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A finitely generated, locally indicable group with no faithful action by C^1 diffeomorphisms of the interval

According to Thurston's stability theorem, every group of C^1 diffeomorphisms of the closed interval is locally indicable (.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the semi-direct product between F_2 an Z^2, although locally indicable, does not embed into Diff_+^1 (]0,1[). (Here F_2 is any free subgroup of SL(2,Z), and its action on Z^2 is the projective one.) Moreover, we show that for every non-solvable subgroup G of SL(2,Z), the semi-direct product between G and Z^2 does not embed into Diff^1_+(S^1).


Introduction
Without any doubt, one of the most striking results about groups of diffeomorphisms is Thurston's stability theorem [19]. In the 1-dimensional context, this theorem establishes that Diff 1 + ([0, 1[) is locally indicable, that is, each of its finitely generated subgroups factors through Z. In the language of the theory of orderable groups, this is equivalent to saying that Diff 1 + ([0, 1[) is C-orderable (see for example [15]). This is essentially the only known algebraic obstruction for embedding an abstract left-orderable group into Diff 1 + ([0, 1[). A good discussion on dynamical obstructions for C 1 smoothability of continuous actions on the interval appears in D. Calegari's nice work [3]. Most of them concern resilient orbits. Indeed, as was cleverly noticed by C. Bonatti, S. Crovisier, and A. Wilkinson, for groups of C 1 diffeomorphisms of the interval, there cannot be a central element without interior fixed points in the presence of resilient orbits [14,Proposition 4.2.25]. In the opposite direction, topologically transversal resilient orbits must appear when the topological entropy of the action is positive [11], or when some sub-pseudogroup acts without invariant probability measure [8].
A new obstruction which does not involve resilient orbits is also given in [3]. Nevertheless, these four conditions do not seem to complete the list of all possible dynamical obstructions. For instance, none of them seems to apply to groups of piecewise affine homeomorphisms, though 'in general' the corresponding actions should be non C 1 smoothable...
Giving a pure algebraic equivalent condition for the existence of a group embedding into Diff 1 + ([0, 1[) also seems very hard (see [9,16] for two interesting particular cases). In this work, we show that local indicability, although necessary, is not a sufficient condition, even for finitely generated groups. For this, we deal with a concrete example, namely the group F 2 ⋉ Z 2 (which is easily seen to be locally indicable), where F 2 is any free subgroup of SL(2, Z) whose action on Z 2 is the projective one.
The interest in considering the group F 2 ⋉ Z 2 comes from at least two sources. The first concerns the theory of orderable groups. Indeed, although C-orderable, this group admits no ordering with the stronger property of right-recurrence. This is cleverly noticed (and proved) in [21], where D. Witte-Morris shows that every finitely generated left-orderable amenable group admits a right-recurrent ordering, and hence every left-orderable amenable group is locally indicable. The second source of interest relies on Kazhdan's property (T). Indeed, from [17,Théorème A] it follows that, if the pair (G, H) has the relative property (T) and H is non-trivial and normal in G -as is the case of (F 2 ⋉ Z 2 , Z 2 ) when F 2 has finite index in SL(2, Z) -, then G does not embed into the group of C 1+α diffeomorphisms of the (closed) interval provided that α > 1/2. It is perhaps possible to use the L p extensions of the (relative) property (T) in [1] to conclude, by a similar method, that F 2 ⋉ Z 2 does not embed into Diff 1+α + ([0, 1]) for any α > 0. However, it does not seem plausible to deal with the C 1 case (even for the closed interval) using this kind of arguments. (Algebraic obstructions for passing from C 1 to C 1+α embeddings exist: see for example [16].) Our proof of Theorem A is strongly influenced by an argument due to J. Cantwell and L. Conlon (namely the proof of the second half of Theorem 2.1 in [6]). It relies on considerations about 'growth' of orbits (perhaps the right invariant to be considered should be the topological entropy associated to all possible actions on the interval). With slight modifications, these techniques also apply to the case of the circle. To motivate the theorem below, notice that SL(2, Z) ⋉ Z 2 embeds into Homeo + (S 1 ) (see §1).
Theorem B. For any non-solvable subgroup G of SL(2, Z), the group G ⋉ Z 2 does not embed into Diff 1 + (S 1 ).
This result provides a first obstruction for group embeddings into Diff 1 + (S 1 ) for subgroups of Homeo + (S 1 ) which does not rely on Thurston's stability theorem. This solves a question raised by J. Franks in a different manner from those of [5,18].
Unfortunately, our approach does not seem to be appropriate to deal with many other interesting groups which do act faithfully on the interval, as for example surface groups or general limit groups in the spirit of [2] (these groups are bi-orderable, which is stronger than being locally indicable). Another interesting question is the possibility of extending Theorem A to the group of germs of diffeomorphisms, where Thurston's theorem still applies (compare [16,Remark 2.13]). Finally, the investigation of similar phenomena related to the higher dimensional versions of Thurston's theorem also seems promising.

