aa r X i v : . [ m a t h . G R ] N ov SCL, SAILS AND SURGERY
DANNY CALEGARI
This paper is dedicated to the memory of John Stallings.
Abstract.
We establish a close connection between stable commutator lengthin free groups and the geometry of sails (roughly, the boundary of the convexhull of the set of integer lattice points) in integral polyhedral cones. Thisconnection allows us to show that the scl norm is piecewise rational linearin free products of Abelian groups, and that it can be computed via integerprogramming. Furthermore, we show that the scl spectrum of nonabelian freegroups contains elements congruent to every rational number modulo Z , andcontains well-ordered sequences of values with ordinal type ω ω . Finally, westudy families of elements w ( p ) in free groups obtained by surgery on a fixedelement w in a free product of Abelian groups of higher rank, and show thatscl( w ( p )) → scl( w ) as p → ∞ . Introduction
Bounded cohomology, introduced by Gromov in [14], proposes to quantify homol-ogy theory, replacing groups and homomorphisms with Banach spaces and boundedlinear maps. In principle, the information contained in the bounded cohomologyof a space is incredibly rich and powerful; in practice, except in (virtually) trivialcases, this information has proved impossible to compute. Burger-Monod [6] wrotethe following in 2000:Although the theory of bounded cohomology has recently foundmany applications in various fields . . . for discrete groups it remainsscarcely accessible to computation. As a matter of fact, almost allknown results assert either a complete vanishing or yield intractableinfinite dimensional spaces.Perhaps the best known exceptions are Gromov’s theorem [14] that the norm ofthe fundamental class of a hyperbolic manifold is proportional to its volume, andGabai’s theorem [12] that the Gromov norm on H of an atoroidal 3-manifold isequal to the Thurston norm. Even these results only describe a finite dimensionalsliver of the (typically) uncountable dimensional bounded cohomology groups ofthe spaces in question.The case of 2-dimensional bounded cohomology is especially interesting, sinceit concerns extremal maps of surfaces into spaces . In virtually every category it isimportant to be able to construct and classify surfaces of least complexity mappingto a given target; we mention only minimal surface theory and Gromov-Wittentheory as prominent examples. In the topological category one wants to minimizethe genus of a surface mapping to some space subject to further constraints (e.g.that the image represent a given homology class, that it be π -injective, that it Date : 11/1/2010, Version 0.20. be a Heegaard surface, etc.) For many applications (e.g. inductive arguments) itis crucial to relativize this problem: given a space X and a (homologically trivial)loop γ in X , one wants to find a surface of least complexity (again, perhaps subjectto further constraints) mapping to X in such a way as to fill γ (i.e. γ becomesthe boundary of the surface). The problem of computing the genus of a knot(in a 3-manifold) is of this kind. On the algebraic side, the relevant (bounded)homological tool to describe complexity in this context is stable commutator length .In a group G , the commutator length cl( g ) of an element g is the least number ofcommutators whose product is g , and the stable commutator length scl( g ) is thelimit scl( g ) = lim n →∞ cl( g n ) /n . Here, until very recently, the landscape was evenmore barren: there were virtually no examples of groups or spaces in which stablecommutator length could be calculated exactly where it did not vanish identically([20] is an interesting exception).The paper [7] successfully showed how to compute stable commutator length in ahighly nontrivial example: that of free groups. The result is very interesting: stablecommutator length turns out to be rational , and for every rational 1-boundary, thereis a (possibly not unique) best surface which fills it rationally, in a precise sense.The case of a free group is important for several reasons:(1) Computing scl in free groups gives universal estimates for scl in arbitrarygroups(2) The category of surfaces and maps between them up to homotopy is afundamental mathematical object; studying scl in free and surface groupsgives a powerful new framework in which to explore this category(3) Free groups are the simplest examples of hyperbolic groups, and are a modelfor certain other families of groups (mapping class groups, Out( F n ), groupsof symplectomorphisms) that exhibit hyperbolic behaviorThe paper [7] gives an algorithm to compute scl on elements in a free group. Re-finements (see [8], Ch. 4) show how to modify this algorithm to make it polynomialtime in word length. Hence it has become possible to calculate (by computer) thevalue of scl for words of length ∼
60 in a free group on two generators. The utilityof this is to make it possible to perform experiments, which reveal the existenceof hitherto unsuspected phenomena in the scl spectrum of a free group. Thesephenomena suggest many new directions for research, some of which are pursuedin this paper.Figure 1 is a histogram of values of scl between 1 and on alternating words oflength 28 in the (commutator subgroup of the) free group on two generators. Herea word is alternating if the letters alternate between one of a ± and one of b ± .The most salient feature of this histogram is its self-similarity . Such self-similarity Figure 1.
Histogram of values of scl on alternating words oflength 28 in F . Heights of the bars at 1 and have been truncatedto fit in the Figure. CL, SAILS AND SURGERY 3 is indicative of a power law roughly of the form freq( p/ q ) ∼ q − δ for some δ whichin this case is about 2 . and “low” spike at ).Unfortunately, the algorithm developed in [8] is not adequate to explain thestructure evident in Figure 1. One reason is that this algorithm reduces the cal-culation of scl on a particular element of F to a linear programming problem, theparticulars of which depend in quite a dramatic way on the word in question. More-over, though the algorithm is polynomial time in word length, it is not polynomialtime in “log exponent word length”, i.e. the notation which abbreviates a word like aaaaaaaba − a − a − baab − b − to a ba − ba b − . This is especially vexing in viewof the fact that experiments suggest a rich structure for the values of scl on familiesof words which differ only in the values of their exponents. This is best illustratedwith an example. Example . In F with generators a, b , experiments suggest a formulascl( aba − b − ab − n a − b n ) = 1 − n − n ≥ Sails.
One surprising thing to come out of this paper is the discovery of a closeconnection between stable commutator length in free groups, and the geometry of sails in integral polyhedral cones. Given an integral polyhedral cone V the sail of V is the convex hull of D + V , where D is the set of integral lattice points incertain open faces of V (this is a generalization of the usual definition of a sail,in which one takes for D the set of all integer lattice points in V except for thevertex of the cone). Sails were introduced by Klein, in his attempt to generalizeto higher dimensions the theory of continued fractions. Given a cone V , define the Klein function κ to be the function on V , linear on rays, that is equal to 1 exactlyon the sail. It turns out that calculating scl on chains in free products of Abeliangroups reduces to the problem of maximizing a function − χ/ − χ/ Stallings. If A and B are groups, one can build a K ( A ∗ B,
1) by wedging a K ( A,
1) and a K ( B,
1) along a basepoint. Given a surface S and a map f : S → K ( A ∗ B,
1) one can try to simplify S and f in two complementary ways: either bysimplifying the part of S that maps to the two factors, or by simplifying the partthat maps to the basepoint. In a precise sense, the first strategy was pursued in [7]whereas the second strategy is pursued in this paper. DANNY CALEGARI
An interesting precursor of this latter approach is John Stallings’ last paper[18], which uses topological methods to factorize products of commutators in a freeproduct of groups into terms which are localized in the factors. It is a pleasureto acknowledge my own great intellectual debt to John, and it seems especiallyserendipitous to discover, in relatively unheralded work he did in the later part ofhis life, some beautiful new ideas which continue to inform and inspire.1.3.
Main results.
We now briefly describe the contents of the paper. In § § sails in integral polyhedralcones (this is explained in more detail in the sequel). As a consequence, we deriveour first main theorem: Rationality Theorem.
