Faces of the scl norm ball

Faces of the scl norm ball

Danny Calegari Department of Mathematics
Caltech
1/22/2008, Version 0.09
Abstract.

Let where is a compact, connected, oriented surface with and nonempty boundary.

1. The projective class of the the chain intersects the interior of a codimension one face of the unit ball in the stable commutator length norm on .

2. The unique homogeneous quasimorphism on dual to (up to scale and elements of ) is the rotation quasimorphism associated to the action of on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on .

These facts follow from the fact that every homologically trivial -chain in rationally cobounds an immersed surface with a sufficiently large multiple of . This is true even if has no boundary.

1. Introduction

An immersed loop in the plane might or might not bound an immersed disk, and if it does, the disk it bounds might not be unique. An immersed loop on a surface might not bound an immersed subsurface, but admit a finite cover which does bound an immersed subsurface — i.e. it might “virtually” bound an immersed surface. Most homologically trivial geodesics on hyperbolic surfaces with boundary do not even virtually bound an immersed surface. However, in this paper, we show that every homologically trivial geodesic in a closed hyperbolic surface virtually bounds an immersed surface, and every homologically trivial geodesic in a hyperbolic surface with boundary virtually cobounds an immersed surface together with a sufficiently large multiple of . This has implications for the structure of the (second) bounded cohomology of free and surface groups, as we explain in what follows.

Given a group and an element , the commutator length of , denoted , is the smallest number of commutators in whose product is , and the stable commutator length of is the limit . Geometrically, if is a space with and is represented by a loop in , the commutator length of is the least genus of a surface mapping to whose boundary maps to . By minimizing number of triangles instead of genus, one can reinterpret scl as a kind of norm on relative (-dimensional) homology. Technically, if denotes the vector space of real-valued (group) -boundaries (i.e. group -chains which are boundaries of group -chains; see § 2.2), there is a well-defined scl pseudo-norm on . The subspace on which scl vanishes always includes a subspace spanned by chains of the form and , for and , and therefore scl descends to a pseudo-norm on the quotient , which we abbreviate by or or even in the sequel. In certain special cases (for example, when is a free group), scl defines an honest norm on , but we will not use this fact in the sequel. More precise definitions and background are given in § 2.

Dual (in a certain sense) to the space with its scl pseudo-norm, is the space of homogeneous quasimorphisms on , i.e. functions for which there is a least real number (called the defect) such that for all and , and for all . The particular form of duality between scl and is called Bavard duality, which is the equality

 scl(∑tigi)=supϕ∈Q(G)∑tiϕ(gi)2D(ϕ)

(see § 2 for details).

The defining properties of a homogeneous quasimorphism can be thought of as an infinite family of linear equalities and inequalities depending on elements and pairs of elements in . The - duality between scl and means that computing scl is tantamount to solving an (infinite dimensional) linear programming problem in group homology (for an introduction to linear programming, see e.g. [9]). In finite dimensions, and norms are piecewise linear functions, and their unit balls are rational convex polyhedra. Broadly speaking, the main discovery of [5] is that in free groups (and certain groups derived from free groups in simple ways), computing scl reduces to a finite dimensional linear programming problem, and therefore the unit ball of the scl pseudo-norm on is a rational convex polyhedron; i.e. for every finite dimensional rational vector subspace of , the unit ball of the scl pseudo-norm restricted to is a finite-sided rational convex polyhedron (compare with the well-known example of the Gromov-Thurston norm on of a -manifold; see [20]).

In a finite dimensional vector space, a rational convex polyhedron is characterized by its top dimensional faces — i.e. those which are codimension one in the ambient space. In an infinite dimensional vector space, a rational convex polyhedron need not have any faces of finite codimension at all. The codimension of a face of a convex polyhedron in an infinite dimensional vector space is the supremum of the codimensions of its intersections with finite dimensional subspaces. Top dimensional faces of the unit ball of the Gromov-Thurston norm on of a -manifold have a great deal of topological significance (see e.g [20], [14], [4] or [18] for connections with the theories of taut foliations, Seiberg-Witten equations, quasigeodesic flows, and Heegaard Floer homology respectively). It is therefore a natural question to ask whether the scl unit polyhedron in has any faces which are codimension one in , and whether some of these faces have any geometric significance. Our first two main theorems answer these questions affirmatively.

