Abelian covers of graphs and maps between outer automorphism groups of free groups
We explore the existence of homomorphisms between outer automorphism groups of free groups . We prove that if is even and , or is odd and , then all such homomorphisms have finite image; in fact they factor through . In contrast, if with coprime to , then there exists an embedding . In order to prove this last statement, we determine when the action of by homotopy equivalences on a graph of genus can be lifted to an action on a normal covering with abelian Galois group.
Key words and phrases:automorphism groups of free groups, group actions on graphs
1991 Mathematics Subject Classification:20F65, 20F28, 53C24, 57S25
The contemporary study of mapping class groups and outer automophism groups of free groups is heavily influenced by the analogy between these groups and lattices in semisimple Lie groups. In previous papers [4, 5, 6] we have explored rigidity properties of in this light, proving in particular that if then any homomorphism has image at most , and that the only monomorphisms are the inner automorphisms. In this paper we turn our attention to the case .
There are two obvious ways in which one might embed in when : most obviously, the inclusion of any free factor induces a monomorphism ; secondly, if is a characteristic subgroup of finite index, then the restriction map is injective (Lemma 2.3). Neither of these constructions sends the group of inner automorphisms into , so there is no induced map . In the second case one can often remedy this problem by passing to a subgroup of finite index in . Thus in Proposition 2.5 we prove that if for some , then has a subgroup of finite index that embeds in ; for example a finite-index subgroup of embeds in . But if we demand that our homomorphisms be defined on the whole of , then it is far from obvious that there are any maps with infinite image when .
As usual, the case is exceptional: maps to with finite kernel, so to obtain a map with infinite image one need only choose elements of order and that generate an infinite subgroup of . Khramtsov  gives an explicit monomomorphism . More interestingly, he proved that there are no injective maps from to . So, for given , for which values of is there a monomorphism , and for which values of do all maps have finite image? These are the questions that we address in this article. In the first part of the paper we give explicit constructions of embeddings, and in the second half we prove, among other things, that no homomorphism can have image bigger than if is even and . This last result disproves a conjecture of Bogopolski and Puga .
In order to construct embeddings, we consider characteristic subgroups , identify with the subgroup of consisting of inner automorphisms, and examine the short exact sequence
We want to understand when this sequence splits. When it does split, one can compose the splitting map with the map induced by restriction, , to obtain an embedding of into .
Bogopolski and Puga  used algebraic methods to obtain a splitting in the case where with odd and coprime to , yielding embeddings when . We do not follow their arguments. Instead we adopt a geometric approach which begins with a translation of the above splitting problem into a lifting problem for groups of homotopy equivalences of graphs. Proposition 2.1 provides a precise formulation of this translation. (The topological background to it is difficult to pin down in the literature, so we explain it in detail in an appendix.)
The following theorem is the main result in the first half of this paper.
Let be a normal covering of a connected graph of genus with abelian Galois group . The action of by homotopy equivalences on lifts to an action by fiber-preserving homotopy equivalences on if and only if with coprime to .
When translated back into algebra, this theorem is equivalent to the statement that if a characteristic subgroup contains the commutator subgroup , then the short exact sequence splits if and only if , where is the subgroup generated by -th powers and is coprime to . The sufficiency of this condition extends Bogopolski and Puga’s theorem to cover the case where is even.
There exists an embedding for any of the form with coprime to .
The negative part of Theorem A also has an intriguing application. It tells us that does not split. Thus this sequence defines a non-zero class in the second cohomology group of with coefficients in the module (i.e. the standard left -module ). The theorem also assures us that this class remains non-trivial when we take coefficients in , provided that is not coprime to . The non-triviality of these classes provides a striking counterpoint to what happens when one takes coefficients in the dual module , as we shall explain in Section 5.
Let be the standard -module and let be its dual. Then , but if .