Let G = ∗ i A i be a free product of finitely many finitelygenerated free Abelian groups. Then scl is a piecewise rational linear function on B H ( G ) . Moreover, there is an algorithm to compute scl in any finite dimensionalrational subspace. In § B H ( F ). This calculation enables us to rigorously verify the existenceof certain phenomena in the scl spectrum that were suggested by experiments.Explicitly, this calculation allows us to prove: Denominator Theorem.
The image of a nonabelian free group of rank at least under scl in R / Z is precisely Q / Z . and Limit Theorem.
For each n , the image of the free group F n under scl contains awell-ordered sequence of values with ordinal type ω ⌊ n/ ⌋ . The image of F ∞ under scl contains a well-ordered sequence of values with ordinal type ω ω . Finally, we obtain a result that explains the existence of many limit points in thescl spectrum of free groups. Let A ∗ B and A ′ ∗ B be free products of free Abeliangroups. A line of surgeries is a family of surjective homomorphisms ρ p : A ∗ B → A ′ ∗ B determined by a linear family of surjective homomorphisms on the factors. Surgery Theorem.
Fix w ∈ B H ( A ∗ B ) and let ρ p : A ∗ B → A ′ ∗ B be a lineof surgeries, constant on the second factor, and surjective on the first factor with rank( A ′ ) = rank( A ) − . Define w ( p ) = ρ p ( w ) . Then lim p →∞ scl( w ( p )) = scl( w ) . Background
This section contains definitions and facts which will be used in the sequel. Abasic reference is [8].
CL, SAILS AND SURGERY 5
Definitions.
The following definition is standard; see [3].
Definition 2.1.
Let G be a group, and g ∈ [ G, G ]. The commutator length of g ,denoted cl( g ), is the smallest number of commutators in G whose product is g . The stable commutator length of g , denoted scl( g ), is the limitscl( g ) = lim n →∞ cl( g n ) n Commutator length and stable commutator length can be extended to finitelinear sums of groups elements as follows:
Definition 2.2.
Let G be a group, and g , g , · · · , g m elements of G whose productis in [ G, G ]. Definecl( g + · · · + g m ) = inf h i ∈ G cl( g h g h − · · · h m − g m h − m − )and scl( g + · · · + g m ) = lim n →∞ cl( g n + · · · + g nm ) n Remark . Note that scl will be finite if and only if the product of the g i is trivialin H ( G ; Q ).The function scl may be interpreted geometrically as follows. Definition 2.4.
Let S be a compact surface with components S i . Define χ − ( S ) = X i min(0 , χ ( S i ))where χ denotes Euler characteristic.In words, χ − ( S ) is the sum of the Euler characteristics over all components of S for which χ is non-positive. Definition 2.5.
Let g , g · · · g m ∈ G be given so that the product of the g i is trivialin H ( G ; Q ). Let X be a space with π ( X ) = G . For each i , Let γ i : S → X represent the conjugacy class of g i in G .Suppose S be a compact oriented surface. A map f : S → X for which there isa diagram ∂S i −−−−→ S ∂f y y f ` i S ` i γ i −−−−→ X where i : ∂S → S is the inclusion map, and ∂f ∗ [ ∂S ] = n [ ` i S ] in H for someinteger n ( S ) ≥
0, is said to be admissible , of degree n ( S ). Remark . The sign of n ( S ) is changed by orienting S oppositely.The geometric definition of scl asks to minimize the ratio of − χ − to degree overall admissible surfaces. Proposition 2.7.
With notation as above, there is an equality scl G ( g + · · · + g m ) = inf S − χ − ( S )2 n ( S ) over all admissible compact oriented surfaces S . DANNY CALEGARI
See [8], Prop. 2.68 for a proof.If f : S → X is admissible, then S and therefore ∂S are oriented. Some compo-nents of ∂S might map by ∂f to ` i S with zero or even negative degree. Boundarycomponents mapping with opposite degree to the same circle can be glued up afterpassing to a suitable cover. Hence the following proposition can be proved. Proposition 2.8. If f : S → X is admissible, there is f ′ : S ′ → X admis-sible such that ∂f ′ : ∂S ′ → ` i S has positive degree on every component, and − χ − ( S ′ ) / n ( S ′ ) ≤ − χ − ( S ) / n ( S ) . See [8], Cor. 4.29 for a proof. An admissible surface with the property discussedabove is said to be positive . In the sequel all the admissible surfaces we discuss willbe positive, even if we do not explicitly say so.
Definition 2.9.
An admissible surface S realizing the infimum for g , · · · , g m (i.e.for which scl( g + · · · + g m ) = − χ − ( S ) / n ( S )) is said to be extremal . Remark . Extremal surfaces are π -injective, and have other useful properties.Given a group G , let ( C ∗ ( G ; R ) , ∂ ) denote the complex of real group chains, whosehomology is the real (group) homology of G (see Mac Lane [17], Ch. IV, § B n ( G ; R ) denote the subspace of real group n -boundaries. By Definition 2.2, we canthink of scl as a function on the set of integral group 1-boundaries. This functionis linear on rays and subadditive, and therefore admits a unique continuous linearextension to B ( G ).Let H ( G ) (for homogeneous ) denote the subspace of B ( G ) spanned by chains ofthe form g n − ng and g − hgh − for all g, h ∈ G and n ∈ Z . Then scl vanishes on H and descends to a pseudo-norm on B ( G ) /H ( G ). For general G this pseudo-normis not a true norm, but in many cases of interest (e.g. for fundamental groups ofhyperbolic manifolds), scl is a genuine norm on B ( G ) /H ( G ). See [8], § B H ( G ) or just B H if G is understood.3. Free products of Abelian groups
The purpose of this section is to prove that scl is piecewise rational linear in B H ( G ), where G is a free product of Abelian groups. Along the way we developsome additional structure which is important for what follows.3.1. Euler characteristic with corners.
We will obtain surfaces by gluing upsimpler surfaces along segments in their boundary. Since ordinary Euler characteris-tic is not additive under such gluing, we consider surfaces with corners . Technically,a corner should be thought of as an orbifold point with angle π/ π ).When two surfaces with boundary are glued along a pair of segments in theirboundary, the interior points of the segments should be smooth, and the endpointsshould be corners. In the glued up surface, the boundary points which result fromidentifying two corners should be smooth.If S is a surface as above, let c ( S ) denote the number of corners. Define the orbifold Euler characteristic of S , denoted χ o ( S ), by the formula χ o ( S ) = χ ( S ) − c ( S )4 CL, SAILS AND SURGERY 7
With this convention, χ o is additive under gluing, and χ o = χ for a surface withno corners. In the sequel we will only consider surfaces with an even number ofcorners, so χ o will always be in Z .3.2. Decomposing surfaces.
