Irrational Stable Commutator Length in Finitely Presented Groups

We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse and others.


INTRODUCTION
Let G be a group, and let a be an element of the commutator subgroup, which we always denote by [G, G]. The commutator length of a, which we denote cl(a), is defined to be the minimum number of commutators whose product is equal to a. That is, cl(a) = min{n|a = [b 1 , c 1 ] · · · [b n , c n ]} cl(a) is a subadditive function, so the limit of cl(a n ) n , as n → ∞, exists. Commutator length and stable commutator length in groups have long been studied, often under the name of the genus problem. If G = π 1 (V ) for some aspherical space V , and γ is a loop in V representing the conjugacy class of an element a, then the commutator length of a is the minimal genus of a surface S for which there is a map f : S → V taking ∂S to γ. More generally, given a class H ∈ H 2 (V, γ), one can ask for the least genus immersed surface in the relative class H. Stabilizing, one obtains a norm on H 2 (V, γ). If G is finitely presented, H 2 (V, γ) is finitely dimensional, and one can try to minimize this stable norm on the subspace which is the preimage of the class of [γ] under the boundary map H 2 (V, γ) → H 1 (γ). This infimum is the stable commutator length of a.
M. Gromov (in [10] 6.C 2 ) asked the question of whether such a stable norm on H 2 (V, γ), or in our content, the stable commutator length in a finitely presented group, is always rational, or more generally, algebraic. The purpose of this note is to give simple (and even natural) examples which show that the stable commutator length in finitely presented groups can be transcendental.
We now state the contents of this note. In §2 we state the fundamental Duality Theorem of C. Bavard [2], which gives a precise relationship between stable commutator length and homogeneous quasimorphisms on groups. In §3 we give our examples and demonstrate that they have the desired properties, which are based on the work of M. Stein [14], D. Calegari [4] and I. Liousse [13]. The examples are central extensions of certain (finitely generated) groups of piecewise linear homeomorphisms of the circle. C. Bavard's duality theorem connects dynamics (rotation numbers, as studied by I. Liousse [13]) with stable commutator length. We include an appendix presenting basic properties of rotation numbers. They are used in §2 and §3.
1.1. Acknowledgements. The fact that these examples have irrational stable commutator length was first conjectured by D. Calegari, who also suggested the problem of trying to calculate stable commutator length exactly in certain finitely presented groups. I am very grateful to Danny Calegari for giving generous support and advice during the preparation of this work.

STABLE COMMUTATOR LENGTH AND QUASIMORPHISMS
for which there is a constant D(φ) ≥ 0 such that for any a, b ∈ G, we have an inequality In other words, a quasimorphism is like a homomorphism up to a bounded error. The least constant D(φ) with this property is called the defect of φ.

Definition 2.2.
A quasimorphism is homogeneous if it satisfies the additional property φ(a n ) = nφ(a) for all a ∈ G and n ∈ Z.
A homogeneous quasimorphism is a class function by its definition. Denote the vector space of all homogeneous quasimorphisms on G by Q(G). Example 2.3. Let Homeo + (S 1 ) = {f ∈ Homeo + (R)|f (x + 1) = f (x) + 1}. It's the set consisting of all possible lifts of elements in Homeo + (S 1 ) under the covering projection π : R → S 1 . We have the central extension: where Z is generated by the unit translation and p : Homeo + (S 1 ) → Homeo + (S 1 ) is the natural projection from its definition. For g ∈ Homeo + (S 1 ), define rot(g) = lim n→∞ g n (0) n With this definition, rot is a homogeneous quasimorphism with defect 1. (See appendix for a proof.) We have the following fundamental theorems of C. Bavard, which state the duality between scl and homogeneous quasimorphisms. See C. Bavard [2] or D. Calegari [5] for a reference. (2) There is an exact sequence Then the canonical map from bounded cohomology to ordinary cohomology Given a group G, if Q(G) is small enough, we can use (1) of Theorem 2.4 to determine scl. For the previous example G = Homeo + (S 1 ), we have dim R Q(G) = 1. Proposition 2.5 (J. Barge andÉ. Ghys [1]). rot : Homeo + (S 1 ) → R is the unique homogeneous quasimorphism which sends the unit translation to 1.
Proof: Suppose τ ∈ Q( Homeo + (S 1 )) is another such map , then we consider which is also a homogeneous quasimorphism, and since any homogeneous quasimorphism on abelian groups, especially Z ⊕ Z, must be a homomorphism, we have . It therefore induces a homogeneous quasimorphism on Homeo + (S 1 ), denote it still by rot − τ : cl is bounded ( [9]), so the induced map is bounded. Since it's homogeneous, it must be a zero map, i.e. rot = τ By Theorem 2.4 and Proposition 2.5 together, we have scl(g) = 1 2 rot(g), for any g in Homeo + (S 1 ). Suppose G is a subgroup of Homeo + (S 1 ) which is uniformly perfect.
LetG be the preimage of G in Homeo + (S 1 ). Then by the same argument, scl(a) = rot(a)/2D for any a inG, where D is the defect of rot restricted toG.
Our goal therefore is to find the subgroups of Homeo + (S 1 ) which are finitely presented and uniformly perfect, and contain elements with interesting rotation numbers. As a first example, consider Thompson's well-known group of dyadic piecewise linear homeomorphisms.
LetG consist of piecewise linear homeomorphisms f of R with the following properties: (1) For each point x i of discontinuity of the derivative of f (hereafter a "break point"), both x i and f (x i ) are dyadic rational numbers (i.e. of the form p2 q , p, q ∈ Z); (2) The derivatives of the restrictions of f to (x i , x i+1 ) are powers of 2 (i.e. of the form 2 m , m ∈ Z); (3) f preserves dyadic rational numbers and f (x + 1) = f (x) + 1. The elements ofG induce piecewise linear homeomorphisms of S 1 ≃ R/Z. The collection of these homeomorphisms is the Thompson group G ( [15], [6]). Thompson group is simple, F P ∞ and uniformly perfect( [6], [7] and [8]), so we have (The defect is still 1. See appendix.) About the rotation numbers of elements in the Thompson group, we have Theorem 2.6 (É. Ghys and V. Sergiescu [8]). rot(a) is rational for any a ∈G.
So scl takes only rational values on the groupG.