Theorem A.

Let be a free group, and let be a compact, connected, orientable surface with and . Let be the -chain represented by the boundary of , thought of as a finite formal sum of conjugacy classes in . Then the projective ray in spanned by intersects the unit ball of the scl norm in the interior of a face of codimension one in .

By Bavard duality, a face of the scl norm ball of codimension one is dual to a unique extremal homogeneous quasimorphism, up to elements of (which vanish identically on ). It turns out that we can give an explicit description of the extremal quasimorphisms dual to the “geometric” faces of the scl norm ball described in Theorem A.

If is a compact, connected, orientable surface with , then admits a hyperbolic structure with geodesic boundary. The hyperbolic structure and a choice of orientation determine a discrete, faithful representation , unique up to conjugacy. Since is free, this representation lifts to where denotes the universal covering group of . There is a unique continuous homogeneous quasimorphism on (up to scale), called the rotation quasimorphism (discussed in detail in § 3.3). This quasimorphism pulls back by to a homogeneous quasimorphism on , which is well-defined up to elements of . Up to scale, this turns out to be the homogeneous quasimorphism dual to the top dimensional face of the scl norm ball described above:

Theorem B.

Let be a free group, and let be a compact, connected, orientable surface with and . Let be the face of the scl unit norm ball whose interior intersects the projective ray of the class . The face is dual to the extremal homogeneous quasimorphism .

These theorems are both proved in § 3.

Theorem A shows how hyperbolic geometry and surface topology manifest in the abstract (bounded) cohomology of a free group. Theorem B is a kind of rigidity result, characterizing the rotation quasimorphism associated to a discrete, faithful representation of into amongst all homogeneous quasimorphisms by the property that it is “extremal” for . In § 3.5 we use these theorems to deduce a short proof of a relative version of rigidity theorems of Goldman [11] and Burger-Iozzi-Wienhard [3], that representations of surface groups into certain Lie groups of maximal Euler class are discrete (it should be stressed that [11, 3] contain much more than the narrow result we reprove).

In light of Theorem A and Theorem B, it is natural to ask whether the projective class of every element intersects the interior of a face of the scl norm ball of finite codimension. In fact, it turns out that this is not the case. We show by an explicit example (Example 3.23) that there are many elements where has rank at least , whose projective classes are contained in faces of the scl norm ball of infinite codimension.

The method of proof is of independent interest. We show that for any homologically trivial geodesic -manifold in a hyperbolic surface , there is a surface and an immersion for which is taken to some multiple of in ; i.e. the -cycle “rationally bounds” an immersed surface. Note that this remains true even if is closed! Explicitly, the statement of the main technical theorem (proved in § 3.4) is as follows:

Theorem C.

Let be a compact, connected orientable surface with , and a finite rational chain in . Then for all sufficiently large rational numbers (depending on ), the geodesic -manifold in corresponding to the chain rationally bounds an immersed surface .

The connection with stable commutator length is as follows: from the main theorem of [5] it follows that in an oriented hyperbolic surface with boundary, a rational -chain bounds an immersed surface if and only if , where is as above (this is Proposition 3.8 below). Hence Theorem C implies that every chain in which is projectively close enough to satisfies ; Theorems A and B follow.

A number of additional corollaries are stated, including a generalization of the main theorem of [6]. Let be a graph of free or (closed, orientable) hyperbolic surface groups amalgamated over infinite cyclic subgroups, and let be a nonzero rational class in . Let be the fundamental classes in of the vertex subgroups which are closed surface groups. Then for all sufficiently big integers , some multiple of the class in is represented by an injective map from a closed hyperbolic surface group to .

2. Background

2.1. Definitions

The following definition is standard; see [1] or [7], § 2.1.

Definition 2.1.

Let be a group, and . The commutator length of , denoted , is the smallest number of commutators in whose product is .

Topologically, if is a space with , and is a loop representing the conjugacy class of , then is the least genus of a compact oriented surface with one boundary component for which there is a map with in the free homotopy class of .