Throughout the remainder of this section, we fix agroup G = A ∗ B where A and B are free Abelian groups, and a finite set Z of(nontrivial) conjugacy classes in G . We are interested in the restriction of scl tothe space h Z i ∩ B H ( G ) of homologically trivial chains with support in Z .Let K A and K B be a K ( A,
1) and a K ( B,
1) respectively (for instance, we couldtake K ( · , A ) to be a torus of dimension equal to the rank of A , and similarly for B )and let K = K A ∨ K B be a K ( G, K by ∗ = K A ∩ K B .A homotopically essential map γ : S → K is tight if it has one of the followingtwo forms:(1) the image of γ is a loop contained entirely in K A or in K B (we call suchmaps Abelian loops ); or(2) the circle S can be decomposed into intervals, each of which is takenalternately to an essential based loop in one of K A , K B .Every free homotopy class of map to K has a tight representative.A (nonabelian) tight loop γ : S → K induces a polygonal structure on S , withone edge for each component of the preimage of K A or K B and vertex set γ − ( ∗ ).By convention, we also introduce a polygonal structure on S when γ is an Abelianloop, with one (arbitrary) vertex and one edge.For each element of Z , choose a tight loop in the correct conjugacy class. Theunion of these tight loops can be thought of as an oriented 1-manifold L (with onecomponent for each element of Z ) together with a map Γ : L → K . As above, Γinduces a polygonal structure on L . Each oriented edge in this polygonal structureis mapped either to K A or to K B . Let T ( A ) denote the set of A -edges and T ( B ) theset of B -edges . Note that each A or B edge can be thought of as a homotopy classof loop in K A or K B , and therefore determines an element of the (fundamental)group A or B .Let f : S → X be an admissible surface. After a homotopy, we assume (in thenotation of Proposition 2.7) that ∂f : ∂S → L is a covering map, and that f istransverse to ∗ (i.e. f − ( ∗ ) is a system of proper arcs and loops). Furthermore, weassume (by Proposition 2.8) that ∂f : ∂S → L is orientation-preserving .Denote by F the preimage f − ( ∗ ) in S ; by hypothesis, F is a system of properarcs and loops in S . In anticipation of what is to come, we refer to the componentsof F as σ -edges . Since f maps F to ∗ , loops of F can be eliminated by compression(innermost first). Since f restricted to ∂S is a covering map, every arc of F isessential. Thus without loss of generality we assume F is a system of essentialproper arcs in S . Cut S along F and take the path closure to obtain two surfaces S A and S B , which are the preimages under f of K A and K B respectively, andsatisfy S A ∩ S B = F .Each component of ∂S A either maps entirely to K A (those which cover Abelianloops) or decomposes into arcs which alternate between components of F and arcswhich map to elements of T ( A ); we refer to the second kind of arcs as τ -edges . Inorder to treat everything uniformly, we blow up the vertices on Abelian loops intointervals, which we refer to as dummy σ -edges. A σ -edge which is not a dummy DANNY CALEGARI edge is genuine . Thus ∂S A can be thought of as a union of polygonal circles, whoseedges alternate between σ -edges and τ -edges.The surface S A and S B naturally have the structure of surfaces with cornersprecisely at points of F ∩ ∂S . In particular, χ o ( S A ) = χ ( S A ) −
12 number of components of F The number of components of F is equal to the number of genuine σ -edges. on S A (which is therefore equal to the number of genuine σ -edges on S B ). Since S has nocorners, χ ( S ) = χ o ( S ) = χ o ( S A ) + χ o ( S B ) = χ ( S A ) + χ ( S B ) − number of components of F Encoding surfaces as vectors.
We would like to reduce the computation ofscl to a finite dimensional linear programming problem. The main difficulty is thatit is difficult to find a useful parameterization of the set of all admissible surfaces.However, in the end, all we need to know about an admissible surface is − χ − anddegree.We need to keep track of two different kinds of information: the number andkind of τ -edges which appear in ∂S A , and the number and kind of σ -edges whichappear. Each oriented σ edge runs from the end of one oriented τ -edge to thestart of another oriented τ -edge, and can therefore be encoded as an ordered pairof τ -edges; i.e. as an element of T ( A ) × T ( A ). However, not every element of T ( A ) × T ( A ) can arise in this way: the only σ -edges associated to Abelian loopsare the “dummy” σ -edges. Consequently we let T ( A ) denote the set of orderedpairs ( τ, τ ′ ) with τ, τ ′ ∈ T ( A ) subject to the constraint that if either of τ, τ ′ is anAbelian loop, then τ = τ ′ .Let C ( A ) denote the R -vector space with basis T ( A ), and C ( A ) the R -vectorspace with basis T ( A ). The oriented surface S A determines a set of oriented σ -edges and therefore a non-negative integral vector v ( S A ) ∈ C ( A ). This vector isnot arbitrary however, but is subject to two further linear constraints which wenow describe.Define a linear map ∂ : C ( A ) → C ( A ) on basis vectors by ∂ ( τ, τ ′ ) = τ − τ ′ ,and extend by linearity. Since each τ -edge is contained between exactly two σ edges, ∂ ◦ v ( S A ) = 0. Similarly define h : C ( A ) → A ⊗ R by h ( τ, τ ′ ) = ( τ + τ ′ )and extend by linearity. By definition, h ◦ v ( S A ) is equal to the image of [ ∂S A ] in H ( K A ) = A . Since ∂S A is a boundary, h ◦ v ( S A ) = 0. Definition 3.1.
Let V A be the convex rational cone of non-negative vectors v in C ( A ) satisfying ∂ ( v ) = 0 and h ( v ) = 0.Note that V A is the cone on a compact convex rational polyhedron. A surface S A as above determines an integral vector v ( S A ) ∈ V . Conversely we will see thatfor every non-negative integral vector v ∈ V A there are many possible surfaces S A with v ( S A ) = v . For such a S A , the number of genuine σ -edges depends only onthe vector v . However, it is important to be able to choose such a surface S A with χ ( S A ) as big as possible. Finding such an S A is an interesting combinatorialproblem, which we now address. Definition 3.2. A weighted directed graph is a directed graph Σ together with anassignment of a non-negative integer to each edge of Σ. The support of a weighted CL, SAILS AND SURGERY 9 directed graph is the subgraph of Σ consisting of edges with positive weights, to-gether with their vertices.Let v ∈ V A be integral. Define a weighted directed graph X ( v ) as follows. Firstlet Σ denote the directed graph with vertex set T ( A ) and edge set T ( A ). Edgesof Σ correspond to basis vectors of C ( A ). Give each edge a weight equal to thecoefficient of v when expressed in terms of the natural basis. Definition 3.3.
Given a graph X ( v ) as above, let supp( v ) denote the support of X ( v ) (so that supp( v ) is a subgraph of Σ), and let | X ( v ) | denote the number ofcomponents of supp( v ). Lemma 3.4.
Let v ∈ V be a non-negative integral vector. Then there is a planarsurface S A with v ( S A ) = v and with | X ( v ) | boundary components. Moreover forany surface S A with v ( S A ) = v , the number of boundary components of S A is atleast | X ( v ) | .Proof. We construct a component of ∂S A for each component of X ( v ). Since ∂ ( v ) =0, the indegree (i.e. the sum of the weights on the incoming edges) and the outdegree(i.e. the sum of the weights on the outgoing edges) at each vertex of X ( v ) areequal. The same is true for each connected component of X ( v ). A connecteddirected weighted graph with equal indegree and outdegree at each vertex admitsan Eulerian circuit ; i.e. a directed circuit which passes over each edge a number oftimes equal to its weight. This fact is classical; see e.g. [4], § I.3. The vertices visitedin such a circuit (in order) determine a sequence of elements of T ( A ). Together withone σ edge (mapping to ∗ ) between each pair in the sequence, we construct a circleand a map to K A . If we do this for each component of X ( v ), we obtain a 1-manifold D and a map to K A . The image of D in H ( K A ) = A is equal to h ( v ) = 0, so D bounds a map of a surface S ′ A to K A . Since A is Abelian, every embedded once-punctured torus in S ′ A has boundary which maps to a homotopically trivial loop in K A , and can therefore be compressed. After finitely many such compressions, weobtain a planar surface S A as claimed.Conversely, if S A is a surface with v ( S A ) = v then every boundary componentdetermines an Eulerian circuit in X ( v ) in such a way that the sum of the degreesof these circuits is equal to the weight. In particular, each component of X ( v ) is inthe image of at least one boundary component, and the Lemma is proved. (cid:3) A vector v in C ( A ) can be thought of as a finite linear combination of elementsof T ( A ). Define | v | to be the sum of the coefficients of v excluding the coefficientscorresponding to Abelian loops . Hence for v = | v ( S A ) | , the number | v | is just thenumber of genuine σ -edges in ∂S A . We conclude that χ o ( S A ) = χ ( S A ) − | v | In order to determine scl we would like to construct surfaces S with a given v ( S )with χ ( S ) as large as possible. The first easy, but key observation is the following: Lemma 3.5.