Existence of actions by homeomorphisms
As is well-known [4,14], there exist faithful group actions of SL(2, Z) ⋉ Z 2 by (orientation preserving) circle homeomorphisms. Indeed, let us consider the canonical action of SL(2, R) by real-analytic circle diffeomorphisms, and let p ∈ S 1 be a point whose stabilizer under this action is trivial. Replace each point f (p) of the orbit of p by an interval I f (where f ∈ SL(2, Z)) in such a way that the total sum of these intervals is finite. Doing this, we obtain a topological circle S 1 p provided with a faithful SL(2, Z)-action (we use affine transformations for extending the maps in SL(2, Z) to the intervals I f ).
Let I = I id be the interval corresponding to the point p, and let {ϕ t : t ∈ R} be a non-trivial topological flow on I. Choose any real numbers t 1 , t 2 which are linearly independent over the rationals, and let h 1 = ϕ t 1 and h 2 = ϕ t 2 . Extend h 1 , h 2 to S 1 p by letting For x in the complement of the union of the I f 's, we simply set h 1 (x) = h 2 (x) = x. The reader will easily check that the group generated by h 1 , h 2 ∼ Z 2 and the copy of SL(2, Z) acting on S 1 p is isomorphic to SL(2, Z) ⋉ Z 2 . If F 2 is a free subgroup of finite index in SL(2, Z), then F 2 ⋉ Z 2 is locally indicable. Thus it acts faithfully by homeomorphisms of the interval [15]. Although no such action arises as the restriction of the action constructed above, a faithful action may be constructed by following a similar procedure. For this, fix two (orientation preserving) homeomorphisms f 1 , f 2 of [0, 1] generating a free group admitting a free orbit. There are many ways to obtain these homeomorphisms. We may take for example a left-ordering on F 2 , and next consider its dynamical realization (see the comment after Example 2.6 in [15]). Another way is to use the fact that the group generated by x → x + 1 and x → x 3 is free [7]. Denoting by p ∈]0, 1[ a point whose stabilizer under the corresponding F 2 -action is trivial, and then proceeding as above, we obtain the desired faithful action of F 2 ⋉ Z 2 on the interval.

Preparation arguments: topological rigidity
Similarly, λs = br + ds. This shows that (r, s) is an eigenvector of f T with eigenvalue λ. Now let f 0 be a hyperbolic element in F 2 so that we have: (i) (r, s) is not an eigenvector of f T 0 , (ii) (r, s) is not orthogonal to an eigenvector of f −1 0 , (iii) neither (1, 0) nor (0, 1) are eigenvectors of f . By Claim 1, f 0 (I) is disjoint from I. Thus, changing f 0 by its inverse if necessary, we may suppose that f 0 (I) is to the left of I. Moreover, changing f 0 by f k 0 for k > 0 sufficiently large, we may suppose that the expanding eigenvalue λ of f −1 0 is greater than 2. For a certain vector (α, β) in the expanding direction of f −1 The first of these equalities easily yields Since (r, s) is not orthogonal to any eigenvector of f −1 0 , the value of t = αr + βs is nonzero. Assume that t is positive (the case where it is negative is similar). Replacing h 1 and h 2 respectively by h k 1 and h k 2 for k > 0 very large, we can ensure that t > 0 is sufficiently large so that we have: Moreover, replacing f 0 by f k 0 for k > 0 large enough, we may suppose that, for all n ∈ N, Let a (resp. b) be the fixed point of f 0 to the left (resp. to the right) of I. Since f 0 normalizes Z 2 , these points are also fixed by Z 2 . In §3.1, we will show that the dynamics of the subgroup H of  Proof. This follows as a direct application of the Müller-Tsuboi's conjugacy trick: it suffices to conjugate by a C ∞ diffeomorphism of [a, b[ whose germ at a is that of x → e −1/x 2 at the origin (see [13,20] for the details).
In what follows, we will consider the dynamics of f 0 and h 1 after the preceding conjugacy, so they are tangent to the identity at a. Remark 3.1. Since h 1 has a sequence of fixed points converging to a, its derivative at this point must equal 1 even for the original action; nevertheless, this was not necessarily the case for the original diffeomorphism f 0 .
Claim 3. For each k > 0, the intervals of the form where ε i ∈ {0, 1}, are two-by-two disjoint.