Definition 2.2.

Let be a group, and . The stable commutator length of , denoted , is the limit

 scl(g)=limn→∞cl(gn)n

Commutator length and stable commutator length can be extended to finite linear sums of groups elements as follows:

Definition 2.3.

Let be a group, and elements of whose product is in . Define

 cl(g1+⋯+gm)=infhi∈Gcl(g1h1g2h−11⋯hm−1gmh−1m−1)

and

 scl(g1+⋯+gm)=limn→∞cl(gn1+⋯+gnm)n

A geometric interpretation of these numbers will be given in § 2.3.

It is a fact that the limit in Definition 2.3 exists, and satisfies and for all and . So it makes sense to define for any and . With this definition, it is immediate that scl is subadditive; i.e.

If we need to emphasize the group we denote cl and scl by and respectively.

One may reduce the calculation of scl on finite sums to a calculation of scl on individual elements, by the following “addition lemma”:

Let have infinite order. Let where is freely generated by . Then

 sclG(g1+⋯+gm)+m−12=sclH(g1x1g2x−11⋯xm−1gmx−1m−1)

This follows from [7], Thm. 2.93 and induction. When the have finite order, the formula must be corrected (in a straightforward way).

2.2. scl as a pseudo-norm

It is convenient to use the language of homological algebra. Given a group , one has the complex of real group chains whose homology is the real (group) homology of ; see Mac Lane ([15], Ch. IV, § 5). A real (group) -chain is a finite formal real linear combination of elements of , so (for instance) a real (group) -chain is just a finite formal real linear combination of elements of . Denote the group of (real) -boundaries by , or for short.

The properties of scl enumerated in § 2.1 imply that the function scl is well-defined, linear and subadditive on finite integral group -boundaries, and therefore admits a unique linear continuous extension to .

Moreover, scl vanishes on the subspace of spanned by chains of the form and for any and . Thus scl defines a pseudo-norm on . See [7], § 2.6 for proofs of these basic facts.

In [5] an algorithm is described to compute scl on elements of where is a free group. The program scallop (source available at [8]) implements a polynomial-time version of this algorithm, described in [7], § 4.1.7–4.1.8. At a number of points in this paper we make assertions about the value of scl on certain chains in ; these assertions are justified using the program scallop.

Remark 2.5.

In the final analysis, our main theorems do not depend logically on the computations carried out with the aid of scallop (also see Remark 3.12). Nevertheless, these computations were an essential part of the process by which these theorems were discovered.

2.3. Extremal surfaces

The definition of stable commutator length can be reinterpreted in geometric terms. Let be a space, and nontrivial free homotopy classes of loops in . Let be a surface for which there is a commutative diagram

 ∂Si−−−−→S∂f⏐⏐↓⏐⏐↓f∐iS1∐iγi−−−−→X

where is the inclusion map, and in for some .

Define to be the sum of Euler characteristic over all components of for which is non-positive. Then there is an equality

 sclG(g1+⋯+gm)=infS−χ−(S)2n

over all compact oriented surfaces as above, where , and represents the conjugacy class of . See [7], Prop. 2.68 for a proof.

Definition 2.6.

The chain is said to rationally bound a surface for which there is a commutative diagram as above. A surface with this property is extremal if every component of has negative Euler characteristic, and there is equality

 scl(g1+⋯+gm)=−χ−(S)2n

Extremal surfaces are -injective ([7], Prop. 2.96).

2.4. Quasimorphisms

Definition 2.7.

Let be a group. A function is a homogeneous quasimorphism if it satisfies for all and , and if there is a least non-negative real number called the defect, such that for all there is an inequality

 |ϕ(g)+ϕ(h)−ϕ(gh)|≤D(ϕ)

The set of homogeneous quasimorphisms on a group is a real vector space, and is denoted . A homogeneous quasimorphism has defect if and only if it is a homomorphism. Thus is a vector subspace of . The defect defines a natural norm on , making it into a Banach space. Any real-valued function on extends by linearity to define a -cochain. There is an exact sequence

 0→H1(G;R)→Q(G)δ→H2b(G;R)→H2(G;R)

where is the coboundary map on -cochains, and denotes bounded (group) cohomology. See [1] or [7], § 2.4 for an explanation of these facts, and Gromov [12] for an introduction to bounded cohomology.