For any v ∈ V A and any positive integer n , the graphs X ( v ) and X ( nv ) have the same number of components.Proof. The graphs are the same, but the weights are scaled by n . (cid:3) It follows that for any v and any ǫ > S A with v ( S A ) = nv and | χ ( S A ) | /n < ǫ . Hence we may take χ o ( S A ) to be projectively as close to − | v | as we like. As far as surfaces with χ ( S A ) ≤ S A with v ( S A ) = v which contain disk components, and it is this which we focus on in thenext section.3.4. Disk vectors and sails.Definition 3.6.
A non-negative nonzero integral vector v ∈ V A with supp( v ) con-nected is called a disk vector .Notice that a disk vector v can contain no Abelian loops. That is, if e in T ( A )is of the form e = ( τ, τ ) where τ ∈ T ( A ) is an Abelian loop, then the coefficientof e in the vector v is necessarily zero. For, the hypothesis that v is a disk vectorimplies that e is the only nonzero coefficient in v . But this implies that h ( v ) is anonzero multiple of h ( τ ) ∈ A . Since A is free and h ( τ ) is nonzero, h ( v ) is nonzero,contrary to the hypothesis that v ∈ V A . This proves the claim.In particular, for v a disk vector, | v | is the ordinary L norm of the vector v ,and is therefore a good measure of its complexity. Definition 3.7.
Let v be a (not necessarily integral) vector in V A . An expressionof the form v = X t i v i + v ′ is admissible if each v i is a disk vector (and, in particular, is integral), each t i ispositive, and v ′ ∈ V A .We are now in a position to define a suitable function χ o on V A . Definition 3.8.
Define χ o on V A by χ o ( v ) = sup X i t i − | v | where the supremum is taken over all admissible expressions v = P t i v i + v ′ ; i.e.expressions where v ′ ∈ V A , the t i >
0, and each v i is a disk vector. Lemma 3.9.
For any surface S A , there is an inequality χ o ( v ( S A )) ≥ χ o ( S A ) .Conversely, for any rational vector v ∈ V A and any ǫ > there is an integer n anda surface S A with v ( S A ) = nv such that | χ o ( S A ) /n − χ o ( v ) | ≤ ǫ .Proof. Let S A be a surface, with disk components D , · · · , D m and S ′ A = S A −∪ i D i .Corresponding to this there is an admissible expression v ( S A ) = X v ( D i ) + v ( S ′ A )Now, χ o ( S ′ A ) = χ ( S ′ A ) − | v ( S ′ A ) | /
2, and since S ′ A contains no disk components, χ ( S ′ A ) ≤
0. Moreover, χ o ( ∪ i D i ) = P i (1 − | v ( D i ) | / χ o ( S A ) = X i (1 − | v ( D i ) | /
2) + χ ( S ′ A ) − | v ( S ′ A ) | / ≤ X i − | v ( S A ) | / ≤ χ o ( v ( S A ))proving the first claim.The idea behind the proof of the second claim is as follows. Since χ o ( S A ) = χ ( S A ) − | v | /
2, to maximize χ o ( S A ) for a given v ( S A ) is to maximize χ ( S A ). Sincecomponents with χ ≤ χ as closeto 0 as desired (by Lemma 3.5), the goal is to (projectively) maximize the numberof disks used. In more detail: if v is rational, and v = P t i v i + v ′ is an admissible CL, SAILS AND SURGERY 11 expression, we can find another admissible expression v = P t ′ i v i + v ′′ where the t ′ i and v ′′ are rational, and P | t i − t ′ i | ≤ ǫ/ ǫ . After multiplyingthrough by a big integer n to clear denominators, we can find a surface S A with v ( S A ) = nv , with P i nt ′ i disk components. Now, it may be that the non diskcomponents have χ negative, but by Lemma 3.5 we can projectively replace thispart of the surface by a planar surface with χ very close to 0. Hence after possiblyreplacing n by a much bigger integer, we can find S A with v ( S A ) = nv such that | χ o ( S A ) /n − ( P t ′ i − | v | ) | ≤ ǫ/
2. This completes the proof. (cid:3)
It remains to study the function χ o . Equivalently, we study κ ( v ) = sup X i t i = χ o ( v ) + | v | / Klein function of V A . Let D A ⊂ V A denote the set of disk vectors,and let D A + V A denote Minkowski sum of D A and V A ; i.e. the set of vectors ofthe form d + v for d ∈ D A and v ∈ V A . Let conv( · ) be the function which assignsto a subset of a linear space its convex hull. Taking convex hulls commutes withMinkowski sum. Note that since V A is convex, conv( D A + V A ) = conv( D A ) + V A . Lemma 3.10.
The Klein function κ is a non-negative concave linear function on V A . The subset of V A on which κ = 1 is the boundary of conv( D A + V A ) .Proof. If v , v are elements of V A , the sum of admissible expressions for v and v is an admissible expression for their sum. This proves concavity. Non-negativity isobvious from the definition. To prove the last assertion, note that an admissibleexpression v = P t i v i + v ′ exhibits v/ ( P t i ) as an element of conv( D A ) + V A =conv( D A + V A ). (cid:3) The following lemma, while elementary, is crucial.
Lemma 3.11.
The sets conv( D A ) and conv( D A + V A ) are finite sided convex closedpolyhedra, whose vertices are elements of D A .Proof. For each open face F of the polyhedron V A (of any codimension ≥ v ) is constant on F . We denote this common support by supp( F ).By definition, D A is the union of the integer lattice points in those open faces F of V A for which supp( F ) is connected.If F is an open polyhedral cone, the convex hull of the set of integer lattice pointsin F is classically called a Klein polyhedron , and its boundary is called a sail . It is aclassical fact, which goes back at least to Gordan [13] that if F is rational, the set oflattice points in the closure of F has a finite basis (as an additive semigroup) whichis sometimes called a Hilbert basis , and the set of lattice points in the interior is afinitely generated module over this semigroup (see e.g. Barvinok [2]). Consequentlythe Klein polyhedron is finite sided, and its vertices are a subset of a module basis.Hence conv( F ∩ D A ) is a finite sided closed convex polyhedron for each F . Since V A has only finitely many faces, the same is true of conv( D A ) and therefore alsofor conv( D A + V A ). The vertices of each conv( F ∩ D A ) are in D A , so the same istrue for conv( D A ) and conv( D A + V A ). (cid:3) Consequently, from Lemma 3.10 and Lemma 3.11 we make the following deduc-tion:
Lemma 3.12.
The function χ o on V A is equal to the minimum of a finite set ofrational linear functions.Proof. This is true for κ , and therefore for κ − | v | / (cid:3) Remark . There is a close connection between vertices of the Klein polyhedronand continued fractions. If F is a sector in R , the sail is topologically a copyof R , and the vertices of the sail are integer lattice points in Z whose ratios arethe continued fraction approximations to the slopes of the sides of F . Klein [16]introduced Klein polyhedra and sails (for not necessarily rational polyhedral cones F ) in an effort to generalize the theory of continued fractions to higher dimensions.In recent times this effort has been pursued by Arnold [1] and his school.3.5. Rationality of scl.
