## Farewell (and what did we miss?)

“I believe in everything until it’s disproved. So I believe in fairies, the myths, dragons.” — John Lennon

Our tour of subgroups of non–positively curved groups has reached its end.

What did we miss? Probably lots — apologies to any aggrieved authors out there.

Here are a couple of corners of the landscape that we are conscious of having failed to explore.

Most likely, there are many more.

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## Cannon-Thurston maps for graphs of groups

Recall from our last post that given an infinite hyperbolic subgroup ${H}$ of a hyperbolic group ${G}$, the extension of the inclusion ${i:H \hookrightarrow G}$ to a continuous ${\hat{i}: \partial H \rightarrow \partial G}$ is called a Cannon-Thurston map. By the continuity of ${i}$, if such a map exists it’s unique.

When ${H}$ is quasi-convex in ${G}$, clearly a Cannon-Thurston map exists: equivalent geodesics in ${H}$ are mapped to equivalent quasi-geodesics in ${G}$. However, the general question of when a Cannon-Thurston map exists if ${H}$ is distorted in ${G}$ is still open. In this post we’ll give three partial results directed at answering this question: the original result of Cannon and Thurston, and two generalizations made by Mitra.

1. Three results

1. Cannon-Thurston:
Given ${G \cong \pi_1(M)}$, where ${M}$ is a hyperbolic closed 3-manifold fibering over a circle, and ${H = \pi_1(S)}$, where ${S}$, a closed 2-manifold, is the fiber surface, there is a Cannon-Thurston map ${\hat{i}: \partial H \rightarrow \partial G}$.

2. Mitra:
Suppose $H$ is an infinite hyperbolic normal subgroup of a hyperbolic group $G$. Then there exists a Cannon-Thurston map ${\hat{i}: \partial H \rightarrow \partial G}$.

3. Mitra:
Let ${\Gamma}$ be a finite graph of groups where all vertex groups ${\Gamma_v}$ and edge groups ${\Gamma_e}$ are hyperbolic, and all injections ${\Gamma_e \rightarrow \Gamma_v}$ (where ${e}$ is incident to ${v}$) are quasi-isometric embeddings. Suppose that ${G = \pi_1(\Gamma)}$ is also hyperbolic. Then for any vertex ${v \in \Gamma}$, there is a Cannon-Thurston map ${\partial \Gamma_v \rightarrow \partial G}$.

2. Key lemma

Both of Mitra’s results above make use of the following key lemma.

Lemma 1. Let ${i:Y \hookrightarrow X}$ be a proper embedding of proper hyperbolic spaces. Suppose, given ${y_0 \in Y}$, that there is a non-negative function ${M(N)}$ such that ${M(N) \rightarrow \infty}$ as ${N \rightarrow \infty}$ and for all geodesic segments ${\lambda \subset Y}$ outside an ${N}$-ball around ${y_0 \in Y}$, any geodesic in ${X}$ joining endpoints of ${i(\lambda)}$ is outside the ${M(N)}$-ball around ${i(y_0)}$ (see the illustration below). Then there’s a Cannon-Thurston map ${\hat{i}: \partial Y \rightarrow \partial X}$.

Proof: If ${i: Y \rightarrow X}$ does not extend continuously, there exist sequences ${(x_m), (y_m)}$ in ${Y}$ such that ${x_m \rightarrow p}$ and ${y_m \rightarrow p}$ for some ${p \in \partial Y}$, but ${i(x_m) \rightarrow u}$ and ${i(y_m) \rightarrow v}$ for ${v \neq u}$ in ${\partial X}$.

Note that ${d_Y(y_0, [x_m,y_m]) \rightarrow \infty}$ as ${m \rightarrow \infty}$. On the other hand, for ${m}$ large enough, ${d_X(i(y_0), [i(x_m),i(y_m)])}$ stays uniformly close to ${d_X(i(y_0),[u,v]) > 0}$ (see the discussion of visibility in the previous post). This contradicts the existence of the function ${M}$. $\Box$

3. Trees of hyperbolic spaces

We’ll sketch part of Mitra’s proof of result number 3 above in the original context in which he established it: that of trees of hyperbolic spaces.

A tree ${T}$ of hyperbolic metric spaces is a metric space ${(X,d)}$ admitting a map ${P: X \rightarrow T}$ onto a simplicial tree ${T}$ such that there exist constants ${\delta, \epsilon, K}$ satisfying

1. For all vertices ${v \in T}$, ${X_v := P^{-1}(v) \subset X}$ is path connected, rectifiable, and ${\delta}$-hyperbolic. We denote the induced path metric by ${d_v}$.
2. All inclusions ${X_v \hookrightarrow X}$ are uniformly proper. That is, there exists a function ${f: \mathbb{N} \rightarrow \mathbb{N}}$ such that for all ${x, y \in X_v}$, if ${d(x,y) \le n}$, then ${d_v(x, y) \le f(n)}$.
3. For all edges ${e \in T}$, ${X_e := P^{-1}(e) \subset X}$ is path connected and rectifiable. The induced path metric is denoted ${d_e}$.
4. For every edge ${e \in T}$ there exists a function ${f_e: X_e \times [0,1] \rightarrow X}$ such that ${f_e|_{X_e \times (0,1)}}$ is an isometry onto ${P^{-1}(\mathrm{int}(e))}$.
5. ${f_e|_{X_e \times \{0\}}}$ and ${f_e|_{X_e \times \{1\}}}$ are ${(K, \epsilon)}$-quasi-isometric embeddings into ${X_{v_1}}$ and ${X_{v_2}}$, where ${e}$ connects ${v_1}$ and ${v_2}$.

[Note: although present in Mitra’s definition, we don’t think condition 2 above is necessary. It is equivalent to

2′. There exists a function ${g: \mathbb{N} \rightarrow \mathbb{N}}$ such that for any ${x, y \in X_v}$, if ${d_v(x,y) \ge n}$, then ${d(x, y) \ge g(n)}$, and $g(n) \rightarrow \infty$ as $n \rightarrow \infty$.

This, however, follows from the structure of the space ${X}$. Let ${D=d_v(x,y)}$. In order to shortcut an ${X_v}$-geodesic from ${x}$ to ${y}$, we can’t use a path that travels to more than ${D/2}$ other vertex spaces ${X_w}$, since the distance between such subspaces is at least 1. The length of a path in ${X}$ between ${x}$ and ${y}$ is bounded (up to a constant) below by $\mathrm{min}_{n < D/2} (2n + D/K^n)$ which goes to infinity like $\log(D)$ as $D$ increases.]

One example of a tree of hyperbolic spaces is the “discrete hyperbolic plane.” It’s constructed by gluing squares of side length 1 in as illustrated in the figure below.

The vertical lines correspond to vertex subspaces ${X_v}$, and each ${X_e}$ is glued to the upper vertex subspace by an isometry and to the lower vertex subspace by a ${(2,0)}$-quasi-isometry stretching the metric by a factor of 2.

Theorem 2. Let ${(X, d, T)}$ be a tree of hyperbolic metric spaces. If ${X}$ is hyperbolic, then there exists a Cannon-Thurston map for all ${X_v \hookrightarrow X}$.