Proof.
Let Let i be the largest index for which ε i = ε ′ i , say ε i = 1 and ε ′ i = 0, and let Notice that each of the maps (f −j 0 h 1 f j 0 ) ε j either fixes all the points in I (when ε j = 0) or moves all of them to the right (when ε j = 1). In particular, W * moves the left endpoint u of J =]u, v[ to a point u * which coincides with or is to the right of f −i 0 h 1 f i 0 (u). By (3), this implies that On the other hand, W ′ * moves v to a point v ′ * which coincides with or is to the left of (f Since µ(J) < t and inequalities (2) and (4) show that v ′ * is to the left of u * . This implies that W * (J) and W ′ * (J) do not intersect, and hence W (J) ∩ W ′ (J) = ∅.
To conclude the proof of the fact that the action of F 2 ⋉ Z 2 is not by C 1 diffeomorphisms of [0, 1[, fix N ∈ N so that, for all x ∈ [a, b[ to the left of f N (I), Fix also a positive lower bound A < 1 for the derivative of f 0 and h 1 to the left of I. By opening brackets in the next expression, one easily checks that the length of each interval of the form |J|.
Since there are 2 k of these intervals, for some constant C > 0, this yields However, this is clearly impossible for a large k, thus completing the proof.
We close this Section by noticing that similar arguments to those above apply to actions by C 1 diffeomorphisms of the interval ]0, 1] instead of [0, 1[.

The case of the open interval
To prove Theorem A in the general case of the open interval, we would like to apply the arguments of the preceding Section. For this, we need to ensure that either a or b actually belongs to ]0, 1[. Indeed, if not, we are not allowed to use the procedure of the Claim 1.
Thus, we need to find a hyperbolic element f ∈ F 2 such that: (i) (r, s) is not an eigenvector of f T . (ii) (r, s) is not orthogonal to any eigenvector of f −1 , (iii) neither (1, 0) nor (0, 1) are eigenvectors of f , (iv) f has fixed points inside ]0, 1[. For this, we begin by noticing that our free group F 2 ⊂ SL(2, Z) must contain a free subgroup F on two generators whose non-trivial elements are hyperbolic and satisfy properties (i), (ii), and (iii) above. Indeed, this can be easily shown using a ping-pong type argument on RP 1 . Now F must contain non-trivial elements having fixed points in ]0, 1[; if not, the action of F on ]0, 1[ would be free, which is in contradiction with Hölder's theorem [10,14]. Therefore, any element f ∈ F having fixed points in ]0, 1[ satisfies (i), (ii), (iii), and (iv), and this concludes the proof of Theorem A.

The case of the circle
Let G be a non-solvable subgroup of SL(2, Z). To show Theorem B, we would again like to apply similar arguments to those of §3.1. However, there are certain technical issues that need a careful treatment.
First of all, notice that, a priori, an irreducible component I for the action of Z 2 is not necessarily an interval: it could coincide with the whole circle. We claim, however, that this cannot happen. Indeed, let (r ′ , s ′ ) be the point in T 2 whose coordinates are the rotation numbers of (1, 0) and (0, 1), respectively. Recall that the rotation number function is invariant under conjugacy. Moreover, its restriction to Z 2 is a group homomorphism into T 2 (see for example [10, §6.6] or [14, §2.2.2]). Since G normalizes Z 2 , for all f = a b c d ∈ G we have (modulo Z) r ′ = ρ(1, 0) = ρ(f (1, 0)) = ρ(a, c) = ar ′ + cs ′ and s ′ = br ′ + ds ′ .
This means that (r ′ , s ′ ) is a fixed point for the action of f T on T 2 . But since G is non-solvable, this cannot hold for every f ∈ G, thus showing that I does not coincide with S 1 . Now let µ be a Z 2 -invariant Radon measure on I. Let τ µ : Z 2 → R be the corresponding translation number homomorphism, and let r = τ µ ((1, 0)) and s = τ µ ((0, 1)). Analogously to the case of §3.2, we need to find a hyperbolic element f ∈ G so that the following conditions are fulfilled: (i) (r, s) is not an eigenvector of f T , (ii) (r, s) is not orthogonal to any eigenvector of f −1 , (iii) neither (1, 0) nor (0, 1) are eigenvectors of f , (iv) f has fixed points on the circle. To obtain the desired element, we need to consider two cases separately.
If G does not preserve any probability measure on S 1 , then Margulis' alternative [12] and its proof provide us with a free subgroup (in two generators) of G whose elements have fixed points. Clearly, some (actually, 'most') of these elements are hyperbolic and satisfy the conditions (i), (ii), and (iii) above.
If G preserves a probability measure on S 1 , then the rotation number function ρ : G → T 1 is a group homomorphism [10,14]. Therefore, the rotation number of all of the elements in [G, G] is zero, and hence these elements must have fixed points. Since G is non-solvable, [G, G] contains free subgroups, which allows arguing as in the preceding case.