We are now in a position to prove the main theorem ofthis section.
Theorem 3.14 (Rationality) . Let G = ∗ i A i be a free product of finitely manyfinitely generated free Abelian groups. Then scl is a piecewise rational linear func-tion on B H ( G ) . Moreover, there is an algorithm to compute scl in any finite di-mensional rational subspace.Proof. We first prove the theorem in the case G = A ∗ B where A and B are freeand finitely generated as above.We have rational polyhedral cones V A , V B in C ( A ) and C ( B ) respectively,which come together with convex piecewise rational linear functions χ o . There is arational subcone Y ⊂ V A × V B consisting of pairs of vectors ( v A , v B ) in V A × V B whichcan be glued up in the following sense. For each co-ordinate ( τ A , τ ′ A ) ∈ T ( A ) whoseentries are not Abelian loops, there is a corresponding co-ordinate ( τ B , τ ′ B ) ∈ T ( B )determined by the property that as oriented arcs in L , the arc τ B is followed by τ ′ A , and τ A is followed by τ ′ B . Say that this pair of co-ordinates are paired . Then Y is the subspace consisting of pairs of vectors whose paired co-ordinates are equal.Define χ on Y by χ ( v A , v B ) = χ o ( v A ) + χ o ( v B ). By Lemma 3.12, the function χ is equal to the minimum of a finite set of rational linear functions on Y . Finally,there is a linear map d : Y → H ( L ) with the property that a surface S = S A ∪ S B with ( v ( S A ) , v ( S B )) = y ∈ Y satisfies ∂f ∗ ( ∂S ) = dy ∈ H ( L ).Define a linear programming problem as follows. For l ∈ H ( L ), define Y l ⊂ Y to be the polyhedron which is equal to the preimage d − ( l ). Then definescl( l ) = − max y ∈ Y l χ ( y ) / Y and therefore Y l are finite sided rational polyhedra, and χ is the minimumof a finite set of rational linear functions on these polyhedra, the maximum of χ on Y l can be found algorithmically by linear programming (e.g. by Dantzig’s simplexmethod [11]), and is achieved precisely on a rational subpolyhedron of Y l . Notethat although maximizing the minimum of several linear functions is ostensiblya nonlinear optimization problem, it may be linearized in the standard way, byintroducing extra slack variables, and turning the linear terms (over which one isminimizing) into constraints . See e.g. Dantzig [11] § V i and a piecewise rational linear function χ o for each A i , and a slightly morecomplicated gluing condition to define the subspace Y , but there are no essentiallynew ideas involved. One minor observation is that one should build a K ( G,
1) by
CL, SAILS AND SURGERY 13 gluing up K ( A i , S i mapping to each factor are glued up along genuine σ -arcs in pairs, and not in more complicated combinatorial configurations. We leavedetails to the reader. (cid:3) Remark . If G = ∗ i A i where each A i is finitely generated Abelian but notnecessarily torsion free, there is a finite index subgroup G ′ of G which is a freeproduct of free Abelian groups. The piecewise rational linear property of scl isinherited by finite-index supergroups. Hence scl is piecewise rational linear on B H ( G ) in this case too. A similar observation applies to amalgamations of suchgroups over finite subgroups.A perhaps surprising corollary of the method of proof is the following: Corollary 3.16.
Let { A i } and { B i } be finite families of finitely generated freeAbelian groups. For each i , let ρ i : A i → B i be an injective homomorphism, andlet ρ : ∗ i A i → ∗ i B i be the corresponding injective homomorphism. Then ρ inducesan isometry of the scl norm. That is, for all chains c ∈ B H ( ∗ i A i ) , there is equality scl( c ) = scl( ρ ( c )) .Proof. An injective homomorphism ρ i : A i → B i induces an injective homomor-phism of vector spaces A i ⊗ R → B i ⊗ R . The only place in the calculation ofscl that the groups A i enter is in the homomorphisms h : C ( A i ) → A i ⊗ R , andthe map h is only introduced in order to determine the subspace h − (0). Since( ρ i ◦ h ) − (0) = h − (0), the linear programming problems defined by chains c and ρ ( c ) are the same, so the values of scl are the same. (cid:3) Example . Corollary 3.16 is interesting even (especially?) in the rank 1 case.Let G = F , freely generated by elements a, b . Then for any non-zero integers n, m the homomorphism ρ : F → F defined by ρ ( a ) = a m , ρ ( b ) = b n is an isometryfor scl. Hence (for instance) every value of scl which is achieved in a free group isachieved on infinitely many automorphism orbits of elements.The composition of an arbitrary alternating sequence of automorphisms and in-jective homomorphisms as above can be quite complicated, and shows that B H ( F )admits a surprisingly large family of (not necessarily surjective) isometries.If G is (virtually) free, every vector in B H ( G ) with positive scl norm rationallybounds an extremal surface, by the main theorem of [7]. However, if G is a freeproduct of Abelian groups of higher rank, extremal surfaces are not guaranteed toexist. For a vector v ∈ V A to be represented by an injective surface it is necessarythat it should be expressible as a sum v = P v i where each v i is in V A , and | X ( v i ) | ≤ i . The v i correspond to the connected components of S A with v ( S A ) = v . Since A is Abelian, for π ( S ) → A to be injective, every component of S A must be either a disk (in which case | X ( v i ) | = 1) or an annulus (in which case | X ( v i ) | = 2). Example . In Z ∗ Z , let the Z factor be generated by a , and let v , v begenerators for the Z factor. The chain c = av a − v − + v + v − v − satisfiesscl( c ) = 1 /
2, but no extremal surface rationally fills c , and in fact, there does noteven exist a π -injective surface filling a multiple of c . To see this, observe thatevery non-negative v ∈ V B has | X ( v ) | ≥
3, and therefore every surface S B with v ( S B ) = v has nonabelian (and therefore non-injective) fundamental group. Let G be the group obtained by doubling Z ∗ Z along c . Notice that G isCAT(0), since a K ( G,
1) can be obtained by attaching three flat annuli to twocopies of S ∨ T along pairs of geodesic loops corresponding to the terms in c .The Gromov norm on H ( G ; Q ) is piecewise rational linear. On the other hand, if α ∈ H ( G ; Q ) is any nonzero class obtained by gluing relative classes on either sidealong c , then no surface mapping to a K ( G,
1) in the projective class of α can be π -injective. Remark . Example 3.18 suggests a connection to the simple loop conjecture in3-manifold topology.
Example . The support of a disk vector cannot include a vertex correspondingto an Abelian loop. This observation considerably simplifies the calculation of sclon certain chains. Consider a chain of the form w = a − α + b − β + w ′ where α and β are positive, and w ′ is either a single word or a chain composed only of the letters a and b (and not their inverses). Suppose further that w ∈ B H ( F ), so that scl( w )is finite. Then by the remark above, there are no disk vectors, so χ o = −| v | / w ) ∈ Z .Explicitly, suppose w = a − α + b − β + P w i where each w i is of the form w i = a α i, b β i, · · · a α i,ni b β i,ni where each α i,j and each β i,j is positive, and P i,j α i,j = α , P i,j β i,j = β . Recallthat Abelian loops do not contribute to | v A | or | v B | . If S is a surface with v ( S ) =( N v A , N v B ) then ∂S wraps around each τ -edge with multiplicity exactly N . Henceeach w i contributes n i to v A and similarly for v B , and therefore | v A | = | v B | = P i n i .In particular, χ is constant on the polyhedron Y l , and scl( w ) = P i n i .In fact, the same argument shows that | v A | and | v B | are constant on Y l , andtherefore we can calculate scl by maximizing κ instead of χ o . We record this factas a proposition: Proposition 3.21.