There is a duality between stable commutator length and quasimorphisms, called Bavard duality. For chains in , this takes the following form

Theorem 2.8 (Bavard duality).

Let be an element of . Then there is an equality

 scl(C)=supϕ∈Q(G)/H1(G)∑tiϕ(gi)2D(ϕ)

See [1] for a proof when is an element of , or [7], Thm. 2.73 for the general case.

Definition 2.9.

A quasimorphism is extremal for a chain if there is equality

 scl(C)=∑tiϕ(gi)2D(ϕ)

Given a chain , the set of extremal quasimorphisms for is a nonempty convex cone, and is closed (away from ) in the natural Banach space topology on , as well as in the topology of termwise convergence in . See [7], Prop. 2.81 for a proof.

3. Immersed surfaces

3.1. Doodles

The question of which immersed loops in surfaces bound immersed subsurfaces is subtle and interesting, and has fascinated many mathematicians (see e.g. [13], p. 47). An immersed loop in the plane which bounds an immersed disk necessarily has winding number , but not every loop with winding number bounds an immersed disk. See Figure 1.

Blank [2] gave an algorithm to determine which immersed loops in the plane bound immersed disks; his algorithm was extended to other surfaces by Francis [10] and others, but the answer is not very illuminating: some curves bound immersed disks, some don’t, and the reason is complicated. Other authors have studied the existence of branched immersions with prescribed boundary, which are much easier to construct.

Milnor [17] gave a well-known example of an immersed loop in the plane which bounds two different immersed disks. See Figure 2.

On a hyperbolic surface , every homotopy class of essential loop contains a canonical geodesic representative. One can therefore ask which conjugacy classes in are represented by geodesics which bound immersed surfaces. The answer turns out to be independent of the choice of hyperbolic structure on , and therefore in principle is a purely “algebraic” problem.

One subtle aspect of the problem is illustrated by the example in Figure 3. This shows an immersed loop (in the isotopy class of a geodesic) in a genus punctured surface which does not bound an immersed surface, but which “virtually” bounds an immersed surface: there is an immersed surface with two boundary components, each of which wraps positively once around .

One is therefore led to study the following question. If is an orientable hyperbolic surface, which conjugacy classes in are represented by geodesics which virtually bound immersed surfaces? That is, when is there an immersed surface in , all of whose boundary components wrap positively around the geodesic representative of ? Only a homologically trivial loop virtually bounds a surface at all, so we restrict attention to in the commutator subgroup. It turns out that we can give a complete and somewhat surprising answer to this question. If is closed, then every homologically trivial geodesic virtually bounds an immersed surface. If has (geodesic) boundary, then a homologically trivial geodesic corresponding to a conjugacy class virtually bounds an immersed surface if and only if the projective class of , thought of as a -boundary, intersects a certain top dimensional face of the unit ball in the scl pseudo-norm on . This is explained in the remainder of this section.

3.2. Positive immersed surfaces

Definition 3.1.

An immersion between oriented surfaces is positive if it is orientation-preserving.

Definition 3.2.

Let be an immersed -manifold in . The -manifold bounds a (positive) immersion if there is a commutative diagram

 ∂Ti−−−−→T∂f⏐⏐↓⏐⏐↓f∐iS1γ−−−−→S

for which is an orientation-preserving homeomorphism. The -manifold rationally bounds a positive immersion if there is some integer , and a commutative diagram as above, for which is a positive immersion (i.e. an orientation-preserving covering map) such that in .

Compare with Definition 2.6.

We are concerned in the sequel with the case that is compact, connected and oriented, possibly with boundary, and satisfying . The surface admits a (nonunique) hyperbolic structure with totally geodesic boundary; we fix such a structure. Let be a chain in where the are integers, and the are primitive.

For each , let be a geodesic loop corresponding to the conjugacy class of , and let be the union of the . We say that rationally bounds a positive immersed surface if there is an integer and a positive immersion as in Definition 3.2 for which in , where .

