) are conjugate subgroups 1 1 in Diff +(S ), if and only if s1 ∈ Cds2. Let S and S+ be the sets of signatures defined at the end of the previous subsection. The elements of S and S+ are representatives of distinct orbits in S under the dihedral and cyclic actions, respectively. We now define: s s RAFF := {Affc (R) | s ∈ S}, and RAFF + := {Affc +(R) | s ∈ S+}.

To summarize the preceding discussion, we have shown that RAFF and RAFF + have the properties described in the end of Section 0.2: s s 1 2 1 1 (1) if s1, s2 ∈ S and Affc (R) is conjugate to Affc (R) in Diff (S ), then s1 = s2; (2) for each s ∈ S, there exists a subgroup H of a dihedral group such that s Affc (R) ' Aff+(R) × H, (3) for each finite dihedral or cyclic group H, there exist infinitely many s ∈ S so s that Affc (R) ' Aff+(R) × H, and the elements of RAFF + have the same properties, with the additional features 1 1 that the conjugacy is taken in Diff +(S ) in (1), and H in (2) and (3) are cyclic. 2. Existence of ramified covers: proof of Proposition 1.1 In this section, we prove the following proposition: Proposition 2.1. For every signature vector s ∈ S, there exists a rational function π : RP 1 → RP 1 that is a ramified cover over 0 with signature s

Note that, to obtain from Proposition 2.1 a rational functionπ ˜ : RP 1 → RP 1 that is a ramiﬁed cover with signature s, but over a diﬀerent point x0 6= 0, one simply 1 translates the function π by x0, if x0 6= ∞, or composes with x 7→ − x , if x0 = ∞. Hence Proposition 1.1 is easily reduced to Proposition 2.1.

Proof of Proposition 2.1. We ﬁrst construct, in Lemma 2.2, a rational function π0 that has all the desired properties,n except that π0 has one extra critical point, ∞, with critical value ∞. In Lemma 2.4, we remove this extra critical point, dividing π0 by a suitably chosen polynomial rε.

Lemma 2.2. Given any signature vector (s1, . . . , sd, o1, . . . , od) and any sequence of 1 1 real numbers a0 < a1 < . . . < a2d−2, there is a rational function π0 : RP → RP such that

(1) π0(±∞) = ±∞, (2) π0 has a zero of order si at a2i−2, for 1 ≤ i ≤ d, and π0 has no other zeroes or critical points, and (3) π0 has a simple pole at a2i+1, for 0 ≤ i ≤ d − 2 and no other poles except at ∞, which has odd order. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 19

(4) The logarithmic derivative of π0 is nonvanishing on R.

Proof: Let s1 s2 sd P (x) = (x − a0) (x − a2) ... (x − a2d−2) and let

Q(x) = (x − a1)(x − a3) ... (x − a2d−3). P (x) The rational function Q(x) has a pole at ∞ of odd order since (s1, . . . , sd, o1, . . . , od) P (x) is a signature vector. Further, Q(x) has simple poles at a2i+1, 0 ≤ i ≤ d − 2, and no other poles except ∞, and has a zero of order si at a2i−2, 1 ≤ i ≤ d, and no other zeroes. There may be other critical points, but there are at most a ﬁnite number of P (x) them. If there are none, then we are done, setting π0(x) = Q(x) . Assume then that P (x) Q(x) has additional critical points. Let h(x) be a polynomial of even degree with no zeros, with critical points of even degree at ai, 0 ≤ i ≤ 2d − 2, and with no other critical points. For example, let Z x h(x) = (x − a0)(x − a1) ... (x − a2d−2) dx + C a0 where C is a large enough constant so that h(x) is never zero. Let P (x) π (x) = [h(x)]N 0 Q(x) where N is a positive integer to be determined later. There are simple poles of order 1 at a2i+1, 0 ≤ i ≤ d − 2 and no other poles except ∞. Note that, since h has even degree, ∞ is a pole of odd order. The zeros of π0 are the points a2i−2, 1 ≤ i ≤ d, and each has order si. There are no other zeros, and we will show that if N is large enough, then there are no other critical points. We use the following elementary fact. Lemma 2.3. If x is a critical point of h(x)k(x), then either h(x)k0(x) = h0(x)k(x) = 0 or the logarithmic derivative of h at x equals the negative of the logarithmic derivative of k at x: d h0(x) k0(x) d ln |h(x)| = = − = − ln |k(x)|. dx h(x) k(x) dx

By this lemma, critical points of π0 occur either where P (x) P 0 N[h(x)]N−1h0[x] = (x)[h(x)]N = 0(7) Q(x) Q 20 LIZZIE BURSLEM AND AMIE WILKINSON or where d P (x) d d (8) ln | | = − ln |[h(x)]N | = −N ln |h(x)|. dx Q(x) dx dx We get no new critical points from the ﬁrst equation, and this is true for any N > 0. 0 Since h(x) is never zero, and h vanishes precisely at a0, a1, . . . , a2d−2, it is easy to see that the only solutions to equation (7) are the critical points of P ; that is, the set of a2i−2 such that si > 1. It remains to show that if N is large enough, then we get no new critical points d from equation (8). Note that for any positive function f, dx ln |f(x)| is positive if 0 d 0 and only if f (x) is, and dx ln |f(x)| is negative if and only if f (x) is also negative. In particular, d ln |h(x)| < 0 on (−∞, a ) dx 0 > 0 on (a2i, a2i+1), 0 ≤ i ≤ d − 2

< 0 on (a2i+1, a2i+2), 0 ≤ i ≤ d − 2

> 0 on (a2d−2, ∞)

= 0 if x = ai, 0 ≤ i ≤ 2d − 2. Since P (x) ln | | = −∞ if P (x) = 0 Q(x) = ∞ if Q(x) = 0 → ∞ as x → ±∞, there is a neighborhood U of {a0, a1, . . . , a2d−2} ∪ {±∞}, disjoint from the other P critical points of Q , such that P (x) ln | | < 0 on (−∞, a ) ∩ U Q(x) 0

> 0 on (a2i, a2i+1) ∩ U, 0 ≤ i ≤ d − 2

< 0 on (a2i+1, a2i+2) ∩ U, 0 ≤ i ≤ d − 2

> 0 on (a2d−2, ∞) ∩ U

undeﬁned at x = ai, 0 ≤ i ≤ 2d − 2. d P (x) d So for any positive N, dx ln | Q(x) |= 6 −N dx ln |h(x)| on U. Since R \ U is compact and contains no critical points of h, we can choose N so that on R \ U, d P (x) d | ln | | | < N | ln |h(x)| |. dx Q(x) dx GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 21

For such an N, we get no new critical points for π0 from equation 8. Note that, since h has even degree, ∞ is a pole of π0 of odd order. Hence the logarithmic derivative of π0 is nonvanishing and has the prescribed zeroes, poles and critical points.

1 1 Lemma 2.4. Let π0 : RP → RP be a rational function given by Lemma 2.2. Let 2m 2m + 1 be the order of the pole ∞ for π0. For ε > 0, let rε(x) = 1 + εx , and deﬁne 1 1 πε : RP → RP by: π0(x) πε(x) = . rε(x) Then, for ε > 0 suﬃciently small,

(1) πε has precisely the same zeroes as π0, of the same order, and no other critical points, (2) πε has precisely the same poles as π0, and all of them are simple, including ∞.

Proof: Let such a π0 be given. We ﬁrst claim that there is a real number M > 0 such that, for all ε > 0, the only pole, zero or critical point of πε in the region |x| > M is a simple pole at ∞. 2m 2m Let Rε(x) = 1/rε(1/x) = x /(x + ε), let Π0(x) = 1/π0(1/x), and let Πε(x) = Π0(x)/Rε(x) = 1/πε(1/x). The claim is equivalent to: Lemma 2.5. There exists δ > 0 such that, for all ε > 0: 0 (1) Πε(0) = 0, Πε(0) 6= 0, and (2) Πε has no other zeroes, poles, or critical points in (−δ, δ).

Proof: First note that, since ∞ is a pole of π0 of order 2m + 1, 0 is a critical point of Π0 of order 2m + 1, and Π0 has no other zeroes, poles or critical points in a δ0- neighborhood of 0, if δ0 > 0 is suﬃciently small. Fix such a δ0. In this neighborhood, we have that 2m+1 Π0(x) = cx + a(x), for some c 6= 0, where a(x)/x2m+1 → 0 as x → 0. It follows that (2m + 1)cx2m + a0(x) Π 0(x)/Π (x) = 0 0 cx2m+1 + a(x) 2m + 1 = + b(x), x

where b(x) is bounded. Thus there exists C > 0 so that for all x ∈ (−δ0, δ0), Π 0(x) 2m + 1 (9) | 0 − | ≤ C. Π0(x) x 22 LIZZIE BURSLEM AND AMIE WILKINSON

We calculate directly that 0 Rε(x) 2mε (10) = 2m . Rε(x) x(x + ε)

By Lemma 2.3, if x is a critical point of Πε, then either 0 0 (11) Rε(x)Π0 (x) = Rε(x)Π0(x) = 0, or R0 (x) Π 0(x) (12) ε = 0 . Rε(x) Π0(x)

Since 0 is the only critical point of Π0 and the only zero of R in (−δ0, δ0), it is also 0 the only solution to (11). But for any ε > 0, we have that Πε(0) = cε 6= 0. Next we show that there exists a δ ∈ (0, δ0) so that, for all ε > 0, equation (12) has no solution in (−δ, δ). Indeed, from (9) and (10), we obtain: 0 0 Π0 (x) Rε(x) 2m + 1 2mε | | − | | ≥ − C − 2m Π0(x) Rε(x) |x| |x|(x + ε) 2mx2m + (1 − C|x|)(x2m + ε) = . |x|(x2m + ε)

This is positive provided |x| < δ = min(δ0, 1/C). Thus for all ε > 0, Πε has no critical points in (−δ, δ). To ﬁnish the proof of Lemma 2.5, note that Πε has no poles or other zeroes in (−δ, δ), since Π0 has no poles or other zeroes in (−δ, δ), and Rε has no other zeroes in (−δ, δ).

Let δ be given by Lemma 2.5, and let M = 1/δ.

Lemma 2.6. For ε > 0 suﬃciently small, πε has exactly the same zeroes, poles, and critical points, of the same order, as π0 in the region [−M,M].

Proof: By Lemma 2.3, if x is a critical point of πε, then either 0 0 (13) rε(x)π0(x) = rε(x)π0(x) = 0, or r0 (x) π0 (x) (14) ε = 0 . rε(x) π0(x) 0 Since rε is everywhere positive, (13) is satisfied only where π0(x) = 0; these critical points of πε have the same order as for π0. π0 (x) By Lemma 2.5, | 0 | is bounded away from 0 on the compact set [−M,M]. On π0(x) 0 the other hand, by finding the critical points of | rε(x) |, we can calculate directly that, rε(x) GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 23 for all x ∈ R, 0 2m−1 rε(x) 2mεx | | = | 2m | rε(x) εx + 1 1 1− 1 ≤ ε 2m (2m − 1) 2m , which tends to 0 as ε → 0. It follows that for ε > 0 sufficiently small, (14) has no solutions in [−M,M].