The structure of the proof is as follows. Given a geodesic segment ${\lambda}$ in ${X_v}$, a quasiconvex set ${B_\lambda \subset X}$ is constructed containing ${\lambda}$ and satisfying the condition that if ${\lambda}$ is outside an ${N}$-ball around ${y_0 \in X_v}$, then ${B_\lambda}$ is outside an ${M}$-ball around ${i(y_0)}$. As ${N \rightarrow \infty}$, so does ${M \rightarrow \infty}$. Applying the key lemma above finishes the proof.

This proof has the same structure as the proof of Mitra’s other result on normal hyperbolic subgroups. The difference lies in the construction of ${B_\lambda}$.

4. Constructing ${B_\lambda}$

Below is a morally correct, albeit simplified, description of how one constructs ${B_\lambda}$ for a given ${X_v}$-geodesic ${\lambda}$ in stages by starting with ${\lambda}$ and adding pieces of other vertex spaces one by one.

Consider the images of ${f_e}$ intersecting ${\lambda}$ for every edge ${e \in T}$ incident to ${v}$. If the intersection is large enough (this constant depends on a number of other constants defining ${X}$), add the geodesic connecting the farthest points of the preimage of ${\lambda}$ in the neighboring vertex space to ${B_\lambda}$. Now take the resulting geodesic segments in (some of) the spaces ${X_w}$, where ${w}$ is distance 1 away from ${v}$ in ${T}$, and apply the same procedure, without backtracking to ${v}$. Propagate throughout the tree ${T}$ without backtracking to previously visited vertices. See the illustration below.

The bulk of Mitra’s proof consists of showing that the resulting ${B_\lambda}$ is indeed quasi-convex in ${X}$.

As an exercise, pick a ${\lambda}$ in one of vertex spaces of the discrete hyperbolic plane described above, construct a ${B_\lambda}$, and convince yourself that it’s quasi-convex, and that it’s far from a given basepoint ${y_0}$ when ${\lambda}$ itself is.

5. From trees of spaces to graphs of groups

How does Theorem 2 imply the graph-of-groups formulation? Given a graph of groups $\Gamma$ with $G=\pi_1(\Gamma)$, consider its universal cover $\tilde{\Gamma}$ (with the corresponding vertex and edge group assignments, so that $\pi_1(\tilde{\Gamma}) = G$; see below for an example).

Taking Cayley graphs of vertex groups, and gluing them using (Cayley graphs of edge groups) $\times [0,1]$ and the corresponding quasi-isometric embeddings, we get a tree of hyperbolic spaces satisfying the hypotheses of Mitra’s theorem.

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## Boundaries of Hyperbolic Groups

In this post and the next, we will discuss Cannon-Thurston maps: (continuous) extensions of inclusions ${H\leq G}$ of hyperbolic groups to the boundaries ${\partial H\rightarrow\partial G}$. It is not evident that such maps should exist when ${H}$ is distorted in ${G}$, but they often do nonetheless. This post will focus on defining the boundary of a hyperbolic group and establishing basic properties. We’ll take a brief look at Cannon-Thurston‘s first (distorted) example. The next post will discuss the progress of Mitra towards the following open question:

Question: If ${H\leq G}$ are hyperbolic groups, does there exist a Cannon-Thurston map ${\partial H\rightarrow\partial G}$?

In particular, he has answered the question in the affirmative for the case of normal hyperbolic subgroups of hyperbolic groups and vertex subgroups of finite hyperbolic graphs of hyperbolic groups with quasi-isometric edge inclusion maps.

1. The boundary of a Gromov-hyperbolic space

In this section, ${X}$ is a geodesic space that is proper and hyperbolic. We’ll see several equivalent definitions for ${\partial X}$ and the closure ${\overline{X}:=X\cup\partial X}$. References include the textbooks of Bridson-Haefliger and Ghys-de la Harpe (in French) and Kapovich-Benakli’s survey paper Boundaries of Hyperbolic Groups.

Here are two examples to keep in mind. Morally, the boundary of the hyperbolic plane should be the circle at infinity in the Poincare disk model. The boundary of the Cayley graph ${\Gamma}$ for ${F_2}$ should be a Cantor set, which can be viewed (under the “obvious” embedding of the Cantor set in the plane – see picture below) as the set of limit points of ${\Gamma}$ not in ${\Gamma}$ itself. Note: the Cantor set is totally disconnected, so the set of ends is the same as the boundary.

Geodesic rays will be parametrized by arclength.

1. Fix a basepoint ${p\in X}$. Then ${\partial(X,p)}$ is the set of equivalence classes of geodesic rays ${c:[0,\infty)\rightarrow X}$ originating at ${p}$ (i.e., ${c(0)=p}$). Ray ${c_1}$ is equivalent to ${c_2}$, written ${c_1(\infty)=c_2(\infty)}$, if ${\sup_t{d(c_1(t),c_2(t))}<\infty}$, or equivalently if the images of ${c_1}$ and ${c_2}$ are a finite Hausdorff distance from each other. [Note: for CAT(0) spaces, each equivalence class contains a single ray since the metric is convex.]
2. ${\partial X}$ is defined the same way, but without requiring that rays originate at ${p}$.

3. ${\partial_qX}$ is the set of equivalence classes of quasigeodesic rays (originating at any point in the space). Here we must use only the finite Hausdorff distance as the equivalence relation. [Note: this definition has no analogue for CAT(0) spaces, where there is no quasigeodesic stability.]

These three sets are in natural bijection: As equivalence classes in (1) are contained in equivalence classes in (2) which are contained in equivalence classes in (3), we need only check that each quasigeodesic equivalence class contains a geodesic ray starting at ${p}$. This follows from the Arzelà–Ascoli theorem and quasigeodesic stability. (Take a sequence ${c_n}$ of generalized geodesic rays representing the points ${c(n)}$, i.e. geodesic segments connecting the basepoint ${p}$ to ${c(n)}$ with domain extended to ${[0,\infty]}$ by the constant map at ${c(n).}$ So ${c_n(\infty)=c(n).}$ A subsequence converges to a geodesic ray from ${p}$ which is Hausdorff-close to ${c}$. See the illustration below.)

To topologize these boundaries, it suffices to topologize ${\partial(X,p)}$. We do this by defining convergence: ${x_n\rightarrow x}$ as ${n\rightarrow\infty}$ for ${x_n,x\in\overline{(X,p)}}$ if they are represented by generalized rays (geodesic rays allowed) ${c_n,c}$ from ${p}$ such that every subsequence of ${\{c_n\}}$ has itself a subsequence converging to ${c}$ pointwise and uniformly on compact sets. It is necessary to take subsequences here because we want the sequence of blue points below with red basepoint to converge (to a single point on the boundary):

Closed sets are defined to be the sets containing all their limit points. For any fixed ${k>2\delta}$, a fundamental system of open sets in ${\overline{X}}$ about ${c\in\partial(X,p)}$ is the collection of ${V_n(c)}$, where a generalized ray ${c'\in V_n(c)}$ if ${d(c(n),c'(n)). (The choice of ${k}$ does not matter since asymptotic rays from a fixed basepoint in a hyperbolic space are uniformly close: within ${2\delta}$ of each other.)