Lemma 3.3.

Suppose chains rationally bound positive immersed surfaces. Then rationally bounds a positive immersed surface.

Proof.

For “generic” chains and there is nothing to prove: the disjoint union of two immersed surfaces is an immersed surface. The issue is that there might be a conjugacy class in the support of both and whose coefficients have different signs. Let be the geodesic in the free homotopy class corresponding to . Positive immersed surfaces with rational boundary and might have boundary components mapping to with different degrees. The following claim shows that we can construct suitable covers of these immersed surfaces such that the boundary components mapping to can be glued up.

Claim.

Let be a connected, oriented surface with and genus at least . Let be a set of boundary components, and let be an immersion, such that the degree of on every component is positive. Let be the degrees of on the components of , and let be a common multiple of the . Then there is a finite cover such that every component of the preimage maps to with degree .

Proof.

An orientable surface with genus at least admits a double cover such that every component of has exactly two preimages in . Let be the components of , and let be the degrees of the map . Let be a common multiple of the . Define a homomorphism as follows. For each , let be the components of in the preimage of , and let be the degree of . Then define and . Since in , the function extends to a (surjective) homomorphism from to . If is the cover corresponding to the kernel of , then every component of , the preimage of , maps to with degree , as desired. ∎

Start with a pair of positive immersed surfaces with rational boundary and . Since the Euler characteristics of these surfaces are negative, they admit finite covers with genus at least one. After passing to a suitable cover (provided by the claim), and gluing up pairs of boundary components which map to geodesics in the common support of and with the same absolute degree but with opposite signs, we obtain a positive immersed surface which rationally bounds. ∎

Remark 3.4.

Compare with the proof of Thm. 3.4 of [6].

We will give a much shorter proof of this Lemma (assuming more machinery) in § 3.3 in the special case that the ambient surface has boundary.

3.3. Rotation quasimorphism

Throughout this section we fix , a compact oriented hyperbolic surface with boundary.

Let be a homologically trivial geodesic in . The geodesic cuts up into connected regions . For each , let be an oriented arc from to which is transverse to , and let be the signed intersection of with . Since is homologically trivial by hypothesis, does not depend on the choice of . Geometrically, if is an oriented surface, and is a smooth map, is a signed count of the preimages of a generic point in .

Definition 3.5.

The algebraic area enclosed by is the sum

 area(γ)=∑ni⋅area(Ri)

The hyperbolic structure and the orientation on determines a discrete faithful representation unique up to conjugacy. Since is free, this representation lifts to , where denotes the universal covering group of . The group acts on the circle at infinity of hyperbolic space, and lets us think of as a subgroup of . The covering group is the preimage of in .

Definition 3.6.

Given , the rotation number of , as defined by Poincaré, is the limit

 rot(g)=limn→∞gn(0)n

where we think of as a subgroup of under the covering projection .

Rotation number pulls back by to define a function rot on . As is well-known, rot is a homogeneous quasimorphism on with . As a function on , the function rot depends on the choice of lift of to . Different lifts are classified by elements of , so rot is well-defined on modulo elements of , and therefore well-defined on the commutator subgroup of independent of the choice of . Though it appears to depend on the choice of hyperbolic structure on , it depends only on the topology of . If we need to stress the dependence of rot on , we write it .

Lemma 3.7 (Area is rotation number).

If , and is a geodesic in corresponding to the conjugacy class of , there is an equality

 area(γ)=2π⋅rotS(g)

This is proved in [7], Lem. 4.58. For the expert, the lemma follows from the fact that the coboundary of the rotation quasimorphism is the (relative) Euler class, and the Gauss-Bonnet theorem.

Note that for every nontrivial , the element is hyperbolic in , and therefore fixes two points in the circle at infinity. It follows that is an integer. An explicit formula for , in terms of an expression of as a reduced word in a standard generating set for , is given in [7], Thm. 4.62.

The functions area and rot extend by linearity and continuity to elements in . This is obvious for the function rot, and straightforward for area: if where the are real numbers and the are primitive conjugacy classes, let be oriented geodesics in in the conjugacy classes of the . The cut up into regions . For each , let be an arc from to transverse to every , and let where denotes signed intersection number. Then .