For any chain w and any l ∈ H ( L ; Z ) , the functions | v A | and | v B | are constant on Y l , and take values in Z , with notation as above. Surgery
In this section we study how scl varies in families of elements, especially thoseobtained by surgery . In 3-manifold topology, one is a priori interested in closed 3-manifolds. But experience shows that 3-manifolds obtained by (Dehn) surgery on afixed 3-manifold with torus boundary are related in understandable ways. Similarly,even if one is only interested in scl in free groups (for some of the reasons suggestedin the introduction), it is worthwhile to study how scl behaves under surgery on freeproducts of free Abelian groups of higher ranks. In this analogy, the free Abelianfactors correspond to the peripheral Z subgroups in the fundamental group of a3-manifold with torus boundary. Definition 4.1.
Let { A i } and { B i } be two families of free Abelian groups. A familyof homomorphisms ρ i : A i → B i induces a homomorphism ρ : ∗ i A i → ∗ i B i . We saythat ρ is induced by surgery . If C ∈ B H ( ∗ i A i ), then we say that ρ ( C ) ∈ B H ( ∗ i B i )is obtained by surgery on C .By Corollary 3.16, it suffices to consider surgery in situations where each ρ i issurjective after tensoring with R . CL, SAILS AND SURGERY 15
One also studies families of surgeries, with fixed domain and range, in which thehomomorphisms ρ i depend linearly on a parameter. Definition 4.2.
With notation as above, let σ i : A i → B i and τ i : A i → B i be two families of homomorphisms. For each p ∈ Z , define ρ i ( p ) : A i → B i by ρ i ( p ) = σ i + pτ i , and define ρ ( p ) : ∗ i A i → ∗ i B i similarly. We refer to the ρ ( p ) asa line of surgeries. If C ∈ B H ( ∗ i A i ), we say the ρ ( p )( C ) are obtained by a line ofsurgeries on C .4.1. An example.
In this section we work out an explicit example of a (multi-parameter) family of surgeries. Given a 4-tuple of integers α , α , β , β we definean element w α ,α ,β ,β in B H ( F ) (or just w α,β for short) by the formula w α,β := a − α − α + b − β − β + a α b β + a α b β We can think of this as a family of elements obtained by surgery on a fixed elementin B H ( Z ∗ Z ). We will derive an explicit formula for scl( w α,β ) in terms of α and β , by the methods of § α and α are coprime, and simi-larly for the β i . We make this assumption in the sequel. Finally, after interchanging a with a − or b with b − if necessary, we assume α and β are strictly positive.The calculation of scl( w ) reduces to a finite number of cases. We concentrate on aspecific case; in the sequel we therefore assume: α > α + α > > α , β > β + β > > β We write F = A ∗ B where A = h a i and B = h b i . The set T ( A ) has threeelements, corresponding to the three substrings of w of the form a ∗ . We denotethese elements 1 , ,
3. Since 1 is an Abelian loop, T ( A ) has five elements; i.e. T ( A ) = { (1 , , (2 , , (2 , , (3 , , (3 , } . Let v ∈ V A have co-ordinates v through v . By the definition of V A , the v i are non-negative. The constraint that ∂ ( v ) = 0is equivalent to v = v . Hence in the sequel we will equate v and v , and write avector in V A in the form ( v , v , v , v ). In this basis, the constraint that h ( v ) = 0reduces to ( α + α )( v − v ) = α v + α v which we rewrite as v = v − ( α v + α v ) / ( α + α )See Figure 2. v v v v v = v Figure 2.
The weighted graph X ( v ) associated to v ∈ V A , where v = v − ( α v + α v ) / ( α + α ) is necessarily non-negative. The cone V A has four extremal vectors ξ i in ( v , v , v , v ) co-ordinates, whichare the columns of the matrix M A := α − α α + α − α α α + α By our assumption of the α i , the entries of this matrix are non-negative (as theymust be). Note that these four vectors are linearly dependent, and V A is the coneon a planar quadrilateral. The cone V A is the image of the non-negative orthant in R under multiplication on the left by M A .A nonzero nonnegative vector ( v , v , v , v ) is a disk vector if and only if it isintegral, if v = 0, and if v = − ( α v + α v ) / ( α + α ) >
0. Since we are assuming α + α >
0, the disk vectors are all contained in the face of V A spanned by ξ and ξ (i.e. the face with v = 0). The set of disk vectors are precisely vectors ofthe form (0 , p, q, − ( p + q ) α /α − q ) where p, q are integers such that q > , p ≥ − α | p + q . Thus the Klein polyhedron conv( D A + V A ) has vertices ξ and d = (0 , − − α , , α − V A .The first face K has vertex ξ , and has extremal rays ξ + tξ and ξ + tξ for t ≥
0. The second face K has vertices ξ and d , and has extremal rays ξ + tξ and d + tξ , as well as the interval from ξ to d . The third face K has vertex d and extremal rays d + tξ and d + tξ . The Klein function κ A has the form κ A = α ( v − α v / ( α + α ) − v ) on the cone of K α (( α + α ) v /α − v − v ) on the cone of K v on the cone of K while on all of V A we have | v | / v + 2 v + v ) / w α,β , we obtain similar expressions for a typical vector( u , u , u , u , u ) in V B . With this notation, the polyhedron Y is the subspace of V A × V B consisting of vectors for which u = v , u = v , u = v and u = v . Thetwo Abelian loops themselves impose no pairing conditions, but since we can writeboth u and v in terms of the other u i , v i (in the same way), the equalities aboveimply u = v .Setting d ( y ) = [ w α,β ] imposes two more conditions on the vectors (at first glanceit looks like it imposes four conditions since there are four terms in w , but twoof these conditions are already implicit in v = v and u = u which were conse-quences of ∂ = 0). These two extra conditions take the form v = v and v = 1 − v .The two conditions give v = 1, v = v = x and v = 1 − x . Making thesesubstitutions, we find that Y [ w ] is the polygon 0 ≤ x ≤ χ on Y [ w ] . In terms of x , the function χ isequal to κ A + κ B − κ A = x/ ( α + α ) if x ≤ ( α + α ) /α /α if ( α + α ) /α ≤ x ≤ ( α − /α − x if x ≥ ( α − /α and similarly for κ B : κ B = x/ ( β + β ) if x ≤ ( β + β ) /β /β if ( β + β ) /β ≤ x ≤ ( β − /β − x if x ≥ ( β − /β Then scl( w ) = 1 − max( κ A ( x ) + κ B ( x )) / Proposition 4.3.
Let w = a − α − α + b − β − β + a α b β + a α b β where the α i andcoprime and similarly for the β i , and they satisfy α > α + α > > α , β > β + β > > β We have the following formulae for scl( w ) by cases: (1) If ( α − /α ≤ ( β + β ) /β then scl( w ) = 1 − (cid:16) α + ( α − α ( β + β ) (cid:17) (2) If ( β − /β ≤ ( α + α ) /α then scl( w ) = 1 − (cid:16) β + ( β − β ( α + α ) (cid:17) (3) Otherwise scl( w ) = 1 − (cid:16) α + β (cid:17) Remark . The program scallop (see [10]) implements an algorithm describedin [8] § B H ( F ), and can be used togive an independent check of Proposition 4.3.Without much more work, we can also treat chains of the form w ′ α,β = a − α − α + b − β − β + a α b β a α b β The cones V A , V B are the same but now the polyhedron Y is slightly different,defined by u = v = u and v = u = v . Hence, in terms of the variable x , thefunction κ A is as before, whereas κ B has the form: κ B = x if x ≤ /β /β if 1 /β ≤ x ≤ − β /β (1 − x ) / ( β + β ) if x ≥ − β /β Hence we have
Proposition 4.5.
