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Stable Commutator Length? Danny Calegari WHAT IS... ? Stable Commutator Length? Danny Calegari Kronecker allegedly once said, “God created the a surface (i.e., the group of self-homeomorphisms natural numbers; all the rest is the work of man.” of a surface, up to isotopy). But to a topologist, the natural numbers are just a Calculating commutator length (even in finite tool for classifying orientable surfaces, by counting groups!) is notoriously difficult. A famous conjec- the number of handles (or genus). ture of Ore from 1951, whose proof was announced Commutator length is the algebraic analogue only very recently, says that in a finite, non-cyclic of “number of handles” in group theory. If G is simple group, cl = 1 for all nonzero elements. So a group, and a,b ∈ G, the commutator of a and instead, one can stabilize the problem. The stable b is the element aba−1b−1 ∈ G. The commutator commutator length of g, denoted scl(g), is the subgroup, denoted [G, G], is the group generated limit cl(gn) by all commutators, and for g ∈ [G, G], the com- scl(g) = lim →∞ mutator length of g, denoted cl(g), is the smallest n n Commutator length is subadditive in [G, G], so number of commutators in G whose product is this limit exists. equal to g. The size of [G, G] is one way of mea- Computing stable commutator length is still suring the extent to which the group G fails to difficult, but feasible in many cases. For instance, obey the commutative law ab = ba. If G is the there now exist fast algorithms to compute scl in fundamental group of a space X, and g ∈ G is free groups. Since every group is a quotient of a represented by a homologically trivial loop γ ⊂ X, free group, calculating scl on elements in a free the commutator length of g is the smallest genus group gives universal upper bounds on scl. of a surface that admits a map to X in such a way Figure 1 plots values of scl by frequency on that the boundary of the surface maps to γ. 64,010 random elements of word length 32 in Estimating minimal genus is important in many a free group on two generators. For simplicity, areas of low-dimensional topology. A knot K in the we restrict attention to a subclass of elements 3-sphere bounds an orientable surface (in its com- for which computation is particularly tractable, plement) of some genus, called a Seifert surface. namely those represented by alternating words. The least such genus is equal to the commutator Some conspicuous features of this plot include length of the longitude of the knot, a certain distin- the following: 3 − guished conjugacy class in the group π1(S K). (1) the existence of a spectral gap between 0 and As another example, given a 3-manifold M, one 0.5, and another gap immediately above 0.5 can try to find the “simplest” 4-manifold W that (2) the non-discrete nature of the set of values bounds it. If M is a certain kind of 3-manifold—for attained instance, a surface bundle over a circle—one can (3) the relative abundance of elements for which ask for W to bea surfacebundle overa surface,and ∈ 1 Z ∈ 1 Z scl 2 , and (to a lesser extent) 6 and so try to estimate (from below) the genus of the base. on to other denominators, revealing a “self- This is tantamount to calculating the commutator similarity” in the histogram, and a power length of an element in the mapping class group of law for the size of the “spikes” of the form freq(p/q) ∼ q−δ, reminiscent of similar power Danny Calegari is professor of mathematics at Caltech. laws that arise in 1-dimensional dynamics (e.g. His email address is [email protected]. the phenomenon of Arnol’d tongues) 1100 Notices of the AMS Volume 55, Number 9 Figure 1. Values of scl on 64,010 alternating words of length 32. The horizontal axis is scl and the vertical axis is frequency. See [2] for a theoretical explanation of some of singular chain groups of a space, and the terms these features; also see the references of [2] and in the bar resolution of a group, are vector spaces [1] for further reading. with canonical bases, and one can use these A fact hinted at in this figure is that the values bases to make these vector spaces into normed of scl attained in a free group are all rational. This spaces. Bounded (co)-homology, introduced by is not a universal phenomenon: there are examples Gromov [3], arises when one studies the natural ∞ of finitely presented groups with irrational scl, but L1 and L norms on these vector spaces using the interestingly enough, no known examples where tools of homological algebra. One can interpret scl is irrational and algebraic. This rationality (or stable commutator length as the infimum of otherwise) has consequences in dynamics. For cer- the L1 norm (suitably normalized) on chains tain groups G of homeomorphisms of the circle, representing a certain (relative) class in group there is a natural central extension Gb of G with homology. 1 ∞ the property that rationality of stable commutator The unit balls in the L and L norms on finite-dimensional vector spaces are rational poly- length in Gb is directly related to the existence of hedra. Computing the L1 norm of a homology periodic orbits in S1 for elements g ∈ G. A simi- class is a kind of linear programming problem. In lar relationship between rationality and dynamics certain groups, computing scl reduces to a finite- exists for certain groups of symplectic matrices. dimensional linear programming problem, which One can learn a lot about an invariant by study- explains the rationality of scl in some cases. The ing when it vanishes. There are many important polyhedral nature of L1 norms is manifest in sever- classes of groups G for which scl is identically al closely related contexts. Most well-known is the zero on [G, G], including Thurston norm on the homology of a 3-manifold, (1) torsion groups which turns up again and again throughout low- (2) solvable groups, and more generally, amenable dimensional topology, in the theories of taut groups foliations, symplectic 4-manifolds, quasigeodesic Z ≥ (3) SL(n, ) for n 3, and many other lattices flows, Heegaard Floer homology, and so on. (uniform and nonuniform) in higher rank Lie Thus stable commutator length gives insight in- groups to bounded (co-)homology of groups and spaces, (4) groups of piecewise-linear homeomorphismsof and conversely. [0, 1]; Thompson’s group of piecewise dyadic rational linear homeomorphisms of the circle Further Reading On the other hand, there are many other class- [1] C. Bavard, Longeur stable des commutateurs, es of groups for which scl is nonzero on typical L’Enseign. Math. 37 (1991), 109–150. elements, including [2] D. Calegari, scl, to be published in the MSJ (1) free groups, hyperbolic groups monograph series; available from the author’s (2) mapping class groups website. (3) groups of area-preserving diffeomorphisms of [3] M. Gromov, Volume and bounded cohomology, 56 surfaces IHES Publ. Math. (1982), 5–99. The problem of computing scl can be recast in homological terms, by counting triangles (or, formally, 2-chains) instead of genus. The (real) October 2008 Notices of the AMS 1101.
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