We can now give necessary and sufficient conditions for a rational chain in a hyperbolic surface to rationally bound an immersed subsurface.

Proposition 3.8.

Let be a compact, connected, oriented hyperbolic surface with geodesic boundary. A rational chain in rationally bounds a positive immersed subsurface if and only if

 scl(C)=rotS(C)/2
Proof.

Given and a surface that it rationally bounds, we can replace by a pleated surface (see e.g. Thurston [21] Ch. 8 for an introduction to the theory of pleated surfaces) and observe that the hyperbolic area of is at least as big as , with equality if and only if the map is an immersion (note that if the original map was already an immersion, then so is its pleated representative). It follows that an immersed surface is extremal, and therefore by Theorem 2.8, if a chain bounds a positive immersed surface, then by Gauss-Bonnet.

Conversely, since has boundary (by assumption) and therefore is free, any rational chain admits an extremal surface, by the Rationality Theorem from [5] (or see [7], Thm. 4.18). Hence if and only if bounds a positive immersed surface. By Lemma 3.7 there is an equality . ∎

In particular, the chain satisfies

 scl(∂S)=area(S)/4π=−χ(S)/2

Hence the surface itself is an extremal surface for .

Remark 3.9.

We can use this fact to give a very short proof of Lemma 3.3 in the case that the ambient surface has boundary. A rational chain in rationally bounds a positive immersed surface if and only if is extremal for , i.e. if . If and rationally bound positive immersed surfaces, then

 scl(C1+C2)≤scl(C1)+scl(C2)=rot(C1)/2+rot(C2)/2=rot(C1+C2)/2

Hence rot is extremal for , and therefore rationally bounds a positive immersed surface. Of course this proof is not “really” short, since it uses the (highly nontrivial) fact that every rational chain in a free group bounds an extremal surface.

Example 3.10.

Let be a once-punctured torus, with standard generators . Let be a nontrivial commutator, and let be the associated geodesic in (necessarily primitive). It is a fact that in a free group, the “standard” once-punctured torus whose boundary is a given nontrivial commutator is always extremal. When does bound an immersed surface? A description of as a cyclically reduced word in determines a polygonal loop in with vertices contained in the integer lattice, as follows. Start at the origin, and read the letters of one by one. On reading (resp. ), take one step to the right (resp. left), and on reading (resp. ), take one step up (resp. down). The polygonal loop can be “smoothed” by rounding the corners where a horizontal and a vertical arc of meet, giving rise to an immersed loop . It turns out that bounds an immersed surface in if and only if the winding number of is . So for example, bounds an immersed surface when , but does not. For details see [7], § 4.2.

3.4. Immersion theorem

We now prove our main technical result (Theorem C) and deduce Theorem A and Theorem B as corollaries.

Theorem C.

Let be a compact, connected orientable surface with , and a finite rational chain in . Then for all sufficiently large rational numbers (depending on ), the geodesic -manifold in corresponding to the chain rationally bounds an immersed surface .

Proof.

Since bounds the immersed surface , it suffices to prove the theorem for a particular positive (depending on ).

Multiply through by a large positive integer to clear denominators, so we can assume the are all integers. Also, by replacing with if necessary, we can assume the are all positive. Pick a hyperbolic structure on , and let be the geodesic loop corresponding to the conjugacy class of . By Scott (LERF for surface groups, [19]) there is a finite cover of in which every component of the preimage of each is embedded. In other words, there is a finite cover , so that if is the union in of all preimages of all components of , then every component of is embedded (although the union typically will not be). The composition of a positive immersion with a covering map is a positive immersion, so it suffices to construct the positive immersion in . Hence without loss of generality we can assume that we are working in the cover, and every individual geodesic is embedded (though of course the union will typically not be).

Let be a standard system of embedded geodesics which are a standard basis for . For each let be integers such that

 [γi]=∑jai,j[αj]+bi,j[βj]−Di

in homology, where is in the image of . Since the entire boundary is homologically trivial, the class is represented (in many different ways) as a positive sum of positively oriented boundary components of .