To complete the proof of Proposition 2.1, we apply Lemma 2.2 to the signature vector s = (s1, . . . , sd, o1, . . . , od) and the sequence ai = i, for i = 0,..., 2d − 2, to obtain a π0 with zeroes of order si at 2i − 2, for i = 1, . . . , d. By Lemma 2.4, there exists an ε > 0 such that πε = π0/rε has zeroes of order si at 2i−2, for i = 1, . . . , d and simple poles at {1, 3,..., 2d − 3, ∞}. It is clear that, restricted to the complement of 1 1 the set {2, 4,..., 2d−2}, the map πε : RP \{2, 4,..., 2d−2} → RP \{0} is a regular cover of degree d. Hence πε is a ramiﬁed cover over 0. Note that every signature vector is determined by its ﬁrst d + 1 coordinates. Since πε|(−∞,0) is orientation-preserving, the signature of πε is equal to s if od = 1, and so setting π = πε, we complete the proof. In the case that od = −1, we set π = −πε.

3. The relation fgf −1 = gλ: the central technical result In this section be analyze the relation fgf −1 = gλ near the origin in [0, ∞). Here f and g are Cr diffeomorphisms with a common fixed point 0. For f and g real-analytic, Nakai [Nak] proved that any such f and g can be locally conjugated into one of the ramified affine groups described at the beginning of Section 1. Nakai’s proof begins by embedding g in an analytic vector field X (such a vector field must first be shown to exist), linearizing f about 0 (again, after proving that |f 0(0)| = κ 6= 1 at that fixed point), and solving the functional equation: X(x) = λκX(κ−1x). Nakai’s result gives a local characterization of diffeomorphisms of S1 satisfying this relation about each common fixed point, and to obtain a global characterization, it is a matter of gluing together these local ones. This was the way Ghys [Gh] proved that every solvable subgroup of Diff ω(S1) is metabelian. To prove Theorems 0.1 and 0.3, we adapt Nakai’s argument to the Cr setting, where additional hypotheses on f and g are required. The initial draft of this paper contained a completely different proof of Nakai’s theorem that does not rely on vector fields and works for Cr diffeomorphisms as well, under the right assumptions. In the Cω and C∞ case, this original proof gives identical results as the vector fields proof, but in the general Cr setting, the proof using vector fields gives sharper results. 24 LIZZIE BURSLEM AND AMIE WILKINSON

At the end of this section, we outline the alternate proof method. The main idea behind this method is to study the implications of the relation fgf −1 = gλ for the Schwarzian derivative of g near 0. Under the assumption that f 0(0) ≤ λ−1 and f and g are C3, one obtains that, in linearizing coordinates for f, the Schwarzian derivative of g must vanish identically in a neighborhood of 0, and so g must be a projective transformation. If f 0(0) > λ−1, one must assume more differentiability of f and g, but then one obtains that, in linearizing coordinates for f, the transformation x 7→ g(x1/m)m is projective, for some m ≥ 1. If g is not the identity, then g(x) = x/(axm + b)1/m, for some a, b, and we recover Nakai’s result. We now state the main technical result of this section. r Let [q, q1) be a half open interval, let r ∈ [2, ∞] ∪ {ω}, and let f, g ∈ Diff ([q, q1)) be diffeomorphisms. Suppose that g has no fixed points in (q, q1). Standing Assumptions. We assume that either A. B. C. or D. holds: A. r = ω, and there exists an integer λ > 1 such that: fgf −1 = gλ. B. r ∈ [2, ∞), and there exists an integer λ > 1 such that fgf −1 = gλ, 1 0 1 r−1 and f (q) ≤ λ . C. r = ∞, f 0(q) < 1, and for some integer λ > 1, fgf −1 = gλ. ∞ D. r ∈ {∞, ω}, g is not infinitely flat, and there is a C flow ϕt :[q, q1) → [q, q1) such that:

(1) ϕ1 = g, and −1 (2) fgf = ϕλ for some positive real number λ 6= 1. Assumptions A. B. and C. will arise in the proof of Theorems 0.1 and 0.3, and assumption D. will arise in the proof of Theorem 0.9. The main technical result that we will use in these proofs is the following. Proposition 3.1. Assume that either A. B. C. or D. holds. Then there is a Cr 1 diﬀeomorphism α :(q, q1) → (−∞, ∞) ⊂ RP such that: (1) for all p ∈ (q, α−1(0)), α(p) = oh(p)s, where h :[q, α−1(0)) → [−∞, 0) is a Cr diﬀeomorphism, s is an integer satisfying 1 ≤ s < r, and o ∈ {±1}. (2) αg(p) = α(p) + 1 (3) αf(p) = λα(p). GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 25

We will write fgf −1 = gλ to describe cases A. B. and C., where λ is an integer, and case D., where λ is a real number. In case D. the notation “gt” is used to denote ϕt, the time-t map of ϕ. We start with a lemma describing which values of f 0(q) and g0(q) can occur. Lemma 3.2. Assume that one of assumptions A.-D. holds. Then g0(q) = 1 , and either (1) g(i)(q) = 0 for 2 ≤ i ≤ r (in particular, neither assumption A. nor D. can hold in this case), or 1 1 0 s (2) f (q) = ( λ ) for some integer 1 ≤ s < r, and g(i)(q) = 0 for 2 ≤ i ≤ s, and g(s+1)(q) 6= 0.

Proof: Since fg = gλf, f 0(g(p)) g0(p) = (gλ)0(f(p)) f 0(p). When p = q, we thus have: g0(q) = (gλ)0(q). But (gλ)0(q) = (g0(q))λ, and so g0(q) = 1. Suppose that f 0(q) = κ 6= 1. Then in local coordinates in a neighborhood of q, identifying q with 0, we can write f(x) = κx, and g(x) = x + axs+1 + o(xs+1) for some s ≥ 1. Then a fgf −1(x) = x + ( )xs+1 + o(xs+1), and κs gλ(x) = x + λaxs+1 + o(xs+1). So either (1) a = 0, and therefore g(i)(q) = 0 for 2 ≤ i ≤ r, or 1 1 s (2) a 6= 0, in which case κ = ( λ ) , and g(i)(q) = 0 for 2 ≤ i ≤ s 6= 0 for i = s + 1. Now suppose that f 0(q) = 1. Then in a neighborhood of q, we can write f(x) = x + bxk+1 + o(xk+1) and g(x) = x + axs+1 + o(xs+1) 26 LIZZIE BURSLEM AND AMIE WILKINSON for some k, s ≥ 1. If k = s, then [f, g](x) := fgf −1g−1(x) = x + o(xs+1) = gλ−1(x) = x + (λ − 1)axs+1 + o(xs+1). So a = 0, and hence g(i)(0) = 0 for 2 ≤ i ≤ r. If k 6= s, then we use the following well known result (see, e.g., [RS]): Lemma 3.3. If f(x) = x + bxk+1 + o(xk+1) and g(x) = x + axs+1 + o(xs+1) and if s > k ≥ 1, then

[f, g](x) = x + (s − k)abxs+k + o(xs+k). Assume that s > k. (If s < k, then the proof is similar). It follows from Lemma 3.3 that [f, g](x) = x + (s − k)abxs+k + o(xs+k) = gλ−1(x) = x + (λ − 1)axs+1 + o(xs+1). So either λ−1 (1) k = 1, and therefore b = s−1 , or (2) k ≥ 2, and therefore a = 0 and g(i)(q) = 0, for 2 ≤ i ≤ r. But if k = 1, then [f 2, g](x) = x + (s − 1)2abxs+1 + o(xs+1) = f 2gf −2g−1(x) = gλ2−1(x) = x + (λ2 − 1)axs+1 + o(xs+1).

λ2−1 λ−1 (i) So b = 2(s−1) = s−1 , which is impossible, since λ 6= 1. Therefore g (q) = 0, for 2 ≤ i ≤ r.

Lemma 3.4. Assume that A. B. C. or D. holds. Then there is a neighborhood r [q, p1) ⊂ [q, q1) and a C map A :[q, p1) → R such that for all p ∈ [q, p1), s r (1) A(p) = oH(p) , where H :[q, p1) → [0, ∞) is a C diﬀeomorphism, 1 ≤ s < r is an integer, and o ∈ {±1}; (2) 1 Af(p) = A(p); λ GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 27

(3) A(p) Ag(p) = . 1 − A(p) 1 1 0 s (s+1) Consequently, f (q) = ( λ ) for some integer 1 ≤ s < r, and g (q) 6= 0.

Proof: We say that a C2 function c :[a, b) → [a, b) is a C2 contraction if c0 is positive on [a, b) and c(x) < x, for all x ∈ (a, b). Since g has no fixed points in (q, q1), either g or g−1 is a C2 contraction. We will assume until the end of the proof that g is a C2 contraction. Replacing g by g−1 does not change the relation fgf −1 = gλ. 1 Since g has no fixed points in (q, q1), there is a a unique C vector field X0 on 1 [q, q1) that generates a C flow ϕt such that g|[q,q1) = ϕ1 (Szekeres, see [Nav1] for a discussion).

−j j Lemma 3.5. For all j ∈ N and x ∈ [q, q1), f gf (x) = ϕ 1 (x). λj

Proof: We will use the following result of Kopell:

2 2 Lemma 3.6. ([Ko] Lemma 1) Let g ∈ Diff [q, q1) be a C contraction that embeds 1 1 in a C flow ϕ, so that g = ϕ1. If h ∈ Diff [q, q1) satisfies hg = gh, then h = ϕt for some t ∈ R. It follows from the relation f jgf −j = gλj that f −jgf j commutes with g, and there- −j j fore Lemma 3.6 implies that f gf = ϕt for some t ∈ R. This relation also implies −j j λj −j j that (f gf ) = g. So f gf = ϕ 1 . λj Let κ = f 0(q). We may assume, by Lemma 3.2, that κ 6= 1, and therefore there r is an interval [q, p1) on which f has no fixed points, and a C diffeomorphism H : [q, p1) → [0, ∞) such that H f H(x) = κx ([ZL] Theorem 2.8). The diffeomorphism H is unique up to multiplication by a constant. Let F = HfH−1 and let G = HgH−1. Since we have assumed that g is a contraction, we have g([q, p1)) ⊆ [q, p1).

Let X be the push-forward of the vector ﬁeld X0 to [0, ∞) under H, and let φt be the semiﬂow generated by X0, so that G = φ1.

1 0 1 r−1 Lemma 3.7. If F (0) ≤ ( λ ) and G is r-ﬂat at 0, then X(x) = 0 on [0, ∞).

