## A finitely presented non-hyperbolic subgroup of a hyperbolic group

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 $F_3$. Hyperbolic groups are always of type $F_3$, so this kernel is not hyperbolic.

Branched covers

A subcomplex $X$ of a piecewise Euclidean (PE for short) cubical complex $K$ is a branching locus if it satisfies the following two conditions:
1) For each $m$-cell $\chi_e:\square_e^m\to e$ of $K$ with $e\cap X\neq \emptyset$, we have that $\chi_e^{-1}(X)$ is a disjoint union of faces of $\square_e^m$.
2) $Lk(e,K) \smallsetminus Lk(e,X)$ is nonempty and connected for each cell $e\subset L$.

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 $K$ be a finite PE cubical complex and $X$ be a branching locus in $K$. For a vertex $v\in X$, the link $Lk(v,X)$ is a full subcomplex of $Lk(v,K)$.

A branched covering $\hat{K}$ is obtained as follows:

1. Take a finite covering of $K \smallsetminus X$.
2. Lift the piecewise Euclidean metric to this covering.
3. Take the completion of the covering with respect to this metric.

Lemma 2. If $\hat{K}$ is a branched covering of a finite PE cubical complex $K$, then $\hat{K}$ is also a finite PE cubical complex and there is a natural continuous surjection $b:\hat{K}\to K$. If $K$ is nonpositively curved and the branching locus is a graph, then $\hat{K}$ is also nonpositively curved.

Denote the following graph by $\theta$ and let $K=\theta^3$.

Let $X$ be the graph $(\theta\times \{0\}\times \{1\})\cup ( \{1\}\times \theta \times \{0\} ) \cup (\{0\} \times \{1\}\times \theta )\subseteq \theta^3$, which is an example of a branching locus.

Let $\Delta=\theta\vee \theta$:

The fundamental group $\pi_1(\triangle,(1,0))$ is free of rank $6$. As a free basis, take the six loops $\bar{a_0}b_0,\bar{a_0}\bar{c_0},\bar{a_0}\bar{d_0},a_1\bar{b_1},a_1c_1,a_1d_1$. (The bars indicate paths traversed in the opposite direction to their orientations.) Let $\rho:\pi_1(\triangle,(1,0))\mapsto \textup{Sym}(5)$ be the homomorphism such that

$\bar{a_0}b_0\mapsto \alpha, \ \ \bar{a_0}\bar{c_0}\mapsto \alpha^2, \ \ \bar{a_0}\bar{d_0}\mapsto \alpha^3, \ \ a_1\bar{b_1}\mapsto \beta, \ \ a_1c_1\mapsto \beta^2, \ \ a_1d_1\mapsto \beta^3$

where $\alpha$ and $\beta$ are the permutations $(2354)$ and $(12345)$, respectively. The essential feature of this construction is that $\rho$ maps all commutators of $\bar{a_0}b_0,\bar{a_0}\bar{c_0}$ or $\bar{a_0}\bar{d_0}$ with $a_1\bar{b_1},a_1c_1$ or $a_1d_1$ to non–trivial five–cycles, since $\alpha \beta \alpha^{-1}=\beta^2$.

For $i=1,2,3$, let $pr_i:\theta^3\smallsetminus X\mapsto \theta^2\smallsetminus\{(0,1)\}$ be the projections $pr_1(x,y,z)=(y,z)$, $pr_2(x,y,z)=(z,x)$ and $pr_3(x,y,z)=(x,y)$. All are continuous, onto, and induce homomorphisms $pr_{i^{*}}:\pi_1(\theta^3\smallsetminus X,(0,0,0))\to \pi_1(\theta^2\smallsetminus \{(0,1)\},(0,0))$.

Let $p$ be a retract $\theta^2\smallsetminus \{(0,1)\}$ to $\theta\vee \theta$. For $i=1,2,3$ define $f_i := \rho\circ p_{*}\circ pr_{i*}:\pi_1(\theta^3\smallsetminus X,(0,0,0))\to \textup{Sym}(5)$.

The map $f=f_{1}\times f_{2}\times f_{3}:\pi_1(\theta^3\smallsetminus X,(0,0,0))\to S_5\times S_5\times S_5\hookrightarrow S_{125}$ defines a $125$-fold covering of $K \smallsetminus X$. Call the completion of this covering $\hat{K}$ and let $g$ be the natural map $\hat{K} \to \theta^3$.

Why $\pi_1(\hat{K})$ is hyperbolic

We know from Lemmas 2 that $\hat{K}$ is a piecewise Euclidean nonpositively curved complex, so its universal cover will be a $\textup{CAT}(0)$ space. By the following two theorems, to establish that $\pi_1(\hat{K})$ is hyperbolic, it suffices to show that the universal cover of $\pi_1(\hat{K})$ contains no isometrically embedded flat planes.

Theorem. Suppose that a group $G$ acts properly, cocompactly, and by isometries on a $\delta$–hyperbolic metric space. Then $G$ is a hyperbolic group.

Theorem. Let $X$ be a $\textup{CAT}(0)$ metric space which has a cocompact group of isometries, and which does not contain any isometric embeddings of the Euclidean plane. Then $X$ is a $\delta$–hyperbolic metric space.

Consider the vertices in $X$. The $1$-skeleton of the link complex around these (missing) vertices in $\theta^3\smallsetminus X$ 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 $\theta^2\smallsetminus \{0,1\}$ under exactly one of the maps $f_1,f_2$ or $f_3$. Such a loop deformation retracts to one of the commutator loops in $\theta\vee \theta$ described above. The homotopy class of these commutator loops are mapped to nontrivial five–cycles by $f$. Therefore in the branched cover, the link structure around these vertices is distorted, i.e. the aforementioned loops lift to $5$–fold copies.

For any isometrically embedded flat plane in the universal cover, there is a special edge $e$ 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 $x$. The link of $x$ in the universal cover can be defined in the usual way. Consider the intersection $I$ of this link with the flat plane.

The projection of $I$ to the link of one of the end points of $e$ is a well defined injective map. The projected images of the loops of $I$ 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 $\pi_1(\hat{K})$ contains a finitely presented non–hyperbolic subgroup

The orientations of the edges in $\theta^3$ 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 $h:\theta^3 \to S^1$.

The composition $p=h\circ g:\hat{K}\to S^1$ lifts to a map from the universal cover of $\hat{K}$ to $\mathbb{R}$. The ascending and descending links of any vertex in this Morse theory setting are $2$–spheres, which are $1$–connected but not $2$–connected. This implies that the kernel of the map $\pi_1(\hat{K})\to \mathbb{Z}$ induced by $p$ is finitely presented but not of type $F_3$, 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 $\hat{K}$ is hyperbolic, but has a finitely presented subgroup which is of type $F_2$ but not of type $F_3$, and so is not hyperbolic.

Further results

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.