The logic of the remainder of the argument is as follows. We will show that for each , the chain

 Ci:=γi−(∑jai,jαj−bi,jβj)+∂i

bounds a positive immersed surface, where is a positive sum of positively oriented boundary components of . This will be enough to prove the theorem. For, since is homologically trivial, we have

 ∑riCi=∑iriγi+R∂S=C+R∂S

Hence by Lemma 3.3, the chain bounds a positive immersed surface for sufficiently big , as claimed.

First, decompose along a union of embedded separating geodesics into a union of genus one subsurfaces such that is a standard basis for . Hence (in particular), for each , the geodesics are simple, and intersect transversely in one point. The first step is to replace each by a union of geodesics, each of which is embedded and contained in a single subsurface .

Claim.

For each there is a chain which bounds a positive immersed surface, where is a positive sum of embedded geodesics disjoint from , and is a positive sum of positively oriented boundary components of .

We remark that either or both of might be empty in the statement of the claim above.

Proof.

If is a component of , then cobounds a positive immersed surface (in fact, an embedded subsurface of ) together with some components of . After adding a sufficiently big multiple of we obtain a chain of the form as above which bounds an immersed positive surface, and we are done in this case.

Otherwise, is in general position. The intersection consists of a collection of arcs or a single embedded loop. Moreover, since and are geodesic, every arc of is essential. Among the components , there are two which intersect each in a single component. Let be one such. Every arc of has endpoints on this single component of . There are two possibilities (not necessarily mutually exclusive):

1. There is an arc of and an arc of on the positive side of such that the interior of is disjoint from , and .

2. There are arcs of and arcs of on the positive sides of such that the interiors of are disjoint from , and .

In the first case we build a positive immersed surface with one boundary component on by attaching a -handle whose core is . In the second case we build a positive immersed surface with one boundary component on by attaching a -handle whose core is one of . See Figure 4.

The result of attaching a -handle with core to a product neighborhood of produces an embedded positive surface. If some component of the boundary of this surface is homotopically trivial, it is necessarily trivial on the positive side, so we cap it off with an embedded disk. Straighten the resulting surface by an isotopy until its boundary components are geodesic. This produces a new geodesic -manifold called a resolution of whose components are all embedded, such that bounds a positive immersed surface, and such that has at least two fewer intersections with than does. In particular, it is certainly true that each component of has fewer intersections with than does.

We resolve the components of in exactly the same way that we resolved above, and so on, inductively. If some of the resulting components are (isotopic to) elements of , they cobound an immersed positive surface with some positive sum of positively oriented boundary components of as above.

After finitely many steps, we obtain chains and , where is a positive sum of embedded geodesics, each disjoint from , where is a positive sum of positively oriented boundary components of , such that bounds a positive immersed surface.

Now, let be a component of adjacent to . By construction, each component of intersects at most one component of in . So the components of the can be iteratively resolved by the method above. By induction, we end up (finally) with a chain of the desired form, proving the claim. ∎

Let be a component of as in the claim. Let be the component of containing , and let be obtained from by filling in all but exactly one boundary component. For notational simplicity, denote the geodesics by respectively, and let be integers such that in .

Since is a geodesic on but not in , some component of might be a bigon which contains one, or several components of .

By pushing repeatedly over such components of we can eliminate such bigons, innermost first. If is the geodesic in obtained from by pushing over one component of , then and are disjoint, and either or bounds a positive embedded subsurface of . By gluing up finitely many such surfaces, we obtain a geodesic in , such that plus some union of boundary components of bounds a positive immersed surface in , and such that is in the isotopy class of a configuration of geodesics for some hyperbolic structure on .

In this way we are reduced to arguing about embedded curves in a once-punctured torus. We will show that rationally bounds a positive immersed surface in . If we can find such an immersed surface, then by drilling out the components of we will obtain a positive immersed surface in bounded by for some suitable which is a positive sum of boundary components of .