Let w ′ = a − α − α + b − β − β + a α b β a α b β where the α i andcoprime and similarly for the β i , and they satisfy α > α + α > > α , β > β + β > > β We have the following formulae for scl( w ) by cases: (1) If − β /β ≤ ( α + α ) /α then scl( w ′ ) = 1 − (cid:18) max (cid:18) β − β β ( α + α ) , α − α α ( β + β ) (cid:19)(cid:19) (2) Otherwise scl( w ′ ) = 1 − (cid:16) α + β (cid:17) The distribution of values of scl for all w, w ′ with α , β ≤
35 (about 3 millionwords) is illustrated in Figure 3.This statement about integral chains in F can be translated into a statementabout elements of the commutator subgroup of F .Let a, b, c, d be generators for a free group F . For each α, β define w ′′ α,β = a − α − α cb − β − β c − da α b β a α b β d −
18 DANNY CALEGARI
13 2314 3415 25 35 4516 5617 27 37 47 57 67
Figure 3.
Histogram of values of scl on w, w ′ By the self-product formula, i.e. Theorem 2.101 from [8] (also see Remark 2.102),we have an equality scl F ( w ′′ α,β ) = scl F ( w ′ α,β ) + 1Consequently, by the multiplicativity of scl under taking powers, we deduce thefollowing theorem: Theorem 4.6 (Denominators) . The image of a nonabelian free group of rank atleast under scl in R / Z is precisely Q / Z . If F , F are free groups containing elements w and w respectively, then by thefree product formula, i.e. Theorem 2.93 from [8], we have an equality scl( w w ) =scl( w ) + scl( w ) + 1 / F ∗ F . Suppose w i is an infinite family of elements in F for which the set ofnumbers scl( w i ) is well-ordered with ordinal type ω . Then we can take two copies F , F of F , and corresponding elements w i, , w i, in each copy, and observe thatthe set of numbers scl( w i, w j, ) is well-ordered with ordinal type ω . Repeatingthis process inductively, we deduce the following theorem: Theorem 4.7 (Limit values) . For each n , the image of the free group F n under scl contains a well-ordered sequence of values with ordinal type ω ⌊ n/ ⌋ . The imageof F ∞ under scl contains a well-ordered sequence of values with ordinal type ω ω . To obtain stronger results, it is necessary to understand how χ varies as a functionof the parameters in a more general surgery family.4.2. Faces and signatures.
We recall the method to compute scl( w ) described in § w ∈ B H ( A ∗ B ). In broad outline, the method has three steps:(1) Construct the polyhedra V A and V B and Y w ⊂ V A × V B (2) Express χ as the minimum of a finite family of rational linear functions(3) Maximize χ on Y w In principle, step (1) is elementary linear algebra. However in practice, even forsimple w the polyhedra V A , V B , Y w become difficult to work with directly, and it isuseful to have a description of these polyhedra which is as simple as possible; wetake this up in § Y w and χ , step (3) is a straightforward linear programming problem, whichmay be solved by any number of standard methods (e.g. Dantzig’s simplex method[11], Karmarkar’s projective method [15] and so on). These methods are generallyvery rapid and practical.The “answers” to steps (1) and (3) depend piecewise rationally linearly on theparameters of the problem, and it is easy to see their contribution to scl on familiesobtained by a line of surgeries.The most difficult step, and the most interesting, is (2): obtaining an explicitdescription of χ as a function of a parameter p in a line of surgeries. Because CL, SAILS AND SURGERY 19 of Proposition 3.21, this amounts to the determination of the respective Kleinfunctions κ on each of V A and V B . This turns out to be a very difficult question toanswer precisely, but we are able to obtain some qualitative results.For a given combinatorial type of V A , it takes a finite amount of data to specifythe set of open faces with connected support (i.e. those faces with the propertythat the integer lattice points they contain are disk vectors). We call this data that signature of V A , and denote in sign( V A ). Evidently the sail of V A depends only onsign( V A ) (a finite amount of data), and the orbit of V A under GL( C ( A ) , Z ).4.3. Combinatorics of V A . In this section we will give an explicit description of V A as a polyhedron depending on w . Recall that V A is the set of non-negativevectors in C ( A ) in the kernel of both ∂ and h . Define W A to be the set of non-negative vectors in C ( A ) in the kernel of ∂ . Hence V A = W A ∩ ker( h ). We firstgive an explicit description of W A .Let Σ denote the directed graph with vertex set T ( A ) and edge set T ( A ). Non-negative vectors in C ( A ) correspond to simplicial 1-chains, whose simplices areall oriented compatibly with the orientation on the edges of Σ. A vector is in thekernel of ∂ if and only if the corresponding chain is a 1-cycle. Hence we can thinkof W A as a rational convex polyhedral cone in the real vector space H (Σ).A 1-cycle in H (Σ) is determined by the degree with which it maps over everyoriented edge of Σ. A 1-cycle φ in W A determines an oriented subgraph Σ( φ ) of Σwhich is the union of edges over which it maps with strictly positive degree. Lemma 4.8.
An oriented subgraph of Σ is of the form Σ( φ ) for some φ ∈ W A if and only if every component is recurrent ; i.e. it contains an oriented path fromevery vertex to every other vertex.Proof. For simplicity restrict attention to one component. A connected orientedgraph is recurrent if and if it contains no dead ends : i.e. partitions of the verticesof Σ into nonempty subsets Z , Z such that every edge from Z to Z is orientedpositively. Since φ is a cycle, the flux through every vertex is zero. If there werea dead end Z , Z the flux through Z would be positive, which is absurd. HenceΣ( φ ) is recurrent.Conversely, suppose Γ is recurrent. For each oriented edge e in Γ, choose anoriented path from the endpoint to the initial point of e and concatenate it with e to make an oriented loop. The sum of these oriented loops is a 1-cycle φ for whichΣ( φ ) = Γ. (cid:3) Lemma 4.8 implies that the faces of W A are in bijection with the recurrent sub-graphs Γ of Σ. The dimension of the face corresponding to a graph Γ is dim( H (Γ)).As a special case, we obtain the following: Lemma 4.9.
The extremal rays of W A are in bijection with oriented embeddedloops in Σ .Example . Given a graph Γ (directed or not), there is a natural graph O (Γ)whose vertices are embedded oriented loops in Γ, and whose edges are pairs oforiented loops whose union has dim( H ) = 2. In the case that Γ is the 1-skeletonof a tetrahedron, the graph O (Γ) is the 1-skeleton of a stellated cube; see Figure 4.The polyhedron W A depends only weakly on the precise form of w . In fact,discounting Abelian loops, the polyhedron W A only depends on the cardinality of Figure 4. T ( A ). To describe V A we need to consider the function h : C ( A ) → A . Recallthat h ( τ, τ ′ ) = ( τ + τ ′ ) / T ( A ) with elements of A by thinking of the T ( A ) as loops in a K ( A, i : T ( A ) → A and think of i as a function on the vertices of Σ, so that if φ is anembedded loop in Σ, then h ( φ ) = P v ∈ φ i ( v ) ∈ A . Since by hypothesis w ∈ B H ( F ),we have h ( φ ) = 0 whenever φ is a Hamiltonian circuit (an embedded loop whichpasses through each vertex exactly once). Moreover for generic w ∈ B H ( F ) andgeneric i , these are the only embedded loops with h = 0. In any case, we obtain aconcrete description of V A , or, equivalently, of the set of extremal rays. Lemma 4.11.