Theorem A exhausts the ways in which one might obtain embeddings by lifting the action of to covering spaces with an abelian Galois group, but one might hope to construct many other embeddings using non-abelian covers. Indeed the construction developed by Aramayona, Leininger and Souto in the context of surface automorphisms  proceeds along exactly these lines and, as they remark, it can be adapted to the setting of . However, in the embeddings obtained by their method, is bounded below by a doubly exponential function of , whereas in our construction we can take if is even. If is odd, then the smallest value we obtain is where is the smallest prime that does not divide ; in Section 2.1 we describe how quickly grows as a function of .
In the second part of this paper we set about the task of providing lower bounds on the value of such that there is a monomorphism , or even a map with infinite image.
Suppose . If is even and , or is odd and , then every homomorphism factors through .
Note how this result contrasts with our earlier observation that has a subgroup of finite index that embeds in when . The key point here is that subgroups of finite index can avoid certain of the finite subgroups in (indeed they may be torsion-free), whereas our proof of Theorem C relies on a detailed understanding of how the finite subgroups of can map to under putative maps . Two subgroups play a particularly important role, namely , the group of symmetries of the -rose , and , the group of symmetries of the -cage, i.e. the graph with vertices and -edges. Indeed the key idea in the proof of Theorem C is to show that no homomorphism can restrict to an injection on both of these subgroups. In order to establish this, we have to analyze in detail all of the ways in which these finite groups can act by automorphisms on graphs of genus at most . In the light of the realization theorem for finite subgroups of , this analysis amounts to a complete description of the conjugacy classes of the finite subgroups in that are isomorphic to and (cf. Propositions 6.7, 6.10 and 6.12). We believe that these results are of independent interest.
Beyond , the analysis of and becomes more complex, but several crucial facts extend well beyond this range (e.g. Lemma 6.5 and Proposition 7.1). Moreover, Dawid Kielak  has recently extended our methods to improve the bound . Thus, at the time of writing, we have no good reason to suppose that the lower bound that Theorem C imposes on the least with is any closer to the truth that the exponential upper bound provided by Theorem A.
We thank Roger Heath-Brown, Dawid Kielak and Martin Liebeck for their helpful comments.
2. Theorem A: Restatement and Discussion
In the appendix to this paper we explain in detail the equivalence of various short exact sequences arising in group theory and topology. In the case of graphs, the basic equivalence can be expressed as follows.
Let be a characteristic subgroup of a free group let be a connected graph with fundamental group , let be the covering space corresponding to , let be the group of free homotopy classes of homotopy equivalences of , and let be the group of fiber-preserving homotopy classes of fiber-preserving homotopy equivalences of . Note that the deck transformations of lie in the kernel of the natural map
The following diagram of groups is commutative and the vertical maps are isomorphisms:
The characteristic subgroups with abelian are the commutator subgroup and , the subgroup generated by and all th powers in . By combining this observation with the preceding proposition, we see that Theorem A is equivalent to the following statement.
Let be a free group of rank and let be a characteristic subgroup with abelian. Then the short exact sequence
splits if and only with coprime to .
Any splitting of the sequence in Theorem 2.2 gives a monomorphism , which we can compose with the restriction map
To complete the proof of Corollary A we need to know that this last map is injective. This follows from the observation below.
If is a finitely generated free group and is a characteristic subgroup of finite index, then the restriction map is injective.
If is the index of in and is an arbitrary element of , then . If is in the kernel of the restriction map , then . But elements in have unique roots, so and is the identity. ∎
2.1. Expected value of
The subgroup has index in so is free of rank . Thus the smallest for which we obtain an embedding from Theorem 2.2 is , where is the smallest prime which does not divide . If is even we can take but for odd the size of as a function of is not obvious. However, it turns out that the expected value of is a constant (which is approximately equal to ). We are indebted to Roger Heath-Brown for the following argument.
For any natural number , let denote the smallest prime number which does not divide and let be the product of all prime numbers strictly less than (with ). An easy consequence of the Prime Number Theorem is that is asymptotically equal to . This implies in particular that the infinite series used to define in the following proposition is convergent.
The expected value
exists and is equal to the constant
where the sum is over all primes .
Note that if and only if divides and does not divide . The first statement implies, taking logs, that , so can be of order at most .
