Surface subgroups from homology
Let be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If is nonzero, then contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on is a finite-sided rational polyhedron.
A famous question of Gromov (see ) asks whether every one-ended non-elementary word-hyperbolic group contains a closed hyperbolic surface subgroup. Almost nothing is known about this question in general. Gordon–Long–Reid  answer the question affirmatively for Coxeter groups and some Artin groups.
Bestvina remarks that Gromov’s question is inspired by the well-known virtual Haken conjecture in -manifold topology. The case of -manifold groups is instructive. If is an aspherical -manifold, every integral homology class in is represented by an embedded surface . If is not -injective, Dehn’s lemma (see , Chapter 4) implies that can be compressed, reducing . By the hypothesis that is aspherical, after finitely many compressions, one obtains a -injective surface representing the given homology class.
For more general classes of groups, no tool remotely resembling Dehn’s lemma exists. Nevertheless one can consider the following strategy. Let be a , and let be a rational homology class in . Suppose one can find a map of a closed surface with no spherical components, representing in for some integer , which realizes the infimum of over all surfaces and all integers . Then is injective. For, otherwise, one could find an essential loop in the kernel of and (by Scott ) find a suitable finite cover of to which lifts as an embedded loop. Then could be compressed along , producing a new surface representing in homology, and satisfying , contrary to hypothesis. This infimal quantity is called the Gromov–Thurston norm of the homology class (see  or  for an introduction to Gromov–Thurston norms and bounded cohomology). In words, if a map from a surface to realizes the Gromov–Thurston norm in a given projective homology class, it is injective.
It is therefore an intriguing question to understand for which groups and which homology classes in one can find maps of surfaces (projectively) realizing the Gromov–Thurston norm. In this paper we show that if is a group obtained as a graph of free groups amalgamated along cyclic subgroups, and is a homology class with nonzero Gromov–Thurston norm, then some map of a surface to a realizes the Gromov–Thurston norm in the projective class of , and therefore contains a closed hyperbolic surface subgroup. The method of proof is to localize the problem to finding norm minimizers for suitable relative homology classes in the free factors. The relative Gromov–Thurston norm (after normalization) turns out to be equal to the so-called stable commutator norm, introduced in , and studied in free groups in . A consequence of the main theorem of  is that extremal surfaces for the stable commutator norm exist in every rational relative homology class in a free group. These extremal surfaces can be glued together to produce extremal (closed) surface subgroups in . A more careful analysis reveals that the Gromov–Thurston norm on is piecewise rational linear, and if is word-hyperbolic, the unit ball is a finite-sided rational polyhedron.
2. The scl norm
2.1. Commutator length
If is a group and , the commutator length of (denoted ) is the smallest number of commutators in whose product is equal to , and the stable commutator length of (denoted ) is the limit
Geometrically, is the least genus of a surface group which bounds homologically. Since genus is not multiplicative under covers but Euler characteristic is, one can derive a formula for scl in terms of Euler characteristic; we give such a formula in Definition 2.2 below.
Stable commutator length is, in a sense to be made precise shortly, a kind of relative Gromov–Thurston norm.
Let be a compact orientable surface. Define
where the sum is taken over connected components of .
Let be a group. Let be elements in (not necessarily distinct). Let be a connected CW complex with . Further, for each , let be a loop in in the free homotopy class corresponding to the conjugacy class of .
If is an orientable surface, a map is admissible of degree for some positive integer if there is a commutative diagram
|.8............................................................................................................................. ......................................................................................................... ........................................................................... ..................................................................................................................................................................................................................................................................................................|
so that the homology class of is equal to times the fundamental class of in .
where the infimum is taken over all admissible maps of surfaces. If no admissible surfaces exist, set .
If has enough room (e.g. if is a manifold of dimension ) then the maps can be taken to be embeddings, and one can speak of the maps and their images interchangeably. In this context, one can think of an admissible map as a map of pairs which wraps around each with total degree .
When , the value of is the same with either definition above.
The function scl can be extended to integral group -chains, by the formula
and extended to rational chains by linearity, and to real chains by continuity. It is finite exactly on group -chains which are boundaries of group -chains; in other words, scl defines a pseudo-norm on the real vector space , hereafter denoted .
Notice that scl is, by construction, a homogeneous class function in each variable separately. If denotes the subspace of spanned by elements of the form and for and , then scl descends to a pseudo-norm on .
2.2. Comparison with Gromov and filling norms
Let be the bar chain complex of a group (see e.g.  Ch. IV, § 5 for details). In the sequel, the coefficient group is understood where omitted. There is a natural basis for in each dimension, and each becomes a Banach space with respect to the natural norm. This norm induces a pseudo-norm on (group) homology, called the Gromov norm (or norm) defined by
where the infimum ranges over all cycles representing a homology class .
If is a , the norm on may be calculated geometrically by the formula
There is a natural norm on , called the (Gersten) filling norm, introduced in , defined by the formula
where denotes the norm on group -chains. Let be the homogenization of ; i.e.
where and . Then fill descends to a function on and satisfies
For for , this is proved in Bavard ; the general case follows basically the same argument, and is found in , § 2.6. The factor of arises because fill counts triangles, whereas scl counts genus. This explains the sense in which scl can be thought of as a relative Gromov–Thurston norm.