1. This difficulty requiring us to take subsequences does not arise in defining the boundary of CAT(0) spaces, since the metric is convex. If ${X}$ is both hyperbolic and CAT(0), the various ${\overline{X}}$ defined above are homeomorphic to the inverse limit of the closed balls ${\overline{B}(p,n)}$ as ${n}$ varies, induced by the projection maps to these complete convex subsets. The topology is the inverse limit topology (the coarsest topology making all the maps ${\overline{X}\rightarrow\overline{B}(p,n)}$ continuous).
2. Let ${\mathcal{C}(X,\mathbb{R})}$ denote the set of continuous functions from ${X}$ to ${\mathbb{R}}$ with the compact-open topology. Identify ${X}$ with a subset of ${\mathcal{C}(X,\mathbb{R})/\sim}$, where ${f\sim g}$ if ${f-g}$ is a constant map, by associating to ${x\in X}$ the equivalence class of the distance to ${x}$ function: ${d(x,\cdot)}$. The point of ${\partial X:=\overline{X}\setminus X}$ associated to a geodesic ray $c$ is represented by the Busemann function ${b_c}$ defined as

$\displaystyle b_c(x)=\limsup_{t\rightarrow\infty}{\left[d(x,c(t))-t\right]}.$

The level sets of Busemann fuctions are called horospheres, agreeing in ${\mathbb{H}^n}$ with the classical horospheres. A Busemann function can be viewed as a sort of “distance” function to a point of ${\partial X}$, but with close points having very small distance: near ${-\infty}$ instead of near 0.

3. Points of the sequential boundary ${\partial_sX}$ are represented by sequences in ${X}$ that converge to a point in ${\partial X}$. To describe this convergence, we’ll need to use the Gromov product for points ${x,y,z\in X}$:

$\displaystyle (x\cdot y)_z=\frac12\left[d(x,z)+d(y,z)-d(x,y)\right].$

In a tree, the Gromov product represents the overlap length of the segments ${[z,x]}$ and ${[z,y]}$, or equivalently the distance from ${z}$ to ${[x,y]}$. In a ${\delta}$-hyperbolic space, the Gromov product gives a good proxy for ${d(z,[x,y])}$ (to within some fixed multiple of ${\delta}$, depending on the definition of ${\delta}$-hyperbolicity used.)

Fix a basepoint ${p\in X}$. A sequence ${x_n}$ is said to converge at infinity if ${(x_n\cdot x_m)_p\rightarrow\infty}$ as ${n,m\rightarrow\infty}$, and two such sequences ${x_n,y_n}$ converge to the same point if ${(x_n\cdot y_m)_p\rightarrow\infty}$. (Transitivity of this relation follows from one of the many equivalent definitions of ${\delta}$-hyperbolicity: ${(x\cdot y)_w\geq\min((x\cdot z)_w,(y\cdot z)_w)-\delta}$ for all ${w,x,y,z\in X}$.) A fundamental system of neighborhoods of a boundary point is given by bounding the Gromov overlap from below by a sequence of numbers approaching infinity.

2. Basic Properties

Most of these facts are taken from Bridson-Haefliger, chapter III.H.3.

Fact: ${\overline{X}}$ is compact. Arzelà–Ascoli gives sequential compactness, and compactness then follows from first countability.

Fact: ${\partial X}$ is visible: for every ${x,y\in\partial X}$, ${x\neq y}$, there is a geodesic ${c:\mathbb{R}\rightarrow X}$ with ${c(\infty)=x}$, ${c(-\infty)=y}$. Indeed, take geodesic rays ${c_x,c_y}$ originating from a basepoint ${p}$ with ${c_x(\infty)=x,c_y(\infty)=y}$. Since ${x\neq y}$, there is a point ${q=c_x(t_0)}$ (shown in blue) a distance more than ${\delta}$ away from ${c_y}$. For ${n>t_0}$, we can choose geodesic segments ${c_n}$ joining ${c_y(n)}$ to ${c_x(n)}$, and extend them to generalized geodesic lines. By ${\delta}$-hyperbolicity, some point of the segment ${c_n}$ is a distance less than ${\delta}$ from ${q}$, which we can assume is ${c_n(0)}$ (shown in red). Then Arzelà–Ascoli gives a subsequence of ${c_n}$ converging to a bi-infinite geodesic ${c}$. By ${\delta}$-hyperbolicity, ${c_n(t)}$ is within ${2\delta}$ of ${c_x}$ for ${t>0}$ and within ${3\delta+d(p,q)}$ of ${c_y}$ for ${t<0}$. (See the diagram below.) So ${c(\infty)=x}$ and ${c(-\infty)=y}$.

Given a map ${f:X\rightarrow X'}$ between hyperbolic spaces, there is at most one way to extend it continuously to the boundary ${f_\partial:\partial X\rightarrow\partial X'}$.

Fact: If ${f:X\rightarrow X'}$ is a quasi-isometric embedding of geodesic, proper, hyperbolic spaces then ${f_\partial}$ is a (continuous) embedding, where ${f_\partial(c)=f\circ c}$ (using the quasigeodesic model ${\partial_qX'}$ of hyperbolic boundary.) If ${f}$ is a quasi-isometry, then ${f_\partial}$ is a homeomorphism. (These statements follow immediately from quasigeodesic stability and the description of a fundamental base of neighborhoods ${V_n(c)}$ above.)

Example: ${\partial\mathbb{H}^n=S^{n-1}}$, so ${\mathbb{H}^n}$ is not quasi-isometric to ${\mathbb{H}^m}$ unless ${n=m}$.

Since the hyperbolic boundary is a quasi-isometry invariant, we can associate a boundary ${\partial G}$ to a hyperbolic group ${G}$.

Example: If ${S}$ is a closed hyperbolic surface and ${M}$ a closed hyperbolic 3-manifold, then ${\partial\pi_1(S)=\partial\mathbb{H}^2=S^1}$ and ${\partial\pi_1(M)=\partial\mathbb{H}^3=S^2}$.

Example: ${\partial F_2}$ is a Cantor set.

By the above fact, the inclusion map from an undistorted hyperbolic subgroup ${H}$ of a hyperbolic group ${G}$ extends to a continuous map (in fact, an embedding) ${\partial H\rightarrow\partial G}$. When ${H}$ is an arbitrary hyperbolic subgroup of a hyperbolic group ${G}$, a continuous (not necessarily injective) extension of the inclusion to ${\partial H\rightarrow\partial G}$ is called a Cannon-Thurston map.

3. Cannon and Thurston’s Example

Thurston proved that the mapping torus ${M=(S\times[0,1])/((x,0)\sim(\phi(x),1))}$ of a hyperbolic surface ${S}$ by a surface automorphism ${\phi}$ is a hyperbolic 3-manifold if and only if ${\phi}$ is pseudo-Anosov. In particular, there exist hyperbolic 3-manifolds where ${\pi_1M}$ is an HNN-extension of a surface group ${\pi_1S}$, with

$\displaystyle 1\rightarrow\pi_1S\rightarrow\pi_1M\rightarrow\mathbb{Z}\rightarrow1.$

Cannon-Thurston showed the inclusion extends to a surjective map ${S^1=\partial H^2=\partial\pi_1S\rightarrow\partial\pi_1M=\partial\mathbb{H}^3=S^2}$, a space-filling curve. Therefore ${\pi_1S}$ is a distorted subgroup of ${\pi_1M}$. This was the first example found of a Cannon-Thurston map involving a distorted subgroup. (Mitra’s work, described in the next post, generalizes this result: he shows that there is a Cannon-Thurston map for any hyperbolic normal subgroup of a hyperbolic group.) We will sketch here why the map is surjective.