As we just observed, the primes that contribute to the above sum have order at most , so
Letting , we get . ∎
Given , the smallest value of for which Corollary A yields an embedding is , and the preceding proposition tells us that “on average” this is no greater than an exponential function of . In the worst case, can be larger but still only on the order of . Indeed the worst case arises when for some , in which case , and since we see that grows like .
2.2. Embedding a subgroup of finite index
Corollary A gives conditions under which the entire group embeds in . If we relax this to require only that a subgroup of finite index of should embed in , we can obtain many more embeddings as follows.
For all positive integers and , there exists a subgroup of finite index and a monomorphism , where .
For the proposition is trivial, and for it follows immediately from the fact that has a free subgroup of finite index. So we assume that and fix an epimorphism from to a wreath product , where is any finite 2-generator centerless group ( for example). Let be the kernel of this epimorphism and let be the kernel of the composition .
The set of subgroups in that have the same index as is finite, as is the set that have the same index as . The action of on each of these sets defines a homomorphism to a finite symmetric group; define to be the intersection of the two kernels. Note that leaves invariant both and . Let be the kernel of the natural map and note that since the center of is trivial, the intersection of with is contained in , and hence in .
Euler characteristic tells us that the rank of the free group is . The restriction map , which is injective as in Lemma 2.3, induces an injection . To complete the proof, it suffices to note that is the image of in , since . ∎
The preceding argument shows that if is the kernel of a map from onto a finite centerless group, then a subgroup of finite index in injects into .
3. Proof of Theorem A: The existence of lifts
where is a 1-vertex graph with loops (a rose) and is the covering space corresponding to , where with coprime to . We work with an explicit presentation of . We take explicit homotopy equivalences of that generate , lift each to a homotopy equivalence of the universal abelian covering of , project down to , and prove that the resulting elements of satisfy the defining relations of our presentation. The case is special: for one can split .
The generators and relations we will use for are based on those given by Gersten in  for . We fix a generating set for . Gersten gives an elegant and succinct presentation using generators with ; here corresponds to the automorphism which sends and fixes all elements of other than and . In Gersten’s paper automorphisms act on on the right and the symbol means . In the current paper we want automorphisms to act on the left to be consistent with composition of functions in , but we would like to use the same commutator convention. Thus for us a Gersten relation of the form becomes or, equivalently, . His relations, then, are the following:
We will need to distinguish between right transvections and left transvections , for , so we rewrite Gersten’s relations using the translation , and .
In terms of the and , Gersten’s first relation is unnecessary and the rest of the presentation for becomes
To get a presentation for we must add a generator , corresponding to the automorphism , and relations
Finally, to get a presentation for we kill the inner automorphisms by adding the relation
We orient the petals of and label them with the generators . If we fix a base vertex of , we may think of as the 1-skeleton of the standard hypercubulation of with vertices in . The lift starting at of the edge labeled is identified with the standard -th basis vector .
Any automorphism of is realized on by a homotopy equivalence sending the petal labeled to the (oriented) path which traces out the reduced word . This has a standard lift to a -equivariant homotopy equivalence of , which sends to the lift starting at of the path labeled by the reduced word . (Since the homotopy equivalence is -equivariant, it suffices to describe its effect on the edges .) This in turn induces a lift to the quotient for each , which is trivial in if and only if is fiberwise-homotopic to a deck transformation by an element of .
Lifting automorphisms to and by these standard lifts does not give a well-defined homomorphism on . This is because the standard lift of the inner automorphism sends to a -shaped path labeled . The extension to all of is freely homotopic to the deck transformation of . Since this deck transformation is not freely homotopic to the identity (even mod for any ), the assignment does not give well-defined map from to (much less to ).
We rectify this situation by choosing lifts which are shifted from the standard lifts by appropriate translations of . Since is coprime to , there are integers and with . We use the standard lift for , but for we choose the lift which shifts the standard lift by , and for we choose the lift which shifts the standard lift by . Thus on the vertices of , acts as a shear parallel to the direction, is a shear composed with a shift, and is reflection across the hyperplane . In particular, each of our lifts induces an affine map , with and . Each edge beginning at a vertex in the direction is sent to the path that begins at and is labeled .
We represent an affine map by the matrix , acting on the vector . Let denote the elementary matrix with one non-zero entry equal to in the position. Thus the action of on the -skeleton of is represented by the matrix with and ; for we have the matrix with and ; and for the matrix with and .
For example, for we have and
An important point to note is that since the relations (1) to (10) hold in and not just , in order to verify that the above assignments respect these relations we need only verify that the appropriate product of matrices is the identity: such a verification tells us that the corresponding product of our chosen lifts acts trivially on the vertices of , and the action on edges (which is defined in terms of the action on labels) is automatically satisfied. This remark does not apply to relation (11), which requires special attention.
For every integer coprime to , the lifts of , of and of define a splitting of the natural map .
We first claim that the maps and (and hence the maps they induce on ) satisfy relations (1) to (10). In each case, the verification is a straightforward calculation, which we illustrate with several examples using and . (In the light of remark 3.1, each verification simply requires a matrix calculation.)
An example of a relation of type (4) is .
To verify relation (6), we first compute the action of (we only need 2 indices),
As an example of relation (8) we verify :
Relation (11) is the only relation which requires some thought. For example, the matrix corresponding to the product , which lifts conjugation by , is
Thus for all , the map on sends the edge starting at in the direction to the -shaped path labeled starting at . Dragging all vertices of one unit along the edge parallel to gives a fiber-preserving homotopy of this map to the deck transformation . This deck transformation induces the identity on . ∎
For the above construction gives an embedding . Here is an explicit description of the images of the and under this embedding, where .
4. Proof of Theorem A: The non-existence of lifts
We begin by proving that for the map does not split when ; this is equivalent to the case in Theorem A. To do this, we consider the cyclic group of order that corresponds to the group of rotations of the marked graph shown in Figure 1.
The inverse image of in is torsion-free, and therefore does not split.
In this section we present three proofs of this fact. The first is a geometric proof that we feel gives the most insight into the non-splitting phenomenon; this is how we discovered Proposition 4.1. The second proof draws attention to a topological criterion illustrated by the first proof; like the first proof, it is executed using the lower sequence in Proposition 2.1. The third proof is purely algebraic. The first and third proofs also lead to a proof of the following proposition, which completes the proof of Theorem 2.2 (and therefore of Theorem A).
Let and let denote the natural map . Then the short exact sequence splits if and only if is coprime to .
4.1. A direct geometric proof
At several points in the following argument we use the elementary fact that if a connected metric graph is a union of (at least two) embedded circuits, then an isometry that is homotopic to the identity is actually equal to the identity.
Let be an integer and let be the graph that has vertices, contains a simple loop of length and has a loop of length at each of its vertices (see Figure 1).
We fix a maximal tree in the graph, label the remaining edge on the long circuit , and label the loops of length in cyclic order, proceeding around the long cycle: . This provides an identification of with .
Consider the maximal abelian cover of , that is the graph . The Galois group of this covering is and it is helpful to visualise the following embedding of in (see Figure 2).
Fix rectangular coordinates on and define to be the union of the following families of lines: family consists of all lines parallel to the -axis that have integer -coordinates for all , while consists of all lines parallel to the axis that have integer coordinates for all with and which have -coordinate an integer that is congruent to .
The action of the Galois group is by translations in the coordinate directions, with acting as translation by a distance in the direction for , and with acting as translation by a distance in the direction.
Now consider the isometry of that rotates the long cycle through a distance , carrying the oriented loop labelled to that labelled for and taking to . This isometry has order .
A lift of to is obtained as follows:
In other words, shifts by unit in the -direction and permutes the positive axes of the other generators cyclically. In particular, is the deck transformation corresponding to , so is not homotopic to the identity. Any power of which is not a multiple of sends the axis for to a translate of the axis for , for some , so is again not homotopic to the identity. This shows that has infinite order in .
If we choose a different lift of , then it differs from by some deck transformation . Then is the deck transformation , which is non-trivial (hence not homotopic to the identity) for any if . Thus has infinite order in . This proves Proposition 4.1.
If we look mod (i.e. work modulo the action of ), then the last deck transformation considered above can become trivial: the equation