2.3. Extremal surfaces
Given an integral chain , an admissible surface is extremal if it realizes
The Rationality Theorem from , is the following:
Theorem 2.5 (Rationality Theorem,  p.15).
Let be a free group.
for all .
Every integral chain in bounds an extremal surface
The function scl is piecewise rational linear on
There is an algorithm to calculate scl on any finite dimensional rational subspace of
In fact, in , bullet (2) merely says that every rationally bounds an extremal surface, but the argument of the proof establishes the more general statement. The method of proof makes this clear: let be a handlebody with , and let be loops in representing the free homotopy classes of the . In  it is shown that there is a simple branched surface , with boundary mapping to , which carries every admissible surface (after compression and homotopy). The function is a rational linear function of weights on , and therefore may be calculated on any rational class by solving a linear programming problem. An extremal vector obtained e.g. by the simplex method will be rational, and after scaling, is represented by an extremal surface.
We will also use the following technical Lemma, which is Lemma 4.2. from :
Let be a connected surface, and an extremal surface rationally bounding . Then there is another extremal surface rationally bounding , for which every component of maps to with positive degree.
The same argument shows that if bounds some collection , one may replace if necessary by another extremal surface for which every map of a boundary component of to every component has positive degree. Such an extremal surface is said to be positive. Hence in the sequel we will assume that all our extremal surfaces are positive.
From our perspective, the importance of extremal surfaces is the following:
Let be an extremal surface for . Then is incompressible and boundary incompressible. That is, is injective, and if is an essential immersed proper arc with endpoints on components of both mapping to , there is no arc so that is homotopically trivial in .
Suppose represents a conjugacy class in the kernel of . Since surface groups are LERF (), there is a finite cover of to which lifts as an embedded loop. The lifted map is admissible, with , so is also extremal. But can be compressed along the (now embedded) loop , reducing while keeping fixed, thereby contradicting the fact that was extremal.
Similarly, suppose is an arc such that is homotopically trivial in . Let be a cover of in which is embedded. Let be obtained from by attaching a -handle to , and let be equal to on , and map the core of to . Then . However, the union of with the core of is an essential embedded loop in which maps to a homotopically trivial loop in . Hence we can compress this loop, obtaining with , thereby contradicting the fact that was extremal. ∎
A similar argument shows that if a closed surface realizes the Gromov–Thurston norm in its homology class, it is injective. In the sequel, by abuse of notation, we will use the phrase “ is injective” to mean that is incompressible and boundary incompressible.
3. Surface subgroups
3.1. Graphs of free groups
A graph of groups is a collection of groups indexed by the vertices and edges of a connected graph, together with a family of injective homomorphisms from the edge groups into the vertex groups. Formally, let be a connected graph. For each vertex there is a vertex group , and for each edge an edge group so that for each inclusion as an endpoint, there is an injective homomorphism .
The fundamental group of a graph of groups (as above) is defined as follows. Let be the group generated by all the groups and an element for each (oriented) edge with relations that each edge element conjugates the subgroup of to the subgroup of , where is the initial vertex of and is the final vertex, with respect to the choice of orientation on . Let be a maximal subtree of . Then define to be the quotient of by the normal subgroup generated by elements corresponding to edges of .
By abuse of notation, we sometimes say that is a graph of groups with graph . See e.g. Serre  § 5.1. for more details.
In the sequel, let be a graph of groups with graph satisfying the following properties:
Every vertex group is free of finite rank
Every edge group is cyclic
The graph is finite
We say that such a group is a graph of free groups amalgamated over cyclic subgroups.
3.2. Hyperbolic groups
A path-metric space is -hyperbolic for some if for every geodesic triangle , the edge is contained in the (metric) -neighborhood of the union of edges .
A group with a finite generating set is word-hyperbolic (or just hyperbolic for short) if the Cayley graph is -hyperbolic as a path metric space, for some finite .
Hyperbolic groups are introduced in , inspired in part by work of Cannon, Epstein, Rips and Thurston. The theory of hyperbolic groups is vast; the only property of hyperbolic groups we will need is that they do not contain or Baumslag–Solitar subgroups. Here the Baumslag–Solitar group () is given by the presentation
Note that as a special case.
3.3. Construction of surface subgroups
We are now in a position to state the main theorem of this paper.
Let be a graph of free groups amalgamated over cyclic subgroups. If is word-hyperbolic, and is nonzero, then contains a closed hyperbolic surface subgroup. Furthermore, the unit ball of the Gromov–Thurston norm in is a finite-sided rational polyhedron.
We build a space with as follows. For each vertex let be a handlebody with . For each edge let be an annulus. For each let be an embedded loop representing the conjugacy class of the generator of , and glue the corresponding boundary component of to along . The Seifert van-Kampen theorem justifies the equality . In fact, since each and is a , and since the edge homomorphisms are all injective, the space itself is a . See e.g. , Theorem 1B.11. p.92. Hence for all coefficient groups .