${\pi_1S}$ acts properly by isometries on ${\mathbb{H}^3}$ via its inclusion in ${\pi_1M}$. The image of the Cannon-Thurston map is the limit set ${\Lambda\pi_1S}$: the set of accumulation points in ${\partial\mathbb{H}^3}$ of any ${\pi_1S}$-orbit in ${\mathbb{H}^3.}$ Since ${\pi_1S}$ is a normal subgroup, elements of ${\pi_1M}$ act on ${\mathbb{H}^3}$ by permuting these orbits, so they fix ${\Lambda\pi_1S}$. Consequently, they fix the convex hull ${K}$ of ${\Lambda\pi_1S}$ in ${\overline{\mathbb{H}^3}}$. Therefore ${K}$ contains a ${\pi_1S}$-orbit which is quasi-isometric to ${\mathbb{H}^3}$ itself, so the closed set ${K}$ contains the whole boundary ${\partial\mathbb{H}^3}$. But ${K\cap\partial\mathbb{H}^3=\Lambda\pi_1S.}$ (The interior of a ball together with some points on its boundary is alway convex, after all.) So ${\Lambda\pi_1S=\partial\mathbb{H}^3}$.

Kapovich and Short give a generalization of this argument to normal subgroups of ${\delta}$-hyperbolic groups.

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## Finitely presented subgroups of hyperbolic groups of cohomological dimension 2

Suppose $G$ is a group with group ring $\mathbb{Z}G$. A projective resolution of a $\mathbb{Z}G$–module $M$ is an exact sequence

$\cdots \to P_i\to P_{i-1}\to...\to P_0\to M\to 0$

where each $P_j$ is a projective $\mathbb{Z}G$–module. Regard $\mathbb{Z}$ as a trivial $\mathbb{Z}G$–module. The cohomological dimension $\textup{cd}(G)$ of a group $G$ is the smallest integer $n$ such that $\mathbb{Z}$ admits a projective resolution

$\cdots \to 0 \to 0 \to P_{n} \to...\to P_0\to \mathbb{Z} \to 0$

of $\mathbb{Z}G$–modules. (If there is no such integer, the cohomological dimension is infinity.) For more details, see Cohomology of groups by Kenneth S. Brown.

In this post we shall discuss the proof of Gersten’s result that hyperbolic groups of cohomological dimension $2$ have the property that their finitely presented subgroups are also hyperbolic. This result appears in Gersten’s paper Subgroups of word hyperbolic groups in dimension 2.

Standard norms on modules

Suppose $H$ is a group and $M$ is a finitely generated $\mathbb{Z}H$–module. Let $F$ be a finitely generated free $\mathbb{Z}H$–module, with a basis $\{\alpha_1,...,\alpha_n\}$, such that there is a surjective homomorphism $\nu:F \to M$ of $\mathbb{Z}H$–modules. Then $\{h\alpha_i\mid h\in H,1\leq i\leq n\}$ is a free $\mathbb{Z}$–basis for $F$. Equip $F$ with the $l_1$–norm

$| \sum_{i \in \{1, \ldots, n\}, h \in H} n_i h \alpha_i |_1 = \sum_{i \in \{1, \ldots, n\}, h \in H} | n_i |$.

We give $M$ the norm $|m|=\textup{min} \{|a|_1\mid a\in F, \nu(a)=m\}$. Such a norm $|.|$ on $M$ is, up to linear equivalence, independent on the choices of $F$, $\nu$ and $\{\alpha_1,...,\alpha_n\}$ — that is, if another norm $|.|^{\prime}$ is obtained from another surjective homomorphism from a different finitely generated free $\mathbb{Z}H$–module $F^{\prime}$, then there is a uniform constant $C$ such that $C^{-1}|m|\leq |m|^{\prime}\leq C|m|$ for all $m\in M$. So $|.|$ is called a standard norm.

Lemma 1. Suppose

$0 \to M \stackrel{i}{\to} N \to P\to 0$

is a short exact sequence of $\mathbb{Z}H$–modules. Assume
1) $M$ is finitely generated and equipped with a standard norm $|.|$,
2) $N$ is free and finitely generated and given an $l_1$ norm $|.|_1$ associated to some basis, and
3) $P$ is projective.
Then there is a retraction $\sigma:N\to M$ for $i$ (that is, a map $\sigma:N\to M$ such that $\sigma \circ i$ is the identity on $M$) and a constant $C>0$ so that $|\sigma(x)|\leq C|x|_1$ for all $x\in N$.

Proof. Since $P$ is projective, we know that $N\cong M\oplus P$. We choose a map $\sigma^{\prime}:N\to M$ such that $i\circ \sigma^{\prime}$ is the identity on $M$. Let $I$ be a finitely generated free submodule generated by a subset of the generators of $N$ which contains the image of $M$ under the map $i$. Let $N=Q\oplus I$. Notice that $Q$ is also free, but not necessarily finitely generated. We can modify the retraction $\sigma^{\prime}$ to $\sigma:N\to M$ where $\sigma\mid_{I}=\sigma^{\prime}\mid_{I}$ and $\sigma\mid_{P}=0$. Let $\pi:F\to M$ be a surjective homomorphism of $\mathbb{Z}H$–modules where $F$ is finitely generated, free and based.

Since $I$ is free, there is a map $\rho:I\to F$ such that $\sigma\mid_I=\pi\circ \rho$. This map $\rho$ can be extended to $I\oplus Q$ by setting the images under $\rho$ of the elements of $Q$ be $0$. It follows this $\sigma=\pi\circ \rho$ and $\rho$ is supported on $I\oplus Q$.

Now $I$ and $F$ are both finitely generated and free. So the map $\rho$ is given by a finite matrix with entries in $\mathbb{Z}H$. Therefore there exists a constant $C>0$ such that $|\rho(u)|_1\leq C|u|_1$ for all $u\in N$. Since the standard norm on $M$ is defined as an infimum over coset representatives, we have that $|\sigma(u)|=|\pi\circ \rho(u)|\leq |\rho(u)|_1\leq C|u|_1$ for all $u\in N$. This completes the proof.

Theorem (Eilenberg–Ganea). Let $\Gamma$ be an arbitrary group and let $n=\textup{max}\{\textup{cd}(\Gamma),3\}$. Then there exists an $n$-dimensional $K(\Gamma,1)$-complex $Y$. If $\Gamma$ is finitely presented and of finite type, then $Y$ can be taken to be finite.

If the cohomological dimension of the group is infinite, then the theorem merely asserts the existence of a $K(\Gamma,1)$-complex.