Rays of V A are in the projective class of two kinds of -cycles: (1) embedded oriented loops φ in Σ with h ( φ ) = 0 (which includes the Hamil-tonian circuits in Σ ) (2) those of the form h ( φ ′ ) φ − h ( φ ) φ ′ where φ, φ ′ are distinct embedded orientedloops whose intersection is connected (and possibly empty), with h ( φ ′ ) > and h ( φ ) < Proof.
The rays of V A are the intersection of the hyperplane h = 0 with the raysand 2-dimensional faces of W A . The rays of W A which satisfy h = 0 are exactly theprojective classes of the oriented loops φ with h ( φ ) = 0. The 2-dimensional facesof W A correspond to the recurrent subgraphs Γ with dim( H (Γ)) = 2. By Mayer-Vietoris, such a Γ is the union of a pair of embedded loops φ, φ ′ whose (possiblyempty) intersection is connected. The hyperplane h = 0 intersects such a face in aray in the projective class of h ( φ ′ ) φ − h ( φ ) φ ′ . (cid:3) Surgery theorem.
In what follows, we fix A ∗ B and a linear family of sur-jective homomorphisms (i.e. a line of surgeries) ρ p : A ∗ B → A ′ ∗ B where A, A ′ , B are free Abelian, and rank( A ′ ) = rank( A ) −
1. Fix w ∈ B H ( A ∗ B ) and denote w ( p ) := ρ p ( w ).Recall that the set of disk vectors D A in V A is the union of the integer latticepoints in those open faces F of V A for which supp( F ) is connected. In a line of CL, SAILS AND SURGERY 21 surgeries, the polyhedra V A ( p ) vary in easily understood ways. For each p , let M ( p ) be an integral matrix whose columns are vectors spanning the extremal raysof V A ( p ). Then M ( p ) has the form M ( p ) = N + pN ′ , where N and N ′ are fixedintegral matrices, depending only on w . As p → ∞ , the cones V A ( p ) converge inthe Hausdorff topology to a rational cone V ′ A ( ∞ ) spanned by the nonzero columnsof N ′ , and the columns of N corresponding to the zero columns of N ′ . The cone V A ( ∞ ) associated to w has codimension one in each V A ( p ), and codimension onein the limit V ′ A ( ∞ ).For each p , let D A ( p ) denote the disk vectors in V A ( p ), and D A ( ∞ ) the diskvectors in V A ( ∞ ). Similarly, let κ p denote the Klein function on V A ( p ), and κ ∞ theKlein function on V A ( ∞ ). Observe that any v that is in D A ( p ) for some p is alsoin D A ( q ) for all q such that v is in V A ( q ) (i.e. the property of being a disk vectordoes not depend on p ). Lemma 4.12.
There is convergence in the Hausdorff topology conv( D A ( p ) + V A ( p )) → conv( D A ( ∞ ) + V ′ A ( ∞ )) Hence κ ∞ = lim p →∞ κ p | V A ( ∞ ) .Proof. The set of integer lattice points is discrete. Since every integer lattice pointis either in every V A ( p ) or in only finitely many, the intersection of D A ( p ) with anycompact subset of W A is eventually equal to the intersection of this compact setwith ∩ p D A ( p ). Since V A ( ∞ ) = ∩ p V A ( p ), we have D A ( ∞ ) = ∩ p D A ( p ).The last claim follows becauseconv( D A ( ∞ ) + V ′ A ( ∞ )) ∩ V A ( ∞ ) = conv( D A ( ∞ ) + V A ( ∞ )) (cid:3) From this discussion we derive the following theorem.
Theorem 4.13 (Surgery) . Fix w ∈ B H ( A ∗ B ) and let ρ p : A ∗ B → A ′ ∗ B be a lineof surgeries, constant on the second factor, and surjective on the first factor with rank( A ′ ) = rank( A ) − . Define w ( p ) = ρ p ( w ) . Then lim p →∞ scl( w ( p )) = scl( w ) .Proof. Denote lim p →∞ κ p = κ ′∞ , thought of as a function on V ′ A ( ∞ ). Denote by Y ′ w ⊂ V ′ A ( ∞ ) × V B the limit as p → ∞ of Y w ( p ) ⊂ V A ( p ) × V B . Since | v A | isconstant on each Y w ( p ), it follows that | v A | is also constant on Y ′ w . Lemma 4.12implies that the only disk vectors in V ′ A ( ∞ ) which contribute to κ ′∞ are thosein V A ( ∞ ). Since | v A | is non-negative on V ′ A ( ∞ ), a level set of | v A | which is asupporting hyperplane for conv( D A ( ∞ ) + V A ( ∞ )) is also a supporting hyperplanefor conv( D A ( ∞ ) + V ′ A ( ∞ )). It follows that χ restricted to Y ′ w is maximized in Y w .The result follows by applying Theorem 3.14. (cid:3) Remark . By monotonicity of scl under homomorphisms one has the inequal-ity scl( w ( p )) ≤ scl( w ) for all p . Thus surgery “explains” the existence of manynontrivial accumulation points in the scl spectrum of a free group. However itshould also be pointed out that equality is sometimes achieved in families, so thatscl( w ( p )) = scl( w ) for all p (for example, under the conditions discussed in Exam-ple 3.20).It is interesting to note that the limit does not depend on the particular surgeryfamily. Example . Let w = a c ba − b − c − ba − c − b − where [ a, c ] = id, and b gener-ates a free summand. Consider the line of surgeries defined by ρ p ( a ) = a , ρ p ( b ) = b and ρ p ( c ) = pa . In this case, w ( p ) = a p ba − b − a − p ba − − p b − . Thenscl( w ( p )) = 4 p + 34 p + 4 if p is odd, and 2 p + 12 p + 2 if p is even.Define σ = (cid:0) (cid:1) and consider the line of surgeries obtained by the same homomor-phisms ρ p precomposed with the automorphism σ of Z . Thenscl( w ( p ) σ ) = 8 p + 38 p + 4 if p is odd, and 4 p + 54 p + 6 if p is even.Both sequences of numbers converge to 1 = scl( w ) as p → ∞ . Note that evenwhen the values of scl( w ( p )) and scl( w ( q ) σ ) agree, the corresponding elements aretypically not in the same Aut orbit in F . Remark . The precise algebraic form of scl( w ( p )) on surgery families is analyzedin a forthcoming paper of Calegari-Walker [9], where it is shown quite generally thatscl( w ( p )) is a ratio of quasipolynomials in p , for p ≫ Computer implementation.
The algorithm described in this paper hasbeen implemented by Alden Walker in the program sss , available from the author’swebsite [19]. This allows computations that would be infeasible with scallop ; e.g.scl( aba − ba − b − + a b ) = 195 / A and B arcs, and the practical limitfor this number appears to be no more than about 5 (excluding Abelian loops).Some theoretical explanation for this difficulty comes from recent work of LukasBrantner [5], who shows (amongst other things) that the problem of decidingwhether a given disk vector d ∈ D A is essential — i.e. cannot be written as d = e + v for e ∈ D A and v ∈ V A − coNP -complete.5. Acknowledgment
I would like to thank Lukas Brantner, Jon McCammond, Alden Walker andthe referee for helpful comments and corrections. While writing this paper I waspartially supported by NSF grants DMS 0707130 and DMS 1005246.
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