Let denote the union of the cores of the annuli . Let and let be a regular neighborhood of . The Mayer–Vietoris sequence contains the following exact subsequence
Since , it follows that an element of is determined by its image in . Geometrically, let be obtained from by crushing each and a cocore of each to a point. Then is a wedge of ’s, one for each . The induced map is an injection, and an element of is determined by the degree with which it maps over each sphere summand of .
Let be a nonzero class in represented by a map of a closed surface . If we make transverse to the core of each and adjust by a homotopy, we can assume that is a union of subsurfaces of each mapping properly to . If is a component of , the degree of is equal to the number of times winds (with multiplicity) around either boundary component of . If some maps to some with degree , compress a suitable subsurface of and push it off by a homotopy.
For each , let denote the set of loops in which are the boundaries of components of the various . The surface maps to with boundary wrapping various times around the various . Let be such that is an admissible surface bounding . Note that the (for various ) are determined by the homology class , and are precisely the coefficients of the element with respect to a basis for consisting of the various .
For each , let be an extremal surface for . Note that represents in for some integer . If the various could be glued together along their boundary components compatibly with , the components of the resulting surface would be injective, and their union would map to , representing a multiple of the class in . If is word-hyperbolic, we will show how to construct suitable covers of the which can in fact be glued up.
Let be an orientable surface with nonempty boundary components . For , let be some function. If then extends to a function whose kernel defines a regular cover of with the property that each boundary component in the preimage of maps to with degree equal to the order of in .
Homomorphisms from to abelian groups are exactly those which factor through the abelianization . The components determine elements of which are subject only to the relation . This follows directly from the exact sequence in relative homology
together with the fact that and .
The other statements are standard facts from the theory of covering spaces. ∎
By invoking Lemma 3.5 repeatedly, we will construct covers of the which can be glued up over the various one by one. Let be an edge, and the end vertices. Let and be the loops along which the boundary components of are attached. Suppose we have surfaces and mapping to and subsets of the boundary components which map to and respectively. By Lemma 2.6 we can assume that each component of maps to with positive degree, and similarly for . Note that we should allow the possibility that .
Assume for the moment that and . If is a surface of negative Euler characteristic, then admits a finite cover with positive genus. Furthermore, if has positive genus, then admits a degree cover for which every boundary component of has exactly two preimages in , each of which maps with degree . So without loss of generality, we can assume that the components of come in pairs which each map to with the same degree, and similarly for the components of .
Let be the least common multiple of the degrees of maps from components of either or to or . We will define homomorphisms and as follows. If are a pair of components of mapping to with the same degree , then define and , and define similarly on pairs of components of . Note that may be extended to have the value on components of and not appearing in or . Let be the corresponding covers. Then by construction, every component of maps to with degree , and every component of maps to with degree , so and can be glued up along and and their maps to extended over . Proceeding inductively, we can construct a surface and a map representing some integral multiple of the class . Since is made by gluing covers of injective maps, the definition of injective and the Seifert van-Kampen theorem implies that every component of is injective.
If for some (possibly intermediate) surface , then consists of a union of annuli. If not every component is being glued up to , one can still take covers of these annuli as above and glue up. The only potentially troublesome case is when , and the free boundary components of are being glued up to each other. But in this case, contains the mapping cylinder of an injective map from to itself; i.e. it contains a or a Baumslag–Solitar group, and is therefore not hyperbolic. This proves the first part of the theorem.
To prove the statement about Gromov–Thurston norms, observe that if in homology, then by construction . Hence , as constructed, realizes the Gromov–Thurston norm in its homology class. Note that this gives another proof that is injective. In fact, for each , let be the subspace spanned by the along which various are attached. The boundary map in the Mayer–Vietoris sequence defines an integral linear injection with components , and by the construction above,
Since each is an integral linear map, and the scl pseudo-norm on each is a rational piecewise-linear function, the norm (i.e. the Gromov–Thurston norm) on is a piecewise rational linear function. Since is hyperbolic, the unit ball is a (nondegenerate) finite sided rational polyhedron. ∎
If is not necessarily word-hyperbolic, it is nevertheless true (essentially by the argument above) that the Gromov–Thurston pseudo-norm on is piecewise rational linear. Moreover, the same argument shows that for any obtained as a graph of free groups amalgamated over cyclic subgroups, and for any homology class , either some multiple of is represented by an injective closed hyperbolic surface, or .
If is a compact -manifold, every integral class in is represented by an embedded surface which realizes the infimum of in its projective class. It follows that the Gromov–Thurston norm (with Gromov’s normalization) takes on values in on . This by itself ensures that the unit ball is a rational polyhedron. However, for a graph of free groups as above, the Gromov–Thurston norm can take on rational values with arbitrary denominators on elements of , so the polyhedrality of the norm is more subtle. See , § 4.1.9.
While writing this paper I was partially supported by NSF grant DMS 0405491. I would like to thank Mladen Bestvina, Benson Farb and Dongping Zhuang for comments and some discussion. The work in this paper builds largely on work done in  which benefited greatly from many discussions with Jason Manning.
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