By induction, and the (well-known) classification of simple curves in a once-punctured torus, it suffices to show that the chain bounds an immersed surface in the once-punctured torus for sufficiently large . In fact, this turns out to be true for . Since the algebraic area in the once-punctured torus enclosed by is , it suffices (by Lemma 3.7 and Proposition 3.8) to show that which can be verified by calculation, e.g. using scallop (also see Figure 5 below).

Applying Lemma 3.3, we conclude that rationally bounds a positive immersed surface, where is some positive sum of boundary components of . By adding on sufficient copies of we obtain a positive immersed surface with boundary , where is a positive sum of boundary components of . But is an arbitrary component of . By the claim and Lemma 3.3, we obtain a positive immersed surface with rational boundary for suitable . Since was arbitrary, this proves the theorem. ∎

Remark 3.11.

Notice that we do not assume that is nonempty, just that . It is only the subsurfaces which are required to have nonempty boundary, which will be the case, since each has genus , and implies that the genus of is at least .

By the results of § 3.3 and § 2.4 we conclude:

Theorem A.

Let be a free group, and let be a compact, connected, orientable surface with and . Let be the -chain represented by the boundary of , thought of as a finite formal sum of conjugacy classes in . Then the projective ray in spanned by intersects the unit ball of the scl norm in the interior of a face of codimension one in .

Theorem B.

Let be a free group, and let be a compact, connected, orientable surface with and . Let be the face of the scl unit norm ball whose interior intersects the projective ray of the class . The face is dual to the extremal homogeneous quasimorphism .

Remark 3.12.

For the sake of completeness, we exhibit a positive immersed surface with rational boundary in Figure 5.

The figure depicts a genus surface with boundary components. The boundary components are (cyclically) labeled by words in (for clarity, and are used in place of and ). The components are decomposed into arcs each labeled by a letter, such that adjacent arcs have opposite labels. It follows that there is a (unique) homotopy class of map where is a once-punctured torus with standard generators for taking each boundary component of to the geodesic in corresponding to the conjugacy class of the boundary label. Two components of each wrap twice around (the boundary of ). Two other components of wrap once each around . One component of wraps twice around , and one component wraps twice around . Hence represents the chain . Since , the homotopy class of is represented by an immersion.

3.5. Remarks and Corollaries

In this section we collect some miscellaneous remarks and corollaries of our main theorems. The first remark is that one can give a new proof of the relative version of rigidity theorems of Goldman and Burger-Iozzi-Wienhard ([11, 3]; also compare Matsumoto [16]) about representations of surface groups with maximal Euler class.

The context is as follows. Let be a compact oriented surface with boundary, and let be a symplectic representation for which the conjugacy classes of boundary elements fix a Lagrangian subspace. In this case, there is a well-defined relative Euler class in associated to . It is well-known in this context that is a bounded cohomology class, and satisfies .

Our methods give a surprisingly short new proof of the following theorem (due to Goldman for and Burger-Iozzi-Wienhard for ). For simplicity we restrict to Zariski dense representations; this restriction can be removed by analyzing various cases, but since the main virtue of our alternate argument is its brevity, it is probably not worth spelling out the details.

Corollary 3.13.

Let be a compact oriented surface with boundary. Let be Zariski dense, and suppose that boundary elements fix a Lagrangian subspace (so that the relative Euler class is defined). If is maximal, is discrete.

Proof.

In what follows, denote by and its commutator subgroup by . Since has boundary, where is a homogeneous quasimorphism on , unique up to elements of . For each , the value of mod is the symplectic rotation number of . The symplectic rotation number lifts to a quasimorphism on the universal cover of with defect . On the other hand, so we can conclude that the defect of on is exactly , and is extremal for . Hence by Theorem B we conclude that the symplectic rotation number of every element of is zero, and therefore (in particular) is not dense in . Since is simple, every Zariski dense subgroup is either discrete or dense (in the ordinary sense). If is dense, then the closure of is normal in . But is simple, and the closure of is a proper subgroup; hence is discrete. ∎

Bavard [1] asked whether scl takes values in in a free group. Though this turns out not to be the case, nevertheless, elements with values in are very common. Theorem C gives a flexible method to construct many elements in free groups with scl in . For example, from Lemma 2.4 we conclude:

Corollary 3.14.

Let denote the free group on two generators and let