The relation module of a finitely presented group

Let $G= \langle a_1,...,a_n\mid r_1,...,r_m\rangle$ be a finitely presented group. So $G=F/R$, where $F$ is freely generated by $a_1,...,a_n$ and $R$ is the normal closure of the relations $r_1,...,r_m$. Let the presentation $2$–complex of this group be $Y$. The relation module for $G$ is a $\mathbb{Z}G$–module defined in the following way. The underlying abelian group is $R_{ab}$, the abelianization of R , and the $G$–action on it is induced by the conjugation of $F$ on $R_{ab}$. Since $R$ acts trivially, the induced action by $G$ is well defined. Hence this gives us a $\mathbb{Z}G$–module structure on $R_{ab}$. This is naturally isomorphic as a $\mathbb{Z}G$ module to $H_1(\tilde{Y}^{(1)})$, where $\tilde{Y}^{(1)}$ is the $1$-skeleton of the universal cover of the presentation complex $Y$ of $G$, or in other words, the Cayley graph of $G$.

Proposition 1. If $G$ is a word hyperbolic group of cohomological dimension $2$, then $G$ is the fundamental group of a finite aspherical $3$–complex $Y$ such that
1) the relation module $H_1(\tilde{Y}^{(1)})$ is finitely generated and free as a $\mathbb{Z}G$ module, and
2) the standard norm on $H_1(\tilde{Y}^{(1)})$ agrees up to linear equivalence with the $l_1$ norm from any basis for the $\mathbb{Z}G$ module $H_1(\tilde{Y}^{(1)})$.

Proof. Since $G$ has cohomological dimension $2$, it is torsion free. Torsion free hyperbolic groups are of finite type, so $G$ is the fundamental group of a finite asherical complex. In particular, by the Eilenberg-Ganea theorem we know that $G$ is the fundamental group of a finite aspherical $3$–complex $Y_1$.

We need the following lemma (page 184 of Cohomology of groups).

Lemma 2. If the cohomological dimension of a $\mathbb{Z}G$–module $M$ is $n$, and if $0\to K\to P_{n-1}\to...\to P_0\to M\to 0$ is an exact sequence of $\mathbb{Z}G$–modules with each $P_i$ projective, then $K$ is projective.

From the resolution,

$0\to H_1(\tilde{Y}^{(1)})\to C_1(\tilde{Y_1})\to C_0(\tilde{Y_1})\to \mathbb{Z}\to 0$

and the facts that $C_i(\tilde{Y}^{(1)})$ are projective and $G$ has cohomological dimension $2$, we get from Lemma 2 that $H_1(\tilde{Y_1}^{(1)})$ is projective.

The exact sequence

$0\to C_3(\tilde{Y_1})\to C_2(\tilde{Y_1})\to H_1(\tilde{Y_1}^{(1)})\to 0$

then splits. So we have an isomorphism of $\mathbb{Z}H$–modules $C_2(\tilde{Y_1})\cong H_1(\tilde{Y_1}^{(1)})\oplus C_3(\tilde{Y_1})$. By definition this means that $H_1(\tilde{Y_1}^{(1)})$ is stably free as a $\mathbb{Z}G$–module. (A $\mathbb{Z}G$–module $M$ is stably free if there exist free $\mathbb{Z}G$–modules $N,P$ such that $M\oplus N\cong P$ as $\mathbb{Z}G$–modules.)

So we can attach a finite number of $2$-discs trivially at the basepoint of $Y_1$ to produce a finite $3$-complex $Y$ with fundamental group $G$ and with the relation module $H_1(\tilde{Y})$ free as a $\mathbb{Z}G$–module. This proves the first part of the result.

The second part of the proposition follows from an application of Lemma $1$.

We are now ready to prove our main result.

Theorem 1. Let $H$ be a finitely presented subgroup of the word hyperbolic group $G$, where $G$ is of cohomological dimension $2$. Then $H$ is word hyperbolic.

Proof. By Proposition $2$, $G$ has a presentation for which the relation module is free and finitely generated.

Using this fact and Schanuel’s Lemma one can establish that for any finite presentation $\langle F \mid R \rangle$ of $G$, the relation module $R_{ab}$ is stably free.

We can choose presentations for $H$ and $G$ such that the respective presentation complexes $X$ and $Y$ satisfy $X\subseteq Y$. Further, by adding trivial letters and corresponding relations to the presentation for $G$ (as in a previous proof) we can assume that $H_1(\tilde{Y}^{(1)})$ is free.

Let $Q=\tilde{Y}^{(1)}/\tilde{X}^{(1)}$. An application of the Snake Lemma can be used to show that $H_1(Q)$ is projective as a $\mathbb{Z}H$–module. The short exact sequence of $\mathbb{Z}H$–modules arising from the Snake Lemma

$0\to H_1(\tilde{X}^{(1)})\to H_1(\tilde{Y}^{(1)})\to H_1(Q)\to 0$

satisfies the conditions of Proposition $1$.

It follows that there is a retraction $p:H_1(\tilde{Y}^{(1)})\to H_1(\tilde{X}^{(1)})$ for $i: H_1(\tilde{X}^{(1)})\to H_1(\tilde{Y}^{(1)})$, such that there is a uniform constant $C>0$ with the property that $|p(x)|\leq C|x|_1$ for all $x\in H_1(\tilde{Y}^{(1)})$. So if $y\in H_1(\tilde{X}^{(1)})$, then $|y|=|p\circ i(y)|\leq C|i(y)|$.

Let $w$ be a circuit in $\tilde{X}^{(1)}$ and let $S$ be a van Kampen diagram for $w$ in $\tilde{X}$ of minimal area. The area of $S$ is $|y|$, where $y$ is the element of $H_1(\tilde{X}^{(1)})$ corresponding to $w$. The area of $w$ as a circuit in $\tilde{Y}^{(1)}$ is $|i(y)|$, where now $|.|$ is in terms of $\mathbb{Z}G$–modules. It is clear that $|i(y)|$ in $H_1(\tilde{Y}^{(1)})$ as a $\mathbb{Z}G$–module is less than or equal to $|i(y)|$ with $H_1(\tilde{Y}^{(1)})$ as a $\mathbb{Z}H$–module. By the discussion in the preceding paragraph, we can conclude that if $G=\pi_1(Y)$ satisfies a linear isoperimetric inequality, then so does $H=\pi_1(X)$. Since $H$ satisfies a linear isoperimetric inequality, it must be word hyperbolic.

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## 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.

Brady’s construction

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.

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## Dehn functions of subgroups of CAT(0) groups

This is a post based on a guest lecture by Pallavi Dani.

Introduction

In this post, we will discuss Dehn functions of subgroups of non-positively curved groups. Let ${G}$ be a finitely presented group and let ${K_G}$ be a presentation complex for some presentation of ${G}$. If ${l}$ is a loop in ${\widetilde{K}^{(1)}_G}$, we define the filling area of ${l}$ to be

$\displaystyle \textup{FArea}(l)=\min \{\textup{Area} (D)| D=\text{disk in } \widetilde{K}^{(1)}_G, \partial D=l\}.$

Furthermore, we define the Dehn function of a finite presentation for ${G}$ by

$\displaystyle \delta_G(x)=\sup\{\textup{FArea}(l)|\; l \text{ is a loop of length } x\}.$

In these definitions length and area are defined combinatorially, that is, by counting cells.