Proof: We will show that for all x ∈ [0, ∞), φ (x) − x lim t = 0. t→0 t Since the limit exists, it is enough to show that it converges to 0 for a subsequence 1 0 ti → 0. We will use the subsequence ti = λi . If κ = F (0), then F (x) = κx and G(x) = x + R(x), 28 LIZZIE BURSLEM AND AMIE WILKINSON

Corollary 3.8. Under any assumption A.-D., g is not r-flat at q, and therefore 1 1 0 s f (q) = ( λ ) , for some integer 1 ≤ s < r. Proof: Clearly g cannot be infinitely flat if A. or D. holds. Under assumption C., 1 0 1 k k f (q) < λ , for some k > 0 and f, g are C , so C. reduces to B. By Lemma 3.7, under assumption B., if g is r-flat at q, then the semiflow φt is tangent to the trivial vector field, X(x) = 0. But then G = id, and therefore g = id on [q, p1), contradicting the assumption that g has no fixed points in (q, q1).

1 1 0 s Lemma 3.9. If F (0) = ( λ ) for some integer 1 ≤ s < r, then for some a < 0, X(x) = axs+1 on [0, ∞).

Proof: As in the proof of Lemma 3.7, it is enough to show that for all x ∈ [0, ∞), 1 and for ti = λi , G (x) − x lim ti = axs+1 i→∞ ti 1 1 0 s for some a ∈ R. If F (0) = ( λ ) for some integer 1 ≤ s < r, then by Lemma 3.2,

1 1 s+1 s+1 F (x) = ( ) s x and G(x) = x + ax + R(x ) λ for some a ∈ R, where R(x)/xr → 0 as x → 0. The value of a depends on the choice

of linearizing map h for f|[q,p1). For all i ∈ N, i x −i i s φti (x) = F GF (x) = λ G i λ s 1 i x s+1 s = x + ax i + λ R i .H λ λ s GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 29

i φti (x) − x 1 x i s+1 s lim = lim λ ax i + λ R i i→∞ ti i→∞ λ λ s i(s+1) s+1 x λ s s+1 = lim ax + R i s+1 x i→∞ λ s x = axs+1 Since G is a contraction on [0, ∞), we may assume that a ≤ 0, and since G 6= id, it follows that a < 0.

1 1 0 s Corollary 3.10. If one of A.-D. holds, then f (q) = ( λ ) for some integer 1 ≤ s < r, and, after a suitable rescaling of the linearizing map H, x G(x) = 1 . (1 + xs) s

δ Proof: Choose H so that a = −1/s. Solving the diﬀerential equation δt φt(x) = s+1 aφt(x) with initial condition φ0(x) = x, we obtain: x φt(x) = 1 ; (1 + txs) s

since G(x) = φ1(x), the conclusion follows.

To complete the proof of Lemma 3.4 assuming that g is a contraction, let s, H be given by Corollary 3.10, and let o = −1. Then Corollary 3.10 implies that A(p) = (H(p)s) satisﬁes the desired conditions. If g is not a contraction, we replace g by −1 g in the proof. Setting o = 1, we obtain the desired conclusions.

Lemma 3.11. Assume one of A.-D. holds. Let A, H, s, and o be given by Lemma 3.4. For p ∈ [q, p1), let −1 −1 h(p) = ; α (p) = . H(p) 0 A(p) r Then α0 extends to a C map α :(q, q1) → (−∞, ∞) satisfying the conclusions of Proposition 3.1.

Proof: Lemma 3.4 implies that for all p ∈ (q, p1), α0g(p) = α0(p) + 1 and α0f(p) = r λα0(p). We can now extend α0 to a C diﬀeomorphism α from (q, q1) to (−∞, ∞) as follows. Since g has no ﬁxed points in (q, q1), given any p ∈ (q, q1), there is some j j j ∈ Z such that g (p) ∈ (q, p1). Let α(p) = α0(g (p)) − j (which is easily seen to be 30 LIZZIE BURSLEM AND AMIE WILKINSON

independent of choice of j). By construction, αg(p) = α(p) + 1 for all p ∈ (q, q1). Since f(p) = (g−j)λfgj(p), we also have:

−j λ −1 −1 j αf(p) = (α(g ) α0 )(α0fα0 )(α0g (p)) = λ(α(p) + j) − jλ = λα(p).

Hence the conclusions of Proposition 3.1 hold. .

3.1. Sketch of an alternate proof. Suppose that f and g 6= id are germs of analytic diffeomorphisms, defined in a neighborhood of 0 in R, both fixing the origin, and satisfying the relation: fgf −1 = gn, for some integer n > 1. Then Proposition 3.1 implies that f and g are, respectively, simultaneously conjugate, via the germ of an analytic diffeomorphism, to the maps x 7→ x/n1/s and x 7→ x/(1 + axs)1/s, for some s ≥ 1 and a ∈ {±1}. Here we describe an alternate method of proof using the Schwarzian derivative. To simplify notation, assume that n is 2. 1 0 1 s First note that, since g is not infinitely flat, Lemma 3.2 implies that f (0) ∈ { 2 | s ≥ 1}. After conjugating f and g by an analytic germ, we may assume, then, that: x f(x) = 1 , 2 s for some s ≥ 1. 1 s 1 s −1 Let F (x) = f(x s ) = x/2 and let G(x) = g(x s ) . Rewriting the relation F GF = G2, we obtain: 1 G(2x) = G2(x); 2 rearranging and iterating this relation, we obtain:

k x (15) G(x) = 2kG2 , 2k for all k ≥ 1. Recall that the Schwarzian derivative of a C3 function H is defined by: H000(x) 3 H00(x)2 S(H)(x) = − , H0(x) 2 H0(x) and has the following properties: (1) S(H)(x) = 0 for all x iff G is Möbius,and (2) for any C3 function K, S(H ◦ K)(x) = K0(x)2S(H)(K(x)) + S(K)(x). Combining these properties with (15), we will show that S(G) = 0, which implies that G is Möbius. Writing g(x) = G(xs)1/s, we obtain the desired result. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 31

The first thing to check is that G is C3. To obtain this, we use a slightly stronger version of Lemma 3.2 (whose proof is left as an exercise), which states that, if g is not infinitely flat, then g(x) = x + axs+1 + bx2s+1 + ··· Performing the substitution G = g(x1/s)s in this series, one finds that G is C3. Equation (15) implies that

1 k x S(G)(x) = S(G2 )( ), 22k 2k for all k ≥ 1. Thus, by the cocycle condition (2) of the Schwarzian, we have:

2k 1 X x x 2 (16) S(G)(x) = S(G)(Gi−1( )) (Gi−1)0( ) 22k 2k 2k i=1 2k 1 X 2 (17) = S(G)(x ) Πi−1 G0(x ) 22k i j=1 j i=1 i−1 x where xi := G ( 2k ). Fix x, and assume without loss of generality that Gj(x) → 0 as j → ∞. Since 3 0 0 C G is C and G (0) = 1, there is a constant C > 0 such that |G (xi)| ≤ 1 + 2k , and k |S(G)(xi)| ≤ C, for all i between 1 and 2 and all k ≥ 1. Combined with (16), this gives us a bound on the Schwarzian of G at x:

2k 2(i−1) C X C |S(G)(x)| ≤ 1 + 22k 2k i=1 C 2k+1 ! C 1 − (1 + k ) ≤ 2 22k C 2 1 − (1 + 2k ) ! 1 e2C − 1 ≤ , 2k C 2 + 2k for all k ≥ 1. Hence S(G)(x) = 0, for all x, which implies that G is M¨obius.

4. Further properties of ramified covers: proofs of Proposition 1.2, Lemma 1.3 and Proposition 1.4. The following lemma follows from a standard lemma in complex dynamics. We have borrowed the proof from ([McM] Theorem 4.14).

Lemma 4.1. Let π : RP 1 → RP 1 be a ramiﬁed covering map over 0, where π−1(0) = {x1, . . . , xd}. Let s = (s(x1), . . . , s(xd), o1, . . . , od) be the signature of π. Then for 32 LIZZIE BURSLEM AND AMIE WILKINSON

all 1 ≤ i ≤ d, there are neighborhoods Ui of xi and V of 0, and analytic charts hi : Ui → R and ki : V → R such that s(xi) kiπ(x) = hi(x) for all x ∈ Ui.

Proof: In local coordinates at xi, identifying xi with 0, we can write π(x) = axs(1 + O(x)) where a > 0 and s = s(xi). Extend π in a neighborhood of 0 to a complex analytic 1 s n n function satisfying π(z) = az (1 + O(z)). Let φn(z) = (π (z)) s , where the root is 0 chosen so that φn(0) > 0, and let hi(z) = ki(z) = limn→∞ φn(z). Then −1 −1 kiπhi (z) = lim φnπφn (z) n→∞ sn 1 = lim [π(z )] sn n→∞ 1 s sn = lim a sn z (1 + O(z )) n→∞ = zs if |z| < 1.

It remains to show that hi is analytic. Since φn is a sequence of holomorphic functions, 1 this will be true if φn converges to hi uniformly. If |z| < 2 , then 1 n 1 ! sn φn+1(z) π(π (z)) s = n φn(z) π (z)

1 1 n n ! sn a s π (z)(1 + O(π (z))) = πn(z)

1 −sn < a sn+1 (1 + O(2 ))

Since this converges to 1 as n → ∞, φn converges uniformly to hi. The restriction of hi = ki to the real line gives the required charts. This lemma motivates the following deﬁnition.

Deﬁnition: A Cr ramiﬁed cover over p ∈ S1 is a map π : S1 → S1 satisfying: −1 (1) π (p) = {q1, q2, . . . , qd}, where q1 < q2 < . . . < qd; (2) the restriction of π to π−1(S1 \{p}) is a regular Cr covering map onto S1 \{p} of degree d ≥ 1, r (3) for all 1 ≤ i ≤ d, there are neighborhoods Ui of qi and V of p, and C charts hi : Ui → R and ki : V → R with hi(qi) = 0 and ki(p) = 0, such that −1 si ki π hi (x) = x

for some integer si > 0. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 33

Remark: By Lemma 4.1, a ramified cover is a Cω ramified cover. We define the signature of a Cr ramified cover in the obvious way.

r Lemma 4.2. Let π1 and π2 be C ramified covering maps of degree d over p1 and p2, −1 −1 respectively. Fix an orientation-preserving identification between π1 (p1) and π2 (p2) r 1 and between E(π1) and E(π2). Suppose that f ∈ Diff +(S ) satisfies f(p1) = p2 r and that for some ζ ∈ Dd, we have s(π2) = ζ(s(π1)). Then there is a unique C diffeomorphism fˆ : S1 → S1 such that ˆ −1 (1) f(q) = ζ(q), for all q ∈ π1 (p1), ˆ (2) f(e) = ζ(e), for all e ∈ E(π1). and (3) the following diagram commutes:

fˆ S1 - S1

π1 π2 ? ? f S1 - S1 ˆ r 1 Moreover, if ζ ∈ Cd, then f ∈ Diﬀ +(S ).

−1 1 Proof: Since the restriction of π1 to π1 (S \{p1}) and the restriction of π2 to −1 1 r π2 (S \{p2}) are both regular C covering maps of degree d, for any ζ ∈ Dd there is r ˆ 1 −1 1 −1 ˆ a unique C diffeomorphism f0 : S \ π {p1} → S \ π {p2} such that f0(e) = ζ(e) for all e ∈ E(π1), and the diagram in (3) commutes on the restricted domains. The ˆ ˆ condition s(π2) = ζ(s(π1)) implies that f0 extends to a unique homeomorphism f such ˆ −1 that f(q) = ζ(q), for all q ∈ π1 (p1) and such that the diagram in (3) commutes. It remains to show that fˆ is a Cr diffeomorphism. ˆ r −1 It suffices to show that f is a C diffeomorphism at each q ∈ π1 (p1). In local coordinates around q and p1, identifying q with 0 and p1 with 0, we can write s(q) π1(x) = x −1 for some integer s(q) ≥ 1. Similarly, in local coordinates near each q ∈ π2 (p2), s(q) π2(x) = x for some integer s(q) ≥ 1. By the commutative diagram, in a neighborhood of q, (fˆ(x))s(fˆ(q)) = f(xs(q)) ˆ Since s(π2) = ζ(s(π1)), we have s(ζ(q)) = s(q). Let j = s(q) = s(f(q)). Then we can write j 1 fˆ(x) = (f(x )) j 34 LIZZIE BURSLEM AND AMIE WILKINSON

where the root is chosen so that fˆ0(0) > 0. But f is Cr and has a ﬁxed point at 0. So 2 r r f(x) = a1x + a2x + ... + x + o(x )

where a1 6= 0. We can assume that the coordinates have been chosen so that a1 > 0. So near x = 0, 1 ˆ j 2j rj rj j f(x) = (a1x + a2x + ... + x + o(x )) j (r−1)j (r−1)j 1 = x (a1 + a2x + ... + x + o(x )) j . ˆ r Since a1 > 0 and r ≥ 2, f is a C diﬀeomorphism at 0. Similarly, we see that if f is analytic, then fˆ is analytic. Finally, we note that since f is orientation preserving, if ˆ ζ ∈ Cd, then f must also be orientation preserving.

r 1 Corollary 4.3. Let f ∈ Diff −(S ) be an orientation reversing diffeomorphism with r a fixed point p, and let π1 and π2 be C ramified covering maps over p. Fix an −1 −1 orientation-preserving identification between π1 (p) and π2 (p) and between E(π1) and E(π2). Suppose that for some ζ ∈ Dd, we have ζ(s(π1)) = I(s(π2)), where I : Sd → Sd is the involution that reverses the sign of the last d coordinates. Then there is a unique Cr diffeomorphism fˆ : S1 → S1 such that ˆ −1 (1) f(q) = ζ(q), for all q ∈ π1 (p), ˆ (2) f(e) = ζ(e), for all e ∈ E(π1) and (3) the following diagram commutes: fˆ S1 - S1

π1 π2 ? ? f S1 - S1 ˆ r 1 Moreover, if ζ ∈ Cd, then f ∈ Diﬀ +(S ).

Proof: Let π1 = f ◦ π1 and π2 = π2. Use Lemma 4.2 with the ramiﬁed covers π1 ˆ ˆ ˆ and π2 to lift the identity map. Deﬁne f to be this lift: f := id.

Notation: Let f : S1 → S1 be a Cr diffeomorphism with fixed point at p, let 1 1 r π : S → S be a C ramified covering map over p with s(π) = s ∈ Sd, and let ζ ∈ Dd, satisfy: r 1 • ζ(s) = s, if f ∈ Diff +(S ), r 1 • ζ(s) = I(s), if f ∈ Diff −(S ). We denote by fˆ(π, ζ) the unique Cr diffeomorphism satisfying: (1) fˆ(π, ζ)(q) = ζ(q) for all q ∈ π−1(p), (2) fˆ(π, ζ)(e) = ζ(e) for all e ∈ E(π), and GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 35

(3) the following diagram commutes: fˆ(π, ζ) S1 - S1 π π ? ? f S1 - S1

r 1 Lemma 4.4. Let f1 and f2 be C diffeomorphisms on S , both with a fixed point at 1 1 r p, let π : S → S be a C ramified covering map over p, and let ζ1, ζ2 ∈ Dd. Suppose that ζ1 and ζ2 satisfy: r 1 • ζi(s) = s, if fi ∈ Diff +(S ), r 1 • ζi(s) = I(s), if fi ∈ Diff −(S ). Then ˆ ˆ f2(π, ζ2) ◦ f1(π, ζ1) = f\2 ◦ f1(π, ζ2 ◦ ζ1).

ˆ ˆ Proof: The map f2(π, ζ2) ◦ f1(π, ζ1)(q) satisﬁes: ˆ ˆ −1 (1) f2(π, ζ2) ◦ f1(π, ζ1)(q) = ζ2 ◦ ζ1(q), for all q ∈ π (p), ˆ ˆ (2) f2(π, ζ2) ◦ f1(π, ζ1)(e) = ζ2 ◦ ζ1(e), for all e ∈ E(π), and (3) the following diagram commutes: ˆ ˆ f1(π, ζ1) f2(π, ζ2) S1 - S1 - S1 π π π ? ? ? f1 f2 S1 - S1 - S1 ˆ ˆ By Lemma 4.2 and Corollary 4.3, we must have f2(π, ζ2)◦f1(π, ζ1) = f\2 ◦ f1(π, ζ2 ◦ ζ1). The following proposition is a Cr version of Proposition 1.2.

r 1 Proposition 4.5. Suppose that G is a group, and that ρ : G → Diff +(S ) is a representation with global fixed point p. Let π : S1 → S1 be a Cr ramified cover over p with signature vector s. Then for every homomorphism h : G → StabDd (s), there is a unique representation ρˆ =ρ ˆ(π, h): G → Diff r(S1) such that, for all γ ∈ G, (1)ρ ˆ(γ)(q) = h(q), for all q ∈ π−1{p}, (2)ρ ˆ(γ)(e) = h(e), for all e ∈ E(π), and (3) the following diagram commutes: 36 LIZZIE BURSLEM AND AMIE WILKINSON ρˆ(γ) S1 - S1 π π ? ? ρ(γ) S1 - S1 r 1 If h takes values in StabCd (s), then ρˆ takes values in Diff +(S ).

Proof: This follows immediately from the previous two lemmas.

The following lemma is a Cr version of Lemma 1.3. Lemma 4.6. Let G be a group, and let ρ : G → Diff ω(S1) be a representation with 1 1 r 1 global fixed point p. Let π1, π2 : S → S be C ramified covers over p ∈ S , with s(π1) = ζ(s(π2)), for some ζ ∈ Dd.

Then for every homomorphism h : G → StabDd (s), the representation ρ˜(π1, h) is −1 r 1 −1 −1 conjugate to ρ˜(π2, ζhζ ) in Diff (S ), where (ζhζ )(γ) := ζh(γ)ζ . If ρ takes ω 1 values in Diff +(S ), if ζ ∈ Cd, and if h takes values in StabCd (s), then ρ˜(π1, h) and −1 r 1 ρ˜(π2, ζhζ ) are conjugate in Diff +(S ).

Proof: This lemma follows from the diagram below, which commutes by Proposi- tion 4.5 and Lemma 4.2.

−1 ρˆ(π2, ζhζ ) S1 - S1

π π idb 2 idb 2

? ? 1 ρ - 1 S S ρˆ(π , h) 1 1 - 1 S S

id id π π 1 1

? ? ρ S1 - S1 Here, idb is the lift of the identity map given by Lemma 4.2 satisfying: ˆ −1 −1 (1) f(q) = ζ (q), for all q ∈ π1 (p), ˆ −1 (2) f(e) = ζ (e), for all e ∈ E(π1). GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 37 ˆ ˆ Consider two lifts f(π1, ζ), f(π2, ζ) of the same diffeomorphism f (or, more generally, of conjugate diffeomorphisms). For purely topological reasons, if these lifts are conjugate by a map with rotation number 0, then s(π1) and s(π2) have the same length 2d, and the final d entries in these vectors must agree. (More generally, if the conjugacy has nonzero rotation number, then the final d entries of the first vector must lie in the Dd-orbit of the final d entries of the second). We now examine the first d entries of both vectors. We show that, under appropriate regularity assumptions on f and on the conjugacy, these entries must also agree, so that s(π1) = s(π2). The next lemma is the key reason for this. Lemma 4.7. Let c : [0, ∞) → [0, ∞) be a C2 contraction. Suppose that, for some m 1/m n 1/n integers m, n > 0, the maps v1(x) = c(x ) and v2(x) = c(x ) are conjugate by a C1 diffeomorphism h : [0, ∞) → [0, ∞). Then m = n.

Proof: Since c is a C2 contraction, the standard distortion estimate (see, e.g. [Ko]) implies that for all x, y ∈ [0, ∞), there exists an M ≥ 1, such that for all k ≥ 0, 1 (ck)0(x) (18) ≤ ≤ M. M (ck)0(y) Assume without loss of generality that n > m and suppose that there exists a C1 diﬀeomorphism h : [0, ∞) → [0, ∞) such that hv1(x) = v2(h(x)), for all x ∈ [0, ∞). Let H(x) = h(x1/m)n. Note that the C1 function H : [0, ∞) → [0, ∞) has the following properties: (1) H0(x) ≥ 0, for all x ∈ [0, ∞), and H0(x) = 0 iﬀ x = 0; (2) for all k ≥ 0, H ◦ ck = ck ◦ H. Then (2) implies that for every x ∈ [0, ∞): (ck)0(x) H0(x) = H0(ck(x)) (ck)0(H(x)) for all k ≥ 0. But (18) implies that (ck)0(x)/(ck)0(H(x)) is bounded independently of 0 0 k k, so that H (x) = limk→∞ H (c (x)) = 0, contradicting property (1).

s1 s2 Corollary 4.8. Let G and H be infinite subgroups of Affc (R) and Affc (R), respec- 1 1 −1 tively, for some s1, s2 ∈ S. If there exists α ∈ Diff (S ) such that αGα = H, then s1 = s2. s s 1 2 + 1 1 If G < Affc + (R) and H < Affc + (R), with s1, s2 ∈ S , and there exists α ∈ Diff +(S ) −1 such that αGα = H, then s1 = s2.

Proof: Let G, H and α be given. Note that s1 and s2 must have the same length 2d, since the global finite invariant sets of G and H must be isomorphic. Let g, h be elements of G and H with rotation number 0 such that h = αgα−1. Since dilations have twice as many fixed points in RP 1 as translations, if g is a ramified lift of a 38 LIZZIE BURSLEM AND AMIE WILKINSON ˆ ˆ translation, then so is h. Assume that g = S(πs1 , id) and h = T (πs2 , id), where S : x 7→ x + s and T : x 7→ x + t are translations with s, t > 0. Let q1, . . . , qd

and α(q1), . . . , α(qd) be the preimages of ∞ under πs1 and πs2 , respectively. In a mi 1/mi neighborhood of qi, the map g is conjugate to x 7→ (S(x) ) and in a neighborhood ni 1/ni of α(qi), h is conjugate to x 7→ (T (x) ) , where mi = s(qi) and ni = s(α(qi)). Since S is a C2 contraction in a neighborhood of ∞ and T is conjugate to S, it follows from Lemma 4.7 that mi = ni for 1 ≤ i ≤ d, which implies that s1 = s2. Suppose instead that g is a ramified lift of a map in Aff(R) conjugate to the dilation D : x 7→ ax, for some a > 1. Since g must have d fixed points with derivative a, so must h, and so h is also a ramified lift of a map in Aff(R) conjugate to D. Around 2 ∞, the map D is a C contraction, and the same proof as above shows that s1 = s2. The proof in the orientation-preserving case is analogous. Proof of Proposition 1.4: Recall that BS(1, n) =< a, b : aba−1 = bn >, ω 1 and the standard action ρn : BS(1, n) → Diff (S ) is generated by

ρn(a) = f : x 7→ nx, and ρn(b) = g : x 7→ x + 1. 1 1 Suppose thatρ ˆn(πs1 , h1) andρ ˆn(πs2 , h2) ∈ V are conjugate by α ∈ Diﬀ (S ): −1 αρˆn(πs1 , h1)α =ρ ˆn(πs2 , h2).

It follows from Corollary 4.8 that s1 = s2. ˆ Let f1 = f(πs1 , h1(f)) and g1 =g ˆ(πs1 , h1(g)) be the generators ofρ ˆn(πs1 , h1) in ω 1 ˆ Diﬀ (S ), and let f2 = f(πs2 , hπ2 (f)) and g2 =g ˆ(πs2 , h2(g)) be the generators of

ρˆn(πs2 , h2). Since h1 and h2 are homomorphisms, to prove that h1 = h2, it is enough −1 to show that h1(f) = h2(f) and h1(g) = h2(g). But for all q ∈ π1 (∞), we have:

αh1(f)(q) = αf1(q) = f2(α(q)) = h2(f)(α(q)),

and for all e ∈ E(π1),

αh1(f)(e) = αf1(e) = f2(α(e)) = h2(f)(α(e)). −1 −1 Since α(πs1 (∞)) = πs2 (∞) and α(E(π1)) = E(π2), it follows that h1(f) = h2(f). Similarly h1(g) = h2(g).

5. Proof of Theorems 0.1 and 0.3 The construction behind this proof is very simple. We are given a Cr representation ρ of BS(1, n). Using elementary arguments, we are reduced to the case where f = ρ(a) and g = ρ(b) have a common finite invariant set, the set of periodic orbits of g. After taking powers of f and g, the points in this invariant set are fixed by both f and g. Assume that the rotation numbers of f and g are both 0 and that f and g satisfy the hypotheses of the theorem. Using the results from Section 3, we obtain a local GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 39 characterization of f and g around each common fixed point. The map that glues together these local characterizations is a Cr ramified covering map π. Around each global fixed point, π(x) = (φ(x))s, for some integer s > 0, where φ is a linearizing map for f, and in between two global fixed points, π = ψ, where ψ is another linearizing map for f. In these local coordinates, g is the projective transformation x 7→ x/x + 1 (about the global fixed point) and x 7→ x+1 (in between global fixed points), and f is x 7→ x/n (about the global fixed point) and x 7→ nx (in between global fixed points). Hence ρ is a ramified cover of the standard representation. It remains to handle the case where the rotation numbers of f and g are not 0, but this is fairly simple to do, since the elements of the standard representation embed in analytic vector fields. We now give the complete proof. Let ρ : BS(1, n) → Diff r(S1) be a representation, where r ∈ [2, ∞], or r = ω. If r < ∞, assume that 1 1 r−1 σ(ρ) ≤ . n If r = ∞, we assume that σ(ρ) < 1. Let f = ρ(a) and g = ρ(b). Since g is conjugate to gn, it follows that τ(g) = ±τ(gn) = ±nτ(g), where τ(h) denotes the rotation number of h ∈ Homeo (S1). Hence g has k rational rotation number, either of the form n−1 , if f is orientation-preserving, or of k the form n+1 , otherwise. Lemma 5.1. f preserves the set of periodic points of g.

Proof: This follows from the relation fg = gnf. If gk(q) = q, then fgk(q) = f(q) = gnk(f(q)). So f(q) is also periodic for g. Suppose that τ(f) is irrational. Then by Lemma 5.1, the periodic points of g are dense in S1, which implies that gk = id, for some k ≤ n + 1. This implies that conclusion (1) of Theorem 0.1 holds. Suppose, on the other hand, that τ(f) is rational. Choose l so that gl and f l are both orientation-preserving and both have rotation number 0. Then f l leaves Fix(gl) invariant. Choose p ∈ Fix(gl). Any accumulation of {f ln(p)} must be a ﬁxed point for f l and for gl. We have shown: Lemma 5.2. f l and gl have a common ﬁxed point.

Note that the fixed points for f l are isolated; if f is not analytic, then σ(ρ) < 1, l which implies that the fixed points for f are hyperbolic. Let w1 < w2 < . . . < wk be the set of fixed points of f l. 40 LIZZIE BURSLEM AND AMIE WILKINSON

l l 0 l Lemma 5.3. If g (wi) = wi and (f ) (wi) > 1, then g = id on [wi−1, wi+1].

l 0 Proof: Suppose that (f ) (wi) = λ > 1, and let α :[wi, wi+1) → [0, ∞) be a C1 linearizing diﬀeomorphism such that αfα−1(x) = λx for all x ∈ [0, ∞). Let −1 −1 l F = αfα , and let G = αgα . If g 6= id on [wi, wi+1), then there is a point x0 ∈ [0, ∞) such that G(x0) 6= x0. Let x0 be any such point. We may assume that k −1 G (x0) → c as k → ∞, for some c < ∞, because this will be true for either G or G . k Since GF −k = F −kGn for all k ∈ N, it follows that −k 0 nk 0 −k (F ) (G (x0)) nk 0 G (F (x0)) = −k 0 (G ) (x0), (F ) (x0)

0 nk 0 for all k ∈ N. But since G (0) = 1 (by Lemma 3.2), this means that (G ) (x0) → 1, as k → ∞ (or k → −∞), for every point x0 that is not ﬁxed by G. Since G is not the identity, this is not possible. Hence g = id on [wi, wi+1]. A similar argument shows that g = id on [wi−1, wi],

l l Corollary 5.4. If g has a ﬁxed point in the interval (wi, wi+1), then g = id on l l [wi, wi+1]. That is, ∂Fix(g ) ⊆ Fix(f ).

l kl Proof: Suppose that g (p) = p for some p ∈ (wi, wi+1), and suppose that f (p) → wi as k → −∞. By Lemma 5.1, f lk(p) is ﬁxed by gl for all k ∈ Z, so by continuity, l l l 0 wi is a common ﬁxed point for f and g . Since (f ) (wi) > 1, Lemma 5.3 implies l kl that g = id on [wi, wi+1]. Similarly, if f (p) → wi+1 as k → −∞, then g = id on [wi, wi+1].

This has the immediate corollary: Corollary 5.5. f l ﬁxes every component of S1 \ Fix(gl).

Remark: Corollary 5.5 also follows from Theorem 0.6. We give a diﬀerent proof here since we also need Lemma 5.3. l Let ∞ ≤ q1 < q2 < ··· < qd < ∞ be the elements of ∂Fix(g ).

l r Lemma 5.6. For each 1 ≤ i ≤ d, either g |[qi,qi+1] = id, or there exists a C diﬀeo- 1 morphism αi :(qi, qi+1) → (−∞, ∞) ⊂ RP with the following properties: (1) There exist Cr diﬀeomorphisms −1 hi :[qi, αi (0)) → [−∞, 0) and ∗ −1 hi :(αi (0), qi+1] → (0, ∞] GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 41

∗ and integers oi ∈ {±1}, si, si ∈ {1, . . . , r − 1} such that: ( si −1 oi(hi(p)) , if p ∈ (qi, αi (0)) αi(p) = ∗ , ∗ si −1 oi(hi (p)) , if p ∈ (αi (0), qi+1), (2) l αi ◦ g = αi + 1 (3) l l αi ◦ f = n αi.

l r Proof: By Proposition 3.1, if g 6= id on [qi, qi+1], then there are C diffeomorphisms ∗ αi, αi :(qi, qi+1) → (−∞, ∞) ∗ ∗ r integers si, si ∈ {1, . . . , r − 1} and oi, oi ∈ {±1}, and C diffeomorphisms hi : −1 ∗ ∗−1 [qi, αi (0)) → [−∞, 0) and hi :(αi (0), qi+1] → (0, ∞], such that: si −1 (1) αi(p) = oihi(p) , for p ∈ (qi, αi (0)), ∗ ∗ ∗ ∗ si ∗−1 (2) αi (p) = oi hi (p) , for p ∈ (αi (0)), qi+1); l ∗ l ∗ (3) αi ◦ g = αi + 1 and αi ◦ g =α ˜i + 1; l l ∗ l l ∗ (4) αi ◦ f = n αi and αi ◦ f = n αi . ∗ ∗ We will show that αi = αi and oi = oi . ∗ l Both αi and αi linearize f on (qi, qi+1). Note that there is only one fixed point l −1 ∗−1 l −1 for f on (qi, qi+1), at αi (0) = αi (0). Moreover, by (3), αi(g αi (0)) = 1 and ∗ l ∗−1 ∗ l ∗−1 αi (g αi (0)) = αi (g αi (0)) = 1. Since a linearizing diffeomorphism is unique up ∗ l −1 to multiplication by a constant, and since αi agrees with αi at g αi (0), we must ∗ ∗ ∗ have αi = αi on (qi, qi+1). Setting αi = αi = αi , it now follows that oi = oi , and we let oi be this common value.

l l l Corollary 5.7. Either g = id, or ∂Fix(g ) = Fix(g ) = {q1, . . . , qd}.

l l Proof: If ∂Fix(g ) = {q1, . . . , qd}= 6 Fix(g ), then there is an interval [qi−1, qi] on l l l −l r which g = id. If g 6= id on [qi, qi+1], then by Lemma 3.4, either g or g is C conjugate on [qi, qi+1) to the map x x 7→ 1 (1 − xm) m

l r for some integer 1 ≤ m < r. But this map is not r-ﬂat at x = 0, so g is not C at qi. l l 1 Therefore g = id on [qi, qi+1], and similarly, g = id on S . 42 LIZZIE BURSLEM AND AMIE WILKINSON

l 1 Corollary 5.8. If g 6= id, then the map π0 : S \{q1, . . . , qd} → R deﬁned by:

π0(p) = αi(p), for p ∈ (qi, qi+1) r 1 1 extends to a C ramiﬁed cover π : S → RP with signature s = (s1, . . . sd, o1, . . . , od). The following diagrams commute: f l gl S1 - S1 S1 - S1 π π π π ? ? ? ? x 7→ nlx x 7→ x + 1 S1 - S1 S1 - S1

∗ ∗ Proof: Fix i and let αi, hi, hi−1, si, and si−1 be given by Lemma 5.6. We will show that, for 1 ≤ i ≤ d; (1) the map ( −1 hi(p), if p ∈ [qi, αi (0)) hi(p) = ∗ −1 hi−1(p), if p ∈ (αi−1(0), qi], is a Cr diﬀeomorphism onto its image in RP 1, and ∗ (2) si = si−1. 1 1 Since oi is determined by si and oi−1, it will then follow that the map π : S → RP deﬁned by

π(p) = αi(p), for p ∈ [qi, qi+1] r is a C ramified covering map over ∞ with signature (s1, . . . , sd, o1, . . . , od). −1 −1 r Recall from Section 3 that hi(p) = H(p) , where H :[qi, α (0)) → [0, ∞) is a C diffeomorphism such that 1 l −1 1 Hf H (x) = ( ) si x nl and l −1 x Hg H (x) = 1 . (1 + xsi ) si ∗ −1 ∗ −1 r Similarly, hi−1(p) = H∗(p) , where H :(αi−1(0), qi] → (−∞, 0] is a C diffeomorphism such that 1 ∗ l ∗−1 1 s∗ H f H (x) = ( ) i−1 x nl and ∗ l ∗−1 x H g H (x) = 1 . s∗ s∗ (1 + x i−1 ) i−1

1 1 1 1 s∗ l 0 si i−1 ∗ So (f ) (qi) = ( nl ) = ( nl ) , and therefore si = si−1. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 43

r −1 −1 The map H extends to a C diﬀeomorphism H :(αi−1(0), αi (0)) → (−∞, ∞), 1 −1 1 l si ∗ such that Hf H (x) = ( l ) x. We will show that H| −1 = H . n (αi−1(0),qi]

l −1 −1 By Lemma 3.9, the restriction of the diffeomorphism Hg H to (αi−1(0), qi] is the ∂ si time-1 map of a semiflow φt which satisfies the equation ∂t φt(x) = aφt(x) for some a ∈ R. So  x 1 , if x ∈ (−∞, 0] l −1  (1+cxsi ) si Hg H (p) = x  1 , if x ∈ [0, ∞), (1+xsi ) si for some c ∈ R. The value of c depends on the choice of linearizing diffeomorphism l l −1 r for f | −1 . Since Hg H is C , and r > si, we must have c = 1 (otherwise the (αi−1(0),qi] ∗ −1 (si +1)th derivative would not be defined at 0), and therefore H = H on (αi−1(0), qi]. r −1 −1 It follows that hi is a C diffeomorphism on (αi−1(0), αi (0)).

It follows from Lemma 4.6 that the representation of BS(1, nl) generated by f l and gl is Cr conjugate to an element of V. In the remainder of this section, we will show that the diffeomorphisms f and g are Cr ramified lifts of the generators of the standard action of BS(1, n) on S1, hence the representation they generate is also Cr conjugate to an element of V. We begin with some lemmas about ramified lifts of flows on S1. Lemma 5.9. Let ϕ : S1 → S1 be an Cr flow with a fixed point at p, and let π : S1 → 1 r r S be a C ramified covering map over p. Let F = ϕc1(π, id) be the C ramified lift of the time-1 map ϕ1 with rotation number zero. Then F embeds as the time-1 map of r 1 a C flow ψ on S , and for all t ∈ R, ψt = ϕbt(π, id).

r Proof: By Lemma 4.2, given any t ∈ R there is a C diﬀeomorphism ψt = ϕbt(π, id) such that −1 (1) ψt(q) = q for all q ∈ π (p), and (2) the following diagram commutes:

ψt S1 - S1 π π ? ? ϕt S1 - S1

By Lemma 4.4, given any s, t ∈ R, ψt ◦ ψs = ψs+t = ψs ◦ ψt. Let X be the Cr−1 vector ﬁeld that generates ϕ, and let Xˆ be the lift of X under π. Locally, this vector ﬁeld takes the form: ˆ −1 X(q) = dπ(q)π X(πq). 44 LIZZIE BURSLEM AND AMIE WILKINSON

This vector field is clearly Cr on S1 \ π−1(p) and clearly generates the flow ψ on S1. It remains to show that X is a Cr−1 vector field on the set π−1(p). Writing the Taylor expansion of X in coordinates identifying p with 0 ∈ R, we have: m m+1 r−1 r−1 X(x) = amx + am+1x + ··· + ar−1x + o(x ), for some 1 ≤ m ≤ r − 1, a 6= 0. Since π is a Cr ramified cover over p, there exist coordinates k, h such that: kπh−1(x) = xs, for some s ≥ 1. In coordinates identifying q ∈ π−1(p) with 0 ∈ R, we therefore have: a xsm + a xm+1 + ··· + a xs(r−1) + o(xs(r−1)) Xˆ(x) = m s(m+1) r−1 sxs−1 a a a = m xs(m−1)+1 + s(m+1) xsm+1 + ··· + r−1 xs(r−2)+1 + o(xs(r−2)+1) s s s which is Cr−1, since s(r −2)+1 ≥ r −1. A similar argument shows that Xˆ is analytic if X and π are. This completes the proof.

1 1 r Lemma 5.10. Let F : S → S be the time-1 map of a C flow ψt, where r ≥ 2. Suppose that F is not r-flat, and τ(F ) = 0. If G is a Cr orientation preserving diffeomorphism such that FG = GF , and if τ(G) = 0, then G = ψt for some t ∈ R.

Proof: Since τ(F ) = 0 and F is not r-flat, F has a finite set of fixed points. Let q1 < . . . < qd be the elements of Fix(F ). If FG = GF , then G permutes the fixed points of F , and since G is orientation preserving and has rotation number zero,

G([qi, qi+1]) = [qi, qi+1] for all qi ∈ Fix(F ). By Lemma 3.6, on [qi, qi+1), G = ψti for some ti ∈ R, and on (qi, qi+1], G = ψsi for some si. Clearly, ti = si. So for 1 ≤ i ≤ d,

G|[qi,qi+1] = ψti , for some ti ∈ R. 0 1 If F (qi) 6= 1 for some qi ∈ Fix(F ), then since G is C at qi, it follows that ti = ti−1. 0 If F (qi) = 1, then in local coordinates in a neighborhood (qi − ε, qi + ε), identifying qi with 0, k k F (x) = ψ1(x) = x + ax + o(x ) for some a 6= 0 and k ≤ r. Therefore

k k G(x) = x + ti−1ax + o(x ) on (qi − ε, qi] k k = x + tiax + o(x ) on [qi, qi + ε) r Since k ≤ r and G is C , ti = ti−1. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 45

Corollary 5.11. Let π : S1 → S1 be a Cr ramified covering map over ∞, and let F = kˆ(π, id) be an orientation preserving π-ramified lift of an affine map k ∈ Aff(R), with τ(F ) = 0. Let s(π) = (s1, . . . , sd, o1, . . . , od), where si ≤ r − 1 for 1 ≤ i ≤ d. By r 1 1 r Lemma 5.9, F embeds as the time-1 map of a C flow φt. If H : S → S is a C orientation preserving diffeomorphism such that FH = HF , and if τ(H) = 0, then H = φt for some t ∈ R.

Proof: By Lemma 5.10, it is enough to show that F is not r-ﬂat. Let π−1(∞) = {q1, . . . , qd}, and suppose that k is a translation. In projective coordinates around ∞, x k(x) = 1 + cx

for some c ∈ R \{0}. For all 1 ≤ i ≤ d, we can identify a neighborhood of qi with the interval (−1, 1) (where the point qi is identiﬁed with 0), and write π(x) = xsi

for all x ∈ (−1, 1) and for some integer si ≤ r − 1. In these coordinates,

−1 x F (x) = πkπ (x) = 1 (1 + axsi ) si

0 (si+1) where the root is chosen so that F (0) = 1. It is easily checked that F (qi) 6= 0. So F is not r-ﬂat. If k is a dilation, then in projective coordinates around ∞, k(x) = nx for some 0 −1 n ∈ R. So F (qi) 6= 1 for all qi ∈ π (∞), and so F is not r-ﬂat.

Proposition 5.12. Let F ∈ Diff r(S1) be a diffeomorphism such that F l is orientation- preserving and τ(F l) = 0, for some l > 0. Suppose that F l = kbl(π, id) is a Cr ramified l lift of k 6= id, where k ∈ Aff+(R), and suppose that s(π) = (s(q1), . . . , s(qd), o1, . . . , od), where s(qi) ≤ r − 1 for 1 ≤ i ≤ d. Then either F is a ramified lift of k or F is a ramified lift of −k.

−1 Proof: Let ζ ∈ Dd be such that ζ(q) = F (q) for all q ∈ π (∞), and ζ(e) = F (e) for all e ∈ E(π). Lemma 5.13. ζ ∈ Stab # (s(π)). Dd

Proof: Given any q ∈ π−1(∞), there exist an interval [q, p) and Cr diﬀeomorphisms h1 :[q, p) → [0, ∞) and h2 :[F (q),F (p)) → [0, ∞) such that 1 l −1 l s s h1F h1 (x) = [k (x )] and 1 l −1 l t t h2F h2 (x) = [k (x )] , 46 LIZZIE BURSLEM AND AMIE WILKINSON

where s = s(q) and t = s(F (q)). We can assume that kl is a contraction on [0, ∞). (If −l −l l l not, then use k and F ). Since F |[q,p) is conjugate by F to F |[F (q),F (p)), Lemma 4.7 implies that s(q) = s(F (q)), and therefore ζ ∈ Stab# (s(π)). Dd By Lemma 5.13, either ζ(s(π)) = s(π), or ζ(s(π)) = I(s(π)). If ζ(s(π)) = s(π), then let α : S1 → S1 be the Cr ramified lift of the identity map defined by α = idb (π, ζ). By Lemma 4.4, α commutes with F l. So F α−1 commutes with F l, and by construction, −1 −1 1 −1 F α fixes every interval (qi, qi+1) ⊆ π (RP \{∞}). So by Lemma 5.10, F α = ψt for some t ∈ R. Lemma 5.9 implies that F α−1 is a ramified lift of the map k, and by Lemma 4.4, F = ψtα = bk(π, ζ). If ζ(s(π)) = I(s(π)), then we let α be the Cr ramified lift of the orientation reversing map −id : x → −x, defined by α = −did(π, ζ). As above, F α−1 is a ramified lift of the map k, and therefore F = ψtα = −ck(π, ζ).

Corollary 5.14. f and g are Cr ramiﬁed lifts under π of the generators of the standard action of BS(1, n) on S1.

l ω 1 Proof: The standard representation ρnl : BS(1, n ) → Diff (RP ) is analytically ω conjugate to the representation κ : BS(1, nl) → Diff (RP 1) with generators κ(al): x 7→ nlx and κ(bl): x 7→ x+l. So there is a Cr ramified covering map π : S1 → S1 over p such that f l = κ[(al)(π, id) and gl = κ[(bl)(π, id). By Proposition 5.12, either f is a ramified lift of ρn(a): x 7→ nx, or f is a ramified lift of −ρn(a): x 7→ −nx. Similarly, g is either a ramified lift of ρn(b): x 7→ x+1, or a ramified lift of −ρn(b): x 7→ −x−1. Since f and g satisfy the relation fgf −1 = gn, the maps that they are lifted from must also satisfy this relation. Given this requirement, the only possibility is that f is a π-ramified lift of ρn(a) and g is a π-ramified lift of ρn(b). Since the generators ρ(a) = f and ρ(b) = g of the representation ρ are ramified lifts under π of the generators ρn(a) and ρn(b), respectively, of ρn, it follows that, for every γ ∈ BS(1, n), there exists a unique h(γ) ∈ Dd (or in Cd if ρ is orientation-preserving) such that: ρ(γ) = ρ[n(γ)(π, h(γ)).

Since ρ(γ1γ2) = ρ(γ1)ρ(γ2), it follows that h : BS(1, n) → Dd (Cd) is a homomor-

phism. Finally, note that h must take values in StabDd (s) (or StabCd (s), if ρ is orientation-preserving). This concludes the proof of Theorems 0.1 and 0.3. Finally, we sketch the proof of Theorem 0.6. Sketch of proof of Theorem 0.6. Let ρ be a Cr representation of BS(1, n), with r ≥ 2, let f = ρ(a) and g = ρ(b). We may assume that f has rational rotation number. To simplify the proof, assume that both f and g have rotation number 0; GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 47 the full proof is not conceptually more difficult. Assume that g is not the identity map. Let J be a component of the complement of Fix(g). Using a distortion estimate and the group relation one shows that J must be fixed by f, as follows. Otherwise, the f-orbit of J must accumulate at both ends on a fixed point of f. The standard C2 distortion estimate shows that there is an M > 1 such that for all x, y in same component of the f-orbit of J, and for all k ∈ Z, 1 (f k)0(x) < | | < M. M (f k)0(y) But, for all k ∈ N, we have that f kgf −k = gnk . Hence, for all p ∈ J, we have:

k 0 k (f ) (g(y)) (gn )0(p) = g0(y) , (f k)0(y) where y = f −k(p). Note that y and g(y) lie in the same component f −k(J), and k g0(y) is uniformly bounded. This implies that for all p ∈ J and all k ∈ N,(gn )0(p) is bounded, so that g = id on J, a contradiction. So f fixes each component of the complement of Fix(g). Let J be such a component. 1 Since g has no fixed points on J, g embeds in a C flow gt, defined on J minus one of its endpoints, that is Cr in the interior of J (see, e.g. [ZL]). Furthermore, for all t, −1 fgtf = gnt (this follows from Kopell’s theorem). Fixing some point p in the interior of J, this flow defines a Cr diffeomorphism between the real line and the interior of J, sending t ∈ R to gt(p) ∈ J. Conjugating by this diffeomorphism, gt is sent to a translation by t, and f is sent to a diffeomorphism F satisfying F (x + t) = F (x) + tn, for all t, x ∈ R. But this means that F 0(x) = n for all x ∈ R. Up to an affine change of coordinates, g is conjugate on J to x 7→ x + 1 and f is conjugate to x 7→ nx.

6. Proof of Proposition 0.5 Let ρ : BS(1, n) → Diff ω(S1) be a π-ramified lift of the standard representation 1 1 r−1 −1 ρn with σ(ρ) = n , for some r ≥ 2. Let Q be the set of all points q ∈ π (∞) 1 1 r−1 satisfying s(q) = r; this set is nonempty since σ(ρ) = n . Lemma 4.1 implies r−1 that π is conjugate in a neighborhood of q to x 7→ oqx , where oq is either 1 or −1 and the conjugacy identifies q with 0. r−1+t2 1 1 For t ∈ (−1, 1) we deform π to obtain a C map πt : S → S with the following properties: −1 −1 • π0 = π and π (p) = πt (p), for all t; ∞ • πt| 1 −1 is a C covering map onto its image; S \πt (p) −1 • about each q ∈ π (∞) \Q, πt is locally conjugate to π; r−1+t2 • about each q ∈ Q, πt is locally conjugate to x 7→ oqx . 48 LIZZIE BURSLEM AND AMIE WILKINSON

A slight modiﬁcation of the proof of Proposition 4.5 also shows that ρn has a lift to a r r 1 C representation ρt : BS(1, n) → Diﬀ (S ) so that the following diagram commutes, for all γ ∈ BS(1, n):

ρt(γ) S1 - S1

πt πt ? ? ρn(γ) S1 - S1 (One merely needs to check that the integer j in the proof of Lemma 4.2 can be replaced by the real number r − 1 + t2). 1 1 r−1+t2 1 Notice that ρt has the property that σ(ρt) = n , so that ρs is not C conjugate to ρt unles s = t. One can further modify this construction by replacing the points of Q by intervals of length εt, extending ρt(b) isometrically across these inter- r vals, and extending ρt(a) in an arbitrary C fashion to these intervals. Since ρt(b) is r-flat (by Lemma 3.2) on Q for t 6= 0 and r − 1 flat for t = 0, the representation ρt r r−1 is C and varies C continuously in t if we choose εt → 0 as t → 0. In this way, one can create uncountably many deformations of ρ. (Note that, in essence, we have r deformed π to obtain a “broken C ramified cover” àla Theorem 0.6).

7. Proof of Theorem 0.9. Let G ⊂ Diff ∞(S1) be a subgroup without infinitely flat elements. We first show that if G contains a non-empty normal abelian subgroup N, then either Gl := {gl | g ∈ G} is abelian, for some integer l ≥ 1 or G is conjugate in Diff ∞(S1) to a subgroup of s ω Affc (R), for some signature vector s. In the case where G ⊂ Diff (S1), the conjugacy we construct will belong to Diff ω(S1). Recall that any solvable group contains a non-empty normal abelian subgroup, namely the terminal subgroup in its derived series. Proceeding inductively, we then show that if G contains a normal subgroup N, and s if either N l is abelian or N or conjugate to a subgroup of Affc (R), for some signature vector s, then either Gm is abelian for some m, or G is also conjugate to a subgroup s of Affc (R). This implies the result. Proposition 7.1. For r ∈ {∞, ω}, let G ⊂ Diff r(S1) be a subgroup without infinitely flat elements. If Gl is not abelian, for any l ≥ 1, and if N < G is a non-empty normal r s abelian subgroup, then G is conjugate in Diff (S1) to a subgroup of Aff (R), for some signature vector s. Proof of Proposition 7.1: Let r ∈ {∞, ω}, and suppose that G ⊂ Diff r(S1) is a group without infinitely flat elements, and that Gm is not abelian, for any m ≥ 1. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 49

Let N < G be a non-empty normal abelian subgroup. We begin by showing that G has a ﬁnite set of global periodic points.

Lemma 7.2. There is an even integer l > 0, and a ﬁnite set {q1, . . . , qd}, with q1 < q2 < ··· < qd, such that: l l (1) for all f ∈ G, f is orientation-preserving, τ(f ) = 0 and f{q1, . . . , qd} = {q1, . . . , qd}; l (2) for all g ∈ N, Fix(g ) = {q1, . . . , qd}.

Proof: If τ(g) is irrational, for some orientation-preserving g ∈ N, then the elements of N are simultaneously conjugate to rotations. But, since N is normal in G, this implies that the elements of G are simultanously conjugate to rotations, which implies that G is abelian, a contradiction. Hence τ(g) ∈ Q/Z, for every g ∈ N. Fix such a g. Note that the set of periodic points of g is finite; – if Fix(gl) is infinite, for some integer l 6= 0, then there is a l l point q ∈ Fix(g ) that is an accumulation point for a sequence {qi} ⊂ Fix(g ). But this implies that g is infinitely flat at p. So let {q1, . . . , qd} be the periodic points of l l l g. Then g is orientation-preserving, τ(g ) = 0 and Fix(g ) = {q1, . . . , qd}, if l = 2d. If h ∈ N is another element of N, then, since h commutes with g, it follows that l l h({q1, . . . , qd}) = {q1, . . . , qd}, and so τ(h ) = 0 and Fix(h ) = {q1, . . . , qd}. Finally, let f ∈ N \G. Then for each g ∈ N, there exists a g ∈ N such that fgf −1 = l l g. This implies that f(Fix(g )) = Fix(g ); that is, f({q1, . . . , qd}) = {q1, . . . , qd}. It l l follows that f is orientation-preserving, τ(f ) = 0 and f{q1, . . . , qd} = {q1, . . . , qd}.

Let {q1, . . . , qd} and l > 0 be given by the previous lemma, labelled so that ∞ ≤ q1 < q2 < ··· < qd < ∞. We will begin by working with the group Gl = {hl : h ∈ G}. Note that every g ∈ Gl is orientation-preserving and ﬁxes every point in the set {q1, . . . , qd}. Throughout this section, we will be working on the intervals (qi, qi+1), and we will assume that qd+1 = q1. Lemma 7.3. Gl contains a non-trivial, inﬁnite, normal abelian subgroup.

Proof: Let N l = {hl : h ∈ N}. It is not hard to show that N l is abelian, and normal in Gl. Moreover, N l 6= Gl, since Gl is not abelian. If N l is a ﬁnite group, then there is some integer k such that gk = id, for all g ∈ N. So N is conjugate to a group of rotations. But since N is normal in G, it follows that G is conjugate to a group of rotations, and hence abelian, a contradiction. 50 LIZZIE BURSLEM AND AMIE WILKINSON

Let M be a maximal normal abelian subgroup of Gl. For the rest of the proof, ﬁx a non-trivial element g ∈ M. Lemma 7.4. Let C(g) = {f ∈ Gl | gf = fg}. Then C(g) 6= Gl.

Proof: A proof of this lemma is essentially contained in [FS]. This lemma is implied by the following theorem, which is classical. Theorem 7.5. (H¨older’s Theorem) If a group H of homomorphisms acts freely on R, then H is abelian.

If f ∈ C(g), then Fix(f) = Fix(g). So on every interval (qi, qi+1), 1 ≤ i ≤ d, no element of C(g) has a ﬁxed point. By Theorem 7.5, the restriction of the action of C(g) to each interval (qi, qi+1) is abelian. Since f(qi) = qi for all f ∈ C(g) and for all l l l qi ∈ Fix(g), C(g) is an abelian subset of G . But G is not abelian, so C(g) 6= G .

Lemma 7.6. Let f ∈ Gl \C(g). Then for each 1 ≤ i ≤ d, there is a positive r 1 real number λi and a C diﬀeomorphism αi :(qi, qi+1) → RP with the following properties: (1) There exist Cr diﬀeomorphisms −1 hi :[qi, αi (0)) → [−∞, 0) and ∗ −1 hi :(αi (0), qi+1] → (0, ∞], ∗ and integers si, si ∈ {1, . . . , r − 1}, oi ∈ {±1}, such that: ( si −1 oihi(p) , if p ∈ (qi, αi (0)) αi(p) = ∗ ∗ si −1 oihi (p) , if p ∈ (αi (0), qi+1), (2) αg(p) = α(p) + 1 (3)

αf(p) = λiα(p).

Remark: To ensure that the conditions λi > 0 (as opposed to λi 6= 0) hold in Lemma 7.6, it is necessary that we chose l to be even. Proof: We use the following fact, proved by Takens: Theorem 7.7. ([Ta], Theorem 4) Let h : [0, 1) → [0, 1) be a C∞ diffeomorphism with unique fixed point 0 ∈ [0, 1). If h is not infinitely flat, then there exists a unique C∞ vector field X on [0, 1) such that h = ϕ1, where ϕt is the flow generated by X. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 51

(i) For each 1 ≤ i ≤ d, Let ϕt :[qi, qi+1) → [qi, qi+1) be the ﬂow given by this (i) (i) theorem with ϕ1 = g|[qi,qi+1), and let ψt :(qi, qi+1] → (qi, qi+1] be the ﬂow with (i) l −1 ψ1 = g|(qi,qi+1]. If f ∈ G \C(g), then since g ∈ M, fgf ∈ M, and therefore fgf −1 ∈ C(g). By Lemma 3.6, for 1 ≤ i ≤ d, (1) fgf −1 = ϕ(i) on [q , q ), for some λ ∈ \{0}, λ 6= 1, and λi i i+1 i R i −1 (i) (2) fgf = ψµi on (qi, qi+1], for some µi ∈ R \{0}, µi 6= 1. Note that, because l is even, the numbers λi and µi must be positive, for all i. So assumption D. of Section 3 holds in each interval [qi, qi+1) and (qi, qi+1]. of each q ∈ {q , . . . , q }. Moreover, since ϕ(i) and ψ(i) coincide on (q , q ), it is not hard to i 1 d λi µi i i+1 see that we must have λi = µi. Now the result follows from Proposition 3.1, as in the proof of Lemma 5.6.

Corollary 7.8. For every f ∈ Gl \C(g), there is a positive real number λ = λ(f) 6= 1 such that λi = λ, for all 1 ≤ i ≤ d, where λi is given by Lemma 7.6. For every i, ∗ si = si−1, where addition is mod d.

∗ Proof: For 1 ≤ i ≤ d, let hi and hi−1 be given by Lemma 7.6. As in the proof of Corollary 5.8, we can show that the map ( −1 −1 , if p ∈ [qi, α (0)) hi(p)) i H(p) = −1 −1 ∗ , if p ∈ (αi−1(0), qi], hi−1(p) is a Cr diﬀeomorphism onto (−∞, ∞), such that

 1 1 ( ) si x, if x ∈ [0, ∞) −1 λi HfH (x) = 1 1 s∗ ( ) i−1 x, if x ∈ (−∞, 0].  λi−1

1 1 s∗ si i−1 −1 −1 −1 λi So λi = λi−1 . Let F = HfH , and let G = HgH . Since F GF = G on [0, ∞), and since G is not inﬁnitely ﬂat, Lemma 3.2 implies that (j) G (0) = 0, for 2 ≤ j ≤ si

6= 0, for j = si + 1. Similarly, since F GF −1 = Gλi−1 on (−∞, 0], (j) ∗ G (0) = 0, for 2 ≤ j ≤ si−1 ∗ 6= 0, for j = si−1 + 1. ∗ Therefore si = si−1, and so it follows that λi = λi−1. The proof of the next corollary is identical to the proof of Corollary 5.8. 52 LIZZIE BURSLEM AND AMIE WILKINSON

l 1 Corollary 7.9. For every f ∈ G \C(g), the map π0 : S \{q1, . . . , qd} → R deﬁned by: π0(p) = αi(p), for p ∈ (qi, qi+1) r 1 1 extends to a C ramiﬁed cover π : S → RP with signature s = (m1, . . . md, o1, . . . , od). The following diagrams commute: f g S1 - S1 S1 - S1 π π π π ? ? ? ? x 7→ λ(f)x x 7→ x + 1 S1 - S1 S1 - S1

r Corollary 7.10. g embeds in a unique C flow ϕt, with g = ϕ1. The elements of C(g) belong in the flow for g and, for each f ∈ Gl \C(g), lie in the ramified lift under πf of the translation group {x 7→ x + β | β ∈ R}. That is, for any h ∈ C(g), there exist real numbers β, t such that the following diagram commutes: h = ϕt S1 - S1

πf πf ? ? 1 x 7→ x + β - 1 RP RP Proof: This corollary follows directly from Corollary 7.9, Lemma 5.9 and Lemma 5.10.

l Lemma 7.11. For any f1, f2 ∈ G \C(g), there exists a real number γ such that the following diagram commutes:

f2 S1 - S1

πf1 πf1 ? ? x 7→ λ(f )x + γ 1 2 - 1 RP RP Proof: The proof is expressed in a series of commutative diagrams. Lemma 7.12. There exists an α ∈ R such that the following diagram commutes: ϕα S1 - S1

πf1 πf2 ? ? 1 id - 1 RP RP GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 53

Proof: The following diagram shows that if idb is the lift of the identity map (with τ(idb ) = 0) in the diagram above, then idb ◦ g = g ◦bid: g S1 - S1

π π idb f2 idb f2

? ? 1 x 7→ x + 1 - 1 RP RP g 1 - 1 S S

id id π π f1 f1

? ? 1 x 7→ x + 1 - 1 RP RP

Since g embeds in a flow ϕt that is a ramified lift of an affine flow, it follows from Corollary 7.10 that there exists an α such that idb = ϕα.

Lemma 7.13. For every t0 ∈ R there exists γ ∈ R such that the following diagram commutes: ϕt S1 0 - S1

πf1 πf2 ? ? 1 x 7→ x − γ - 1 RP RP Proof: Let α be given by the previous lemma. Let γ be the real number such that

ϕt0−α is a πf1 - ramiﬁed lift of x 7→ x − γ. The proof follows from the following diagram:

ϕt0−α idb = ϕα S1 - S1 - S1

πf1 πf1 πf2 ? ? ? 1 x 7→ x − γ- 1 id - 1 RP RP RP

The composition of the maps on the top row is ϕα ◦ ϕt0−α = ϕt0 . The composition of the maps on the bottom row is x 7→ x − γ.

−1 Lemma 7.14. For all t ∈ R, f2ϕtf2 = ϕλ(f2)t. 54 LIZZIE BURSLEM AND AMIE WILKINSON

Proof: The proof follows from the following diagram:

−1 f ϕt f2 S1 2 - S1 - S1 - S1

πf2 πf2 πf2 πf2 ? 1 ? ? ? x 7→ λ(f ) x x 7→ λ(f )x 1 2 - 1 x 7→ x + t- 1 2 - 1 RP RP RP RP The composition of the maps on the bottom row gives x 7→ x + λ(f2)t. By unique- −1 ness (Lemma 5.9), f2ϕtf2 = ϕλ(f2)t for all t ∈ R.

Let α be given by Lemma 7.12, and let γ be given by Lemma 7.13, with t0 = λ(f2)α. From the following diagram: ϕλα S1 - S1

πf πf f2 1 f2 2

? ? 1 x 7→ x − γ - 1 RP RP id = ϕ 1 b α - 1 S S

F π π x 7→ λ(f )x f1 f2 2

? ? 1 id - 1 RP RP it follows that F (x) = λ(f2)x + γ, completing the proof of Lemma 7.11.

Proposition 7.15. Fix f ∈ Gl \C(g). Then for each h ∈ G, there exists F ∈ Aﬀ(R) such that the following diagram commutes: h S1 - S1

πf πf ? ? F S1 - S1

Proof: By Corollary 7.10 and Lemma 7.11, we have that for each h ∈ G, there exists l l k ∈ Aff+(R), so that h is a πf -ramified lift of k . Therefore, by Proposition 5.12, h is a πf -ramified lift of either k or −k. GLOBAL RIGIDITY OF SOLVABLE GROUP ACTIONS ON S1 55

By Lemma 4.6, this completes the proof of Proposition 7.1.

Proposition 7.16. For r ∈ {∞, ω}, let G ⊂ Diff r(S1) be a subset without infinitely flat elements. Suppose that G contains a non-empty normal subgroup N, and that either (1) N l is abelian, for some l ≥ 1 or r s (2) N is conjugate in Diff (S1) to a subgroup of Aff (R), for some signature vector s ∈ S. Then either Gm is abelian, for some m ≥ 1 or G is also conjugate in Diff r(S1) to a s subgroup of Aff (R) Proof: If N l is abelian, for some integer l > 0, then G contains a normal abelian subgroup, namely N l. Moreover, if N l is a finite set, then G is abelian (see Lemma 7.3). If not, then by Proposition 7.1, either Gm is abelian, for some m ≥ 1 or there exists s a signature vector s ∈ S such that G is Cr conjugate to a subgroup of Aff (R). Case (2) follows from Case (1): if N is conjugate in Diff r(S1) to a subgroup of s Affc (R) for some s ∈ S, then G contains a normal sovable subgroup. But then G must contain a normal abelian subgroup, namely the derived group A for N, which is characteristic. To complete the proof of Theorem 0.9, we see that, by induction, if G is eventually solvable, then either Gm is abelian, for some m ≥ 1, so G is virtually abelian, or G is r s conjugate in Diff (S1) to a subgroup of Affc (R), for some s ∈ S. Acknowledgments Many useful conversations with Gautam Bharali, Christian Bonatti, Keith Burns, Matthew Emerton, Benson Farb, Giovanni Forni, John Franks, Ralf Spatzier, Jared Wunsch and Eric Zaslow are gratefully acknowledged. We thank AndrésNavas for corrections to an earlier version of this work, and we thank Navas and Etienne Ghys for pointing out several references to us. Finally, we thank Benson Farb for reminding us that the BS groups are often interesting. The second author was supported by an NSF grant.

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Lizzie Burslem ([email protected]) Department of Mathematics, University of Michigan 2074 East Hall, Ann Arbor, MI 48109-1109 USA Amie Wilkinson ([email protected]) Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730 USA