It is natural to inquire whether hyperbolicity is inherited by subgroups. This question is easily seen to have a negative answer; finitely generated free groups of rank at least two are hyperbolic but have subgroups, such as their commutator subgroups, which are infinitely generated and so cannot be hyperbolic since hyperbolic groups are finitely presentable. Rips constructed the first example of a finitely generated non-hyperbolic subgroup of a hyperbolic group — see this post. The next natural question, then, is whether there are finitely presentable examples.
The answer is yes. Noel Brady gave the first example in his article Branched coverings of cubical complexes and subgroups of hyperbolic groups. (He gave a further exposition of the example in his 2007 CRM notes.)
Brady’s construction, which we will explain in detail, involves cubical complexes, their branched covers, and Bestvina–Brady Morse theory. He constructs a finite branched covering of a three–dimensional cubical complex which has a hyperbolic fundamental group. Then he defines a map from the branched covering to the unit circle which lifts to a Morse function on the universal covers. Combinatorial Morse theory is used to show that the map from the branched covering to the unit circle has a kernel which is finitely presented but not of type . Hyperbolic groups are always of type , so this kernel is not hyperbolic.
A subcomplex of a piecewise Euclidean (PE for short) cubical complex is a branching locus if it satisfies the following two conditions:
1) For each -cell of with , we have that is a disjoint union of faces of .
2) is nonempty and connected for each cell .
A subcomplex is full if whenever a set of vertices in the subcomplex spans a simplex in the complex, the simplex is itself in the subcomplex.
Lemma 1. Let be a finite PE cubical complex and be a branching locus in . For a vertex , the link is a full subcomplex of .
A branched covering is obtained as follows:
- Take a finite covering of .
- Lift the piecewise Euclidean metric to this covering.
- Take the completion of the covering with respect to this metric.
Lemma 2. If is a branched covering of a finite PE cubical complex , then is also a finite PE cubical complex and there is a natural continuous surjection . If is nonpositively curved and the branching locus is a graph, then is also nonpositively curved.
Denote the following graph by and let .
Let be the graph , which is an example of a branching locus.
The fundamental group is free of rank . As a free basis, take the six loops . (The bars indicate paths traversed in the opposite direction to their orientations.) Let be the homomorphism such that
where and are the permutations and , respectively. The essential feature of this construction is that maps all commutators of or with or to non–trivial five–cycles, since .
For , let be the projections , and . All are continuous, onto, and induce homomorphisms .
Let be a retract to . For define .
The map defines a -fold covering of . Call the completion of this covering and let be the natural map .
Why is hyperbolic
We know from Lemmas 2 that is a piecewise Euclidean nonpositively curved complex, so its universal cover will be a space. By the following two theorems, to establish that is hyperbolic, it suffices to show that the universal cover of contains no isometrically embedded flat planes.
Theorem. Suppose that a group acts properly, cocompactly, and by isometries on a –hyperbolic metric space. Then is a hyperbolic group.
Theorem. Let be a metric space which has a cocompact group of isometries, and which does not contain any isometric embeddings of the Euclidean plane. Then is a –hyperbolic metric space.
Consider the vertices in . The -skeleton of the link complex around these (missing) vertices in is a complete bipartite graph where each component has four vertices (along with some half edges i.e. edges with one end–point missing). Each loop is this graph that consists of four edges maps to the boundary of a neighborhood of a point in under exactly one of the maps or . Such a loop deformation retracts to one of the commutator loops in described above. The homotopy class of these commutator loops are mapped to nontrivial five–cycles by . Therefore in the branched cover, the link structure around these vertices is distorted, i.e. the aforementioned loops lift to –fold copies.
For any isometrically embedded flat plane in the universal cover, there is a special edge in the universal cover intersecting this plane that is not parallel to it, which maps to an edge in the branching locus. Let the intersection of the flat plane with this edge be . The link of in the universal cover can be defined in the usual way. Consider the intersection of this link with the flat plane.
The projection of to the link of one of the end points of is a well defined injective map. The projected images of the loops of are loops of more than four edges, so there is a natural “angle sum” barrier to the local embeddability of such a flat plane around this edge.
Why contains a finitely presented non–hyperbolic subgroup
The orientations of the edges in give rise to a map to the oriented unit circle mapping the edges to the cirlce respecting the orientation. This extends linearly to a map .
The composition lifts to a map from the universal cover of to . The ascending and descending links of any vertex in this Morse theory setting are –spheres, which are –connected but not –connected. This implies that the kernel of the map induced by is finitely presented but not of type , and in particular not hyperbolic. For further details of such Combinatorial Morse theory, see this earlier blog post.
We conclude that the fundamental group of is hyperbolic, but has a finitely presented subgroup which is of type but not of type , and so is not hyperbolic.
Gersten and Short proved that Brady’s subgroup satisfies a polynomial isoperimetric inequality. Its exact Dehn function remains unknown.
Brady, Clay and Dani have used a variant of the construction set out above to give a hyperbolic group with a finitely presented subgroup that contains infinitely many conjugacy classes of finite–order elements (and so is not hyperbolic). Predecessors to this example were discussed in this post.