Ends and boundaries of groups

 After being asked what manifolds can arise as boundaries of hyperbolic groups I found the survey of Kapovich and Benakli which I enjoyed reading. I highlight some of my favourites from the theory, which isn't an easy choice at all. 

A classical fact is that the group of isometries of the hyperbolic plane, with the disc model, is isomorphic to $\mathrm{PSL}_2(\mathbb{R})$, given by Mobius transformations, and that these isometries are elliptic (there is a fixed point in the plane), parabolic (there is a fixed point on the boundary), or loxodromic (there are two fixed points on the boundary), corresponding to whether $|\mathrm{tr}|=2, <2$, or $>2$.

Since the notion of boundary was so useful, one might ask when boundaries can be constructed, and whether these can be meaningfully attached to groups, since a philosophy of geometric group theory is that groups and the spaces they act on geometrically (properly and cocompactly by isometries) should have the same coarse geometry.

In the case of the hyperbolic plane, the points on the boundary happen to correspond very nicely to endpoints of geodesics, so one can attempt to turn this into a definition: the boundary is the set of equivalence classes of geodesics in some Cayley graph of the group, for some definition of equivalence. We will return to this below, but this turns out to be a coarse geometric invariant in general only when the group is hyperbolic. What one can do in general is define the notion of ends.

Ends

Ends capture the notion of the number of connected components at infinity, which in particular does give rise to an invariant of groups. Let $n\geqslