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 an- alytic 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 differ- entiability 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}.