As we have seen in previous posts [1] and [2], the Dehn functions of hyperbolic and CAT(0) groups are quite well understood:

1. ${G}$ is hyperbolic if and only if ${\delta_G(x)\simeq x}$, or equivalently if and only if ${\delta_G(x)}$ is subquadratic. This result is due to Gromov; the absence of Dehn functions between linear and quadratic is called the “Gromov Gap”.
2. If ${G}$ is CAT(0) then ${\delta_G(x)\preceq x^2}$. (On account of the Gromov gap, this means that ${\delta_G(X) \simeq x}$ or ${x^2}$.)

Thus, the possibilities for Dehn functions of hyperbolic and CAT(0) groups are very limited, and it is natural to ask whether subgroups of such groups inherit this property. More specifically:

1. Which functions arise as Dehn functions of subgroups of CAT(0) groups?
2. Are there gaps in the corresponding isoperimetric spectrum?
3. If so, do they correspond to something algebraic/geometric?

Subgroups of CAT(0) groups with large Dehn functions

It is possible to construct subgroups of CAT(0) groups with “large” Dehn functions. Here, by “large” we mean strictly greater than quadratic. We will discuss a 3-step scheme for constructing such subgroups, called the Bieri doubling trick. This idea originated in the work of Bieri, and it was explored extensively by Baumslag-Bridson-Miller-Short in the context of automatic groups.

The construction involves first finding distorted subgroups of CAT(0) groups. We denote the distortion of a subgroup ${H}$ in ${G}$ by ${\text{disto}_H^G(x)}$.

Step 1. Find ${H such that ${1\rightarrow H\rightarrow G\rightarrow \mathbb{Z}\rightarrow 1}$ is a short exact sequence, ${G}$ is CAT(0), and ${H}$ is distorted, with ${\text{disto}_H^G(x)\succcurlyeq x^2}$.

Example. Here is an example of such a pair constructed by N. Brady (for details, see section 2.3.3 of The Geometry of the Word Problem for Finitely Generated Groups). The group ${G}$ is isomorphic to ${F_2 \rtimes \mathbb{Z}}$ and is defined by ${G=\langle a,b,t\,|\, a^t=\phi (a), b^t=\phi(b)\rangle}$. The automorphism ${\phi\colon F(a,b)\rightarrow F(a,b)}$ is given by ${a\mapsto a}$ and ${b\mapsto ab}$. Let ${H}$ be ${F(a,b)}$, the free group generated by ${a}$ and ${b}$. You can see that ${\text{disto}_{H}^G(x)\succcurlyeq x^2}$ by considering the family of words ${t^nb^nt^{-n}}$ (with length on the order of ${n}$ in ${G}$) and noting that

$\displaystyle t^nb^nt^{-n}=\phi^n(b^n)=\phi^{n-1}(ababab\ldots ab)=$

$\displaystyle =\phi^{n-2}(aabaabaab\ldots aab)=aaa\ldots aabaaa\ldots aab\ldots\ldots aaa\ldots aab.$

The final word has ${n}$ ${a}$‘s in between every pair of consecutive ${b}$‘s, giving a total of about ${n^2}$ letters, and it is clearly reduced in ${H}$. Hence, the distortion is at least ${x^2}$.

To see that ${G}$ is CAT(0), note that ${G}$ is isomorphic to ${\langle \alpha, \beta, t|\; \alpha^t=\alpha, t^{\beta}=\alpha\rangle}$, where ${\alpha=at^{-1}, \beta=bt}$. The corresponding presentation complex built out of squares can be shown to satisfy the link condition.

Step 2. Form the double ${\Delta (G,H)}$ of ${G}$ along ${H}$ by amalgamating two copies of ${G}$ along ${H}$. So ${\Delta =\Delta(G,H) :=G\ast_{H} G}$.

Bridson showed that ${\delta_{\Delta}(x)\succeq \text{disto}_H^G(x)}$. Alternatively, see Bridson-Haefliger.

Example. In fact, for ${G}$ and ${H}$ in Brady’s example above, we can do better than Bridson’s lower bound: we have ${\delta_{\Delta (G,H)}(x)\succeq x^3}$, for a proof see Theorem 6.20 in Bridson-Haefliger.

To see this, write ${\Delta=\langle a,b,t,s\;|\;a^t=a^s=\phi(a), b^t=b^s=\phi(b)\rangle}$. Now consider the following family of embedded disks in ${\widetilde {K}_\Delta}$:

These disks have boundary length on the order of ${n}$, and Area ${\succeq n^3}$. Moreover, the boundary loops of these disks do not admit fillings with smaller area, since ${\widetilde {K}_\Delta}$ is ${2}$-dimensional and aspherical. (In such spaces the embedded filling is the most efficient among the fillings of any loop.) This shows that ${x^3}$ is a lower bound on the Dehn function.

Step 3. The double ${\Delta(G,H)}$ embeds in ${G\times F_2}$. Since ${G}$ and ${F_2}$ are CAT(0), so is ${G\times F_2}$.

Example. We illustrate this for our example. Write ${G\times F_2}$ as ${(F(a,b)\rtimes \langle t \rangle) \times F(u,v)}$. Then the subgroup ${\langle a, b , tu, tv\rangle}$ of ${G\times F_2}$ is isomorphic to ${\Delta (G, F(a,b))}$. (It is easy to see that the relations ${(tu)a(tu)^{-1}=\phi(a)}$ and ${(tv)a(tv)^{-1}=\phi(a)}$, and the corresponding relations involving ${b}$, are satisfied, since ${u}$ and ${v}$ commute with the other generators. Further, one can show that there are no additional relations among these generators — see Baumslag-Bridson-Miller-Short.)

Our example shows that ${x^3}$ occurs as the Dehn function of a subgroup of a CAT(0) group. By varying the pair ${(G,H)}$ in Step 1, one can construct other examples.

Sources for Step 1 and results.

1. Brady constructed a family of pairs ${F_n < G_n}$, where ${G_n\cong F_n\rtimes \mathbb{Z}}$ is a CAT(0) group, and ${\text{disto}_{F_n}^G(x) \simeq x^n}$. (The example above is the case n=2.) Then the double ${\Delta}$ embeds in ${G_n \times F_2}$, and ${\delta_{\Delta}(x)\simeq x^{n+1}}$. Thus, ${x^n}$ occurs as the Dehn function of a subgroup of a CAT(0) group for all integers ${n \geq 3}$.

2. Let ${M^3}$ be a hyperbolic surface bundle over a circle. Then one has a short exact sequence ${1\rightarrow \pi_1(S)\rightarrow \pi_1(M^3)\rightarrow \mathbb{Z}\rightarrow 1}$, and ${\pi_1(M^3)}$ is a CAT(0) group. Bridson and Haefliger prove that if ${H}$ is a finitely generated, infinite index subgroup of a hyperbolic group ${G}$ then ${\text{disto}_H^G(x)\succcurlyeq e^x}$. Hence, ${\Delta(\pi_1(M^3),\pi_1(S))\hookrightarrow \pi_1(M^3)\times F_2}$ and ${\delta_{\Delta}(x)\succeq e^x}$. In fact it is ${\simeq e^x}$. Thus, ${e^x}$ occurs as the Dehn function of a subgroup of a CAT(0) groups.

There are other sources of examples of ${H \leq G}$ to use in Step 1, for example due to Barnard-Brady and Samuelson. However, these do not appear to lead to additional examples of Dehn functions. For example, it remains unknown whether ${x^{\alpha}}$ for ${\alpha \notin \mathbb{Z}}$ or functions growing faster than ${e^x}$ can occur.

Dehn functions of kernels

The Bestvina-Brady groups, discussed in previous posts, have been a good source of counterexamples, and so one might expect them to be candidates for subgroups of CAT(0) groups with large Dehn functions. It turns out that their Dehn functions are relatively well-behaved:

• W. Dison showed that if ${H}$ is a Bestvina-Brady group then ${\delta_{H}(x)\preceq x^4}$.
• N. Brady constructed a Bestvina-Brady group ${H}$ with ${\delta_{H}(x)\succeq x^4}$. The lower bound was proved in a manner similar to the example above, by constructing a sequence of embedded diagrams in the level set for ${H}$, which is ${2}$-dimensional and contractible. Abrams-Brady-Dani-Duchin-Young generalized this example to produce a large class of Bestvina-Brady groups whose Dehn functions attain the general upper bound of ${x^4}$.
• For a while it was thought that Stallings group might have Dehn function of the form ${x^\alpha}$, where ${\alpha}$ is not an integer, but Dison-Elder-Riley-Young showed that its Dehn function is in fact quadratic.

One can also consider other kernels of particular classes of homomorphisms from particular classes of CAT(0) groups. For instance, Dison studies kernels of maps from direct products of free groups to free abelian groups. He shows, for example, that the kernel of a certain surjective homomorphism ${\phi\colon F_2\times F_2\times F_2 \rightarrow \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}$ has Dehn function ${\succeq x^3}$.

Brady-Guralnik-Lee define a new class of CAT(0) groups, which they call perturbed right-angled Artin groups, and consider kernels of homomorphisms to ${\mathbb Z}$. Using these in Step 1 of the Bieri doubling trick, they produce a class of examples of subgroups of CAT(0) groups with Dehn functions of the form ${x^n}$, where ${n\geq 4}$ is an integer. The kernels are of type ${F_2}$ but not ${F_3}$, unlike the original Bieri doubles, which all have finite ${2}$-dimensional ${K(G,1)}$‘s. Another interesting direction is to study higher dimensional Dehn functions of CAT(0) groups and their subgroups. We won’t go into the technicalities here, but just say that the ${k}$-dimensional Dehn function ${\delta^{(k)}}$ measures the difficulty of filling ${k}$-cycles with ${(k+1)}$-chains (or ${k}$-spheres with ${(k+1)}$-balls) in a suitable model space for ${G}$.

By a result of Gromov, if ${G}$ is a CAT(0) group, then ${\delta_G^{(k)}\preceq x^{(k+1)/k}}$. (As in the case ${k=1}$, the group ${\mathbb Z^n}$ attains this general upper bound.) Abrams-Brady-Dani-Duchin-Young generalized Dison’s result above and showed that for a Bestvina-Brady group ${H}$, one has

$\displaystyle x^{(k+1)/k}\preceq \delta_H^{(k)}(x)\preceq x^{(2k+2)/k}.$

Moreover, these inequalities are sharp.

Hyperbolic groups

In all the examples above, the ambient CAT(0) groups are not hyperbolic. In fact hardly anything is known about Dehn functions of subgroups of hyperbolic groups. The problem is that there is essentially only one example (due to N. Brady) of a finitely presented subgroup of a hyperbolic group which is not itself hyperbolic. (A variant of this example with infinitely many conjugacy classes of finite order elements with was discussed in this post.) The Dehn function of Brady’s example is not linear, since the group is not hyperbolic. It follows from a result of Gersten-Short that its Dehn function is bounded above by a polynomial. However, the Dehn function is not known explicitly.

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## Fibre products and the membership problem

Baumslag, Bridson, Miller and Short (BBMS) provide criteria for what they call the fibre products to be finitely presented. They then leverage these results to show that, amongst other things, there exists a torsion-free hyperbolic group ${G}$ with a fibre product subgroup ${P}$, such that the membership problem for ${P}$ in ${G}$ is undecidable. To connect this to our pursuit of the distortion functions of finitely generated subgroups, we recall that a finitely generated subgroup ${H}$ of a finitely generated group ${G}$ (with solvable word problem) has a solvable membership problem if and only if the distortion function of ${H}$ in ${G}$ is bounded above by a recursive function.

Theorem 1. There exists a torsion free word hyperbolic group ${\Gamma}$ and a finitely presented subgroup ${P< \Gamma\times\Gamma}$ such that there is no algorithm to decide membership of ${P}$ and the conjugacy problem of ${P}$ is unsolvable. Additionally, ${\Gamma}$ can be chosen to be the fundamental group of a compact negatively curved 2-complex.

The group ${\Gamma\times\Gamma}$ has a solvable conjugacy problem (Gersten-Short).

1. Fibre products

The central construction of BBMS is similar to the Mihailova construction we’ve covered before. To any short exact sequence

$\displaystyle 1 \rightarrow N \rightarrow \Gamma \stackrel{p}{\rightarrow} Q \rightarrow 1. \ \ \ \ \ (1)$

we have the associated fibre product ${P<\Gamma\times\Gamma}$ where

$\displaystyle P=\{(g_1,g_2)\ |\ p(g_1)=p(g_2)\}. \ \ \ \ \ (2)$

If ${Q}$ is finitely presented and ${\Gamma}$ is finitely generated then ${P}$ is finitely generated. We can think of ${P}$ as the graph of the relation ${=_Q}$, and so questions about equality in ${Q}$ are questions about membership of ${P}$.

When ${\Gamma}$ is free and ${Q}$ is finitely presented with undecidable word problem, ${P}$ has many undecidable properties (Miller), but ${P}$ will generally not be finitely presented in this case (see the Mihailova construction). Using a refined version of the Rips construction, encapsulated by the following theorem, allows one to get finitely presented ${P}$.

1-2-3 Theorem. Suppose

$\displaystyle 1 \rightarrow N \rightarrow \Gamma \stackrel{}{\rightarrow} Q \rightarrow 1. \ \ \ \ \ (3)$

is exact, and let ${P}$ be the associated fibre product. If ${N}$ is finitely generated (type ${F_1}$), ${\Gamma}$ is finitely presented ( type ${F_2}$), and ${Q}$ is of type ${F_3}$, then ${P}$ is finitely presented.

1.1. Type ${F_3}$ and ${\pi_2}$

The major conceptual underpinning of Theorem 1 is a connection between type ${F_3}$ and ${\pi_2}$. Let ${X}$ be an Eilenberg-Maclane space for ${Q}$ with finite 3-skeleton. We will assume that ${X}$ has a single vertex. This means that its 2-skeleton, ${X^{(2)}}$, can be identified with to a finite presentation ${\mathcal{P}_X}$ of ${Q}$ whose generators are given by the 1-cells and whose relations are given by the attaching maps of 2-cells in ${X^{(2)}}$. Since ${\pi_2\mathcal{P}_X:=\pi_2X^{(2)}}$ is finitely generated as a ${Q}$-module, ${X^{(3)}}$ is finite. This can be seen by taking the attaching maps of the 3-cells as the generators of the module. This identification allows one to find a nice presentation for ${\Gamma}$ and thus a nice presentation for ${P}$.

Obtaining this presentation is rather technical and takes a major portion of BBMS, and we omit the details. However, we will sketch their approach. Let ${\mathcal{P}=\langle \mathcal{X}\ |\ \mathcal{R}\rangle}$ be a presentation for a group ${G}$. Let ${\sigma=(c_1,..,c_m)}$ be a sequence of words of the form ${c_i=w_ir_iw_i^{-1}}$ where ${r_i\in \mathcal{R}^{\pm1}}$ and ${w_i}$ is some word over ${\mathcal{X}}$. We call such a sequence an identity sequence if ${\mathcal{P}rod c_i}$ is freely equal to the empty word in ${F(\mathcal{X})}$. An equivalence relation is given on identity sequences, and we consider the action of ${F(\mathcal{X})}$ on these sequences given by ${w\cdot(c_1,..,c_m)=(wc_1 w^{-1},...,wc_m w^{-1})}$. This action naturally induces a ${G}$-action on the equivalence classes of identity sequences. We can view the identity sequences equipped with this action as a ${G}$-module is isomorphic to ${\pi_2\mathcal{P}}$. Thus ${\pi_2\mathcal{P}}$ being finitely generated as a ${G}$ module means that there is a finite set of identity sequences such that any identity sequence can be reduced under the equivalence relation to finitely many of these sequences in this set. This identification can then be used to determine a presentation for ${\Gamma}$ entirely from the presentations of ${N}$ and ${Q}$.

2. Proof of Theorem 1

Theorem 1 relies on an extension of the enhanced Rips construction.

Theorem 3 (Modified Rips Construction). There is an algorithm that associates to any finite group presentation ${\mathcal{Q}}$, a compact, negatively curved, piecewise hyperbolic 2-dimensional complex ${K}$ and a short exact sequence

$\displaystyle 1 \rightarrow N \rightarrow \Gamma \rightarrow Q \rightarrow 1, \ \ \ \ \ (4)$

such that

1. ${\mathcal{Q}}$ presents the group ${Q}$,
2. ${K}$ has a single vertex ${x_0}$,
3. the 2-cells of ${K}$ are right angled hyperbolic pentagons (each side of which crosses several 1-cells),
4. ${\Gamma=\pi_1(K,x_0)}$,
5. ${N}$ has a finite generating set ${A}$ of cardinality at least 2,
6. each of the 1-cells in ${K}$ is the unique closed geodesic in its homotopy class,
7. the homotopy class of each ${a\in A}$ is represented by one of the 1-cells of ${K}$,
8. ${G}$ is torsion-free,
9. and each ${a\in A}$ generates its centralizer.

The first seven items follow from a construction in Bridson-Haefliger and Wise. Item (8) comes from the fact that the fundamental group of any compact non-positively curved space is torsion free. Item (9) follows from (7). ${\Gamma}$ is hyperbolic and torsion free, so the centralizer of every non-trivial element in ${\Gamma}$ is cyclic. Thus, if ${a}$ were a proper power, its homotopy class would not correspond to a simple closed geodesic.

Theorem ${1}$ follows from the next result.

Theorem 4. There exists a compact negatively curved 2-complex ${K}$ and a finitely presented subgroup ${P<\pi_1(K\times K)=\Gamma\times\Gamma}$ such that

1. the membership problem for ${P}$ is unsolvable, and
2. ${P}$ has unsolvable conjugacy problem.

This theorem utilizes the existence of groups of type ${F_3}$ with unsolvable word problems. Collins and Miller constructed a group ${Q}$ with a finite 2-dimensional ${K(Q,1)}$ and unsolvable word problem. The enhanced Rips construction is applied to to ${Q}$ for ${\Gamma=\pi_1K}$, where ${K}$ is a compact negatively curved 2-complex.

Following the notation of the Rips construction, take ${\{x_1,..,x_n,a_1,..,a_m\}}$ where ${a_i\in A}$ and the ${x_i}$ are lifts of generators of ${Q}$. We take ${\{(x_i,1),(1,x_i),(a_j,1),(1,a_j)\}}$ for a generating set for ${\Gamma}$. The 1-2-3 Theorem implies that the fibre product ${P}$ is finitely presented.

To see that the membership problem for ${P}$ is unsolvable we restrict to specific words. A word on the generators ${(x_i,1)}$ is in ${P}$ if and only if the same word in the ${x_i}$ is equivalent to the identity in ${Q}$. Thus the unsolvability of the word problem for ${Q}$ implies the unsolvability of the membership problem for ${P}$.

The unsolvability of the conjugacy problem follows from the next lemma.

Lemma 5 Let ${H be finitely generated groups. Suppose ${H\lhd G}$ and there exists ${a\in H}$ such that ${C_G(a). If there is no algorithm to decide membership of ${P}$, then ${P}$ has an unsolvable conjugacy problem.

The thrust of the lemma is that, given a word in ${G}$, we consider ${waw^{-1}}$ where ${a}$ is a generator of ${H}$, and this word being conjugate to ${a}$ in ${P}$ is equivalent to ${w}$ being in ${P}$. This lemma is applied with ${H=N\times N}$, as item (9) of Theorem 3 tells us that the centralizer of ${(a_i,a_i)}$ is ${\langle (a_i,1),(1,a_i)\rangle}$, and thereby gives us unsolvability of the conjugacy problem.

3. Isomorphism problem

BBMS also addressees the isomorphism problem for subgroups of fundamental groups of non-positively curved spaces. These results take us away from subgroup distortion, but are worth noting for their own sake.

Corollary of Modified Rips construction. Let ${K,\ \Gamma}$ and ${A}$ be as above. Let ${a\in A}$. Then ${\hat\Gamma=\langle\Gamma,t\ |\ t^{-1}at=a\rangle}$ is the fundamental group of a compact non=-positively curved squared complex, and is thus biautomatic (Niblo-Reeves).

Proof: Re-metrize ${K}$ by subdividing each pentagonal face by introducing a new vertex in the middle of each face and side and using these to break the pentagons into hyperbolic quadrilaterals. These quadrilaterals are replaced by Euclidean squares without changing side length. The resulting space is still non-positively curved and the original 1-cells are still geodesics. We refine these squares until their side lengths are half that of the original 1-cells in ${K}$. We attach an annulus (union of two Euclidean squares) by gluing the boundary circles to the loop representing ${a}$ by an isometry. $\Box$

Theorem 7. There exists a non-positively curved 4-dimensional complex ${K}$ with biautomatic fundamental group ${G}$, and a (countable) recursive class of finitely presented subgroups ${H_n such that there is no algorithm to determine whether or not ${H_n}$ is isomorphic to ${H_1}$.

The group in this theorem is a direct product ${\hat\Gamma \times \Gamma}$ where ${\Gamma}$ is a torsion free word hyperbolic group and ${\hat\Gamma}$ is an HNN-extension of ${\Gamma}$ given by ${\langle \Gamma,\tau\ |\ \tau^{-1}a_1\tau=a_1\rangle}$ where ${a_1\in\Gamma}$ generates a maximal cyclic subgroup.

Corollary 8. There exists a non-positively curved manifold of dimension 9 and a recursive class of finitely presented subgroups of ${\pi_1M}$ such that there is no algorithm to determine homotopy equivalence between covering spaces corresponding to these subgroups.

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