Failure of Separation by Quasi-homomorphisms in Mapping Class Groups

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conjugate to their inverses cannot be separated by quasi-homomorphisms. This note was motivated by the work of Polterovich and Rudnick [12, 13], who used the geometry of the hyperbolic plane to show that on SL2(Z) quasi-homomorphisms exist in abundance and have interesting properties. One of their results is the following: Theorem 1 ([12]). Let g ∈ SL2(Z) be an element of infinite order not conjugate to its inverse. Then there exists a homogeneous quasi-homomorphism φ : SL2(Z) → R with φ(g) 6= 0. Since homogeneous quasi-homomorphisms satisfy φ(g) = −φ(g) and are constant on conjugacy classes, the assumption of the theorem is clearly necessary. In view of Brooks’s classical constructions of quasi-homomorphisms, this result is not suprising. In fact, Polterovich and Rudnick [12] pointed out that such a result essentially holds in much greater generality, because for all non-elementary Gromov hyperbolic groups it can be deduced from the work of Epstein and Fujiwara [5]. In [13], Polterovich and Rudnick generalized Theorem 1 to the following “separation theorem”: Theorem 2 ([13]). Let g ∈ SL2(Z) be a primitive element of infinite order not conjugate to its inverse, and g1, . . . , gn ∈ SL2(Z) any finite number of elements not conjugate to any power of g. Then there exists a homogeneous quasi-homomorphism φ : SL2(Z) → R with φ(g) 6= 0 and φ(g1) = . . . = φ(gn) = 0. Thinking of SL2(Z) as the mapping class group of the two-torus, one naturally wonders whether Theorems 1 and 2 can be generalized to the mapping class groups of higher-genus surfaces. Several years ago we proved that there are non-trivial homogeneous quasi-homomorphisms on mapping class groups [4]. Bestvina and Fujiwara [1] then showed that the space of such quasihomomorphisms is infinite-dimensional, and Polterovich asked whether it might be possible to prove a separation theorem in the spirit of Theorem 2 for mapping class groups. On the one hand, mapping class groups are perfect if the genus of the underlying surface is at least three [14], so they certainly have no homomorphisms to Abelian groups. On the other hand, they are residually Date: June 2, 2006; MSC 2000: primary 20F65, secondary 20F12, 20F69, 57M07. The second author would like to thank L. Polterovich for a conversation raising the question whether a separation theorem for mapping class groups of higher genus surfaces holds, and K. Fujiwara and J. McCarthy for useful comments. Support from the Deutsche Forschungsgemeinschaft and from JSPS Grant 18540083 is gratefully acknowledged. 1The reader can consult [8] for background on quasi-homomorphisms. 1 2 H. ENDO AND D. KOTSCHICK finite [7], thus their elements can be separated by homomorphisms to finite groups, and by linear representations. It is our purpose here to show that elements in mapping class groups cannot be separated by quasi-homomorphisms, by showing that the analogue of Theorem 1 and, a fortiori, the analogue of Theorem 2 fail for mapping class groups of surfaces of genus ≥ 2. We shall prove the following: Theorem 3. For every closed oriented surface of genus at least 2 there exist primitive elements g of infinite order in its mapping class group of orientation-preserving diffeomorphisms such that g is not conjugate to g for all k 6= 0, but all powers of g are products of some fixed number of torsion elements. It follows from the boundedness of the torsion lengths t(g) that the stable torsion length ||g||T = lim n→∞ t(g)