GENERALIZED THOMPSON GROUPS
Our definition of generalized Thompson groups is from [14] by M. Stein. Let P be a multiplicative subgroup of the positive real numbers and let A be a ZP -submodule of the reals with P A = A. Choose a number l ∈ A, l > 0. Let F (l, A, P ) be the group of piecewise linear homeomorphisms of [0, l] with finitely many break points, all in A, having slopes only in P . Similarly define T (l, A, P ) to be the group of piecewise linear homeomorphisms of [0, l]/{0, l} (the circle formed by identifying endpoints of the closed interval [0, l]) with finitely many break points in A and slopes in P , with the additional requirement that the homeomorphisms send A ∩ [0, l] to itself. In these notations, T (1, Z[ 1 2 ], 2 ) is the Thompson group. In our study of generalized Thompson groups, we always assume that P is generated by the set of positive integers {n 1 , n 2 , . . . , n k } and A = Z[ 1 n1 , 1 n2 , . . . , 1 n k ], here {n 1 , n 2 , . . . , n k } forms a basis for P . An important theorem in studying generalized Thompson groups is the following Bieri-Strebel criterion (See [14] appendix for a proof.). Theorem 3.1 (R. Bieri and R. Strebel [3]). Let a, c, a ′ , c ′ be elements of A with a < c and a ′ < c ′ . Then there exists f , a piecewise linear homeomorphism of R, with slopes in P and finitely many break points, Here IP * A is the submodule of A generated by elements of the form (1 − p)a, where a ∈ A and p ∈ P . Let d = gcd(n 1 − 1, . . . , n k − 1) and from now on, we assume that d = 1. In this case, IP * A = P (dZ) = P Z = A, so the Bieri-Strebel criterion from Thm 3.1 is vacuously satisfied.
Take an arbitrary f ∈ T = T (l, A, P ) with the assumptions above. Choose points a < b, c < d ∈ respectively. Construct g ∈ T as follows.  (For a proof, see M. Stein [14].) Let's further assume that the slope group P has rank 2, i.e. P = p, q , A = Z[ 1 p , 1 q ], p, q ∈ Z + and d = gcd(p − 1, q − 1) = 1. F p,q , T p,q are the corresponding groups. M. Stein, in [14], computed the homology groups of F p,q by using its action on a complex.  H 1 (F p,q ) is a free abelian group with rank 2(d + 1), where d = gcd(p − 1, q − 1).
By assumption d = 1, rk Z (H 1 (F p,q ) p,q can be written as a product of commutators of F p,q . So is g i , which is conjugate to h i in T p,q . So overall we proved that T p,q is perfect, i.e. T p,q = T ′ p,q .
Let's compute the set Q(T p,q ). For this purpose, we need the theorem of D. Calegari [4] about the scl of elements in subgroups of P L + (I).
Proof: By the same argument in Proposition 2.5 with Lemma 3.6 replacing the uniformly perfectness. By Bavard's Theorem 2.4, if gcd(p − 1, q − 1) = 1, then for all a inT p,q we have an equality scl(a) = rot(a)/2D. In the appendix we show that D = D(rot) = 1 on the groupsT p,q , and therefore this simplifies to scl(a) = 1 2 rot(a). So it remains to determine the rotation numbers of elements in generalized Thompson groups.
f lies in the class of the homeomorphisms in P L + (S 1 ), satisfying the property D [13]. I. Liousse showed that for any f ∈ P L + (S 1 ) with the property D, there exists h ∈ Homeo + (S 1 ) such that h −1 f h is a rotation, i.e. in SO(2). Thus we have a measure µ on S 1 that is invariant under f and furthermore log(Df )dµ = 0 here Df is the function of the derivative of f , which is a piecewise constant function, so to write down the left side of the equality, we only need to get the measures of the intervals between break points. This can be obtained by the assumed property D, without any knowledge of the conjugate function h. This equality gives a linear equation of the rotation number of f , so in this way, we can get the rotation number.

APPENDIX
In this appendix, we will justify the claims in previous sections on defects of rotation numbers as homogeneous quasimorphisms. We will use a lemma by C. Bavard on defect estimation. See C. Bavard [2] or D. Calegari [5] for a proof. Proof: Refer to [11] for basic properties of rotation numbers.
Proof: Take an arbitrary f in Homeo + (S 1 ) and any ǫ > 0. f is uniformly continuous, so we can choose 0 = x 0 < x 1 < . . . < x l < 1, x i 's are in A that is dense in S 1 = [0, 1]/{0 = 1}, such that |f (x i ) − f (x i−1 )| < ǫ 3 . Since A is dense in [0, 1], we can find y 0 < y 1 < . . . < y l < 1, y i 's are in A and |y i − f (x i )| < ǫ 3 . By the Bieri-Strebel criterion(Theorem 3.1), there exists g ∈ T such that g(x i ) = y i . And from the choice of x i 's and y i 's, it's easy to see that f − g C 0 < ǫ.
Thus we also have thatT ⊆ Homeo + (S 1 ) is dense. On the other hand, the map rot, thought of as a function from Homeo + (S 1 ) to R, is continuous in the C 0 topology [11]. So we have that the defect of rotation number D(rot), restricted toT , is also 1.