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?WHAT IS... a Quasi-morphism? D. Kotschick

As far as I know, the notion of a quasi-morphism n ·|ϕ(x)| = |ϕ(xn)| does not have much to do with theory. This n n ≤ lS (x ) · C(ϕ, S)+(lS (x ) − 1) · D(ϕ) . very natural idea underlies several interesting re- cent developments at the crossroads of algebra, If ϕ(x) =0 and ϕ is bounded on S, then one , geometry, and dynamics. Other, perhaps obtains a positive lower bound for the S-length n n better, names currently in use for the same concept lS (x ) of x , which is linear in n. Hence one obtains are quasi- and pseudo-character. a positive lower bound for the stable S-length n Let G be a . A f : G → R is called a ||x|| = lim lS (x ) of x: S →∞ n quasi-morphism if its deviation from being a n |ϕ(x)| homomorphism is bounded; in other words, there (1) ||x||S ≥ . exists a constant D(f ), called the defect of f, such C(ϕ, S)+D(ϕ) that There are many papers in algebra and topology |f (xy) − f (x) − f (y)|≤D(f ) discussing these kinds of estimates in the case when S is taken to be the of commutators in ∈ for all x, y G. The most obvious examples of G; see [1] and the papers cited there and [3] for quasi-morphisms are of course some more recent developments. (Note that gen- and arbitrary bounded maps. To avoid trivialities uine homomorphisms are useless for bounding associated with the latter and to make subsequent the commutator length, but quasi-morphisms are arguments neater, one usually passes to homoge- not.) Moreover, these estimates are useful for many neous quasi-morphisms. Every quasi-morphism other length problems, where S does not have to can be homogenized by defining be the set of commutators; and other algebraic f (xn) problems, which are not a priori length problems, ϕ(x) = lim . n→∞ n can be attacked using quasi-morphisms. These include the question whether G is boundedly gen- Then ϕ is again a quasi-morphism, is homoge- erated and questions about the width of subgroups. neous in the sense that ϕ(xn)=nϕ(x), and is Here are some examples of quasi-morphisms. constant on conjugacy classes. The first one is due to Brooks: It turns out that homogeneous quasi-morphisms Example 1. Let G = F = x, y be a free group on share enough properties with homomorphisms 2 two generators and w a cyclically reduced word in that they can be used in similar ways. For exam- these generators. Define ϕ (g) to be the number ple, let S ⊂ G be an arbitrary subset. The S-length w of nonoverlapping copies of w minus the number l (x) of x ∈ G is the minimal number of factors S of nonoverlapping copies of w −1 in g. This is a in a factorization of x into elements of S. For a quasi-morphism on F . For w = x or y one obtains homogeneous quasi-morphism ϕ: G → R set 2 the two homomorphisms generating the first C(ϕ, S) = sups∈S |ϕ(s)| . Then cohomology of F2, but for more complicated words these quasi-morphisms are not homomorphisms. D. Kotschick is professor of mathematics at the Ludwig- Maximilians-Universität München. His email address is Many authors have generalized this construction, [email protected]. among them Fujiwara and his collaborators. They

208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 have shown, for example, that the vector space of the disc is a quasi-morphism admitting many quasi-morphisms is infinite dimensional if G is a generalizations, some of which have been studied free group or any nonelementary word-hyperbolic recently by Gambaudo and Ghys. group, or if G admits a suitably weakly hyperbolic Many other interesting quasi-morphisms are action on a Gromov-hyperbolic space. The same turning up on groups of . For conclusion holds if G = AC B such that C has example, for the groups of closed index at least 2 in both A and B and has at least surfaces, one can implicitly obtain the existence of three double cosets in at least one of the factors, nontrivial quasi-morphisms using Seiberg–Witten or if G has infinitely many ends. theory on symplectic four-manifolds; cf. [3]. For Our second example is very classical. It shares, certain symplectomorphism groups Entov and and perhaps explains, the hyperbolic flavor of the Polterovich have constructed nontrivial quasi- first one. morphisms related to the Calabi homomorphism by using quantum cohomology. Z Example 2. Consider the action of G = SL2( ) by There is an important relationship between quasi- Möbius transformations of the upper half-plane. The morphisms and bounded cohomology in the sense translates of a piece of the unit circle connecting of Gromov [2]. The definitions show that the cobound- ∈ H2 ∈ H2 i to a primitive third root of unity ξ form ary of a homogeneous quasi-morphism is a bounded a tree, all of whose vertices are trivalent. The 2-cocycle on G and so represents a bounded degree H2 orientation of defines a cyclic ordering of the edges 2 cohomology . The of this class in the meeting at each vertex. There is a unique l with- usual group cohomology is trivial, as the cocycle is ∈ out backtracking from i to g(i) for any g G. Define a coboundary by definition, although it is not the ϕ(g) to be the number of left turns minus the num- coboundary of any bounded . Thus the vec- ber of right turns that one makes when travelling tor space of homogeneous quasi-morphisms mod- Z along l. This defines a quasi-morphism on SL2( ). ulo the space of homomorphisms to R is identified This is essentially the Rademacher φ-function, with the of the comparison map which can be defined purely arithmetically. It is between bounded and usual group cohomology in related to Dedekind sums, to the signature defects degree 2. of torus-bundles, to Maslov indices, and to eta- Recent progress in the theory of bounded coho- invariants of 3-manifolds and their adiabatic mology due to Burger and Monod in many situations limits. It admits many generalizations and varia- implies that the space of quasi-morphisms is triv- tions, some of which have turned up in work of ial, or at least finite dimensional, most notably for Barge, Gambaudo, and Ghys on lattices in higher rank groups. The tension between groups of surfaces and in work of Polterovich and these results and the existence theorems for (infi- Rudnick in dynamics. nitely many) quasi-morphisms mentioned above leads to interesting conclusions when comparing Example 3. The universal covering of the group various groups arising in geometry and dynamics of orientation-preserving homeomorphisms of S1 with algebraic groups. consists of continuous strictly monotonically Finally, it is sometimes useful to consider quasi- increasing functions f : R → R with the property morphisms with values in groups other than R. For f (x +1)=f (x)+1. For such an f the example, quasi-morphisms with values in Z play a role f n(x) − x in the classification of representations in the home- t(f ) = lim omorphism group of the circle. There is also an in- n→∞ n teresting construction of the real numbers as the exists and is independent of x. This is the trans- space of quasi-morphisms Z → Z modulo bounded lation number of f, defining a homogeneous maps, due to A´Campo. quasi-morphism. In his 1958 paper on flat connections in plane References bundles over surfaces, Milnor explicitly noted that, [1] C. BAVARD, Longeur stable des commutateurs, Enseign. Math. 37 (1991), 109–150. letting SL2(R) act on the circle by projective trans- formations, the translation number is a quasi- [2] M. GROMOV, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99. R morphism on SL2( ). The bound on Euler numbers [3] D. KOTSCHICK, Quasi-homomorphisms and stable of flat bundles that Milnor proved is obtained by lengths in mapping class groups, Proc. Amer. Math. combining two estimates: the lower bound (1) for Soc. (to appear). the (stable) commutator length given by this quasi-morphism and an upper bound for its defect. This was probably the first application of a quasi- morphism. Later Wood generalized the argument 1 to Homeo+(S ) . In a similar spirit, the Ruelle in- variant of area-preserving of

FEBRUARY 2004 NOTICES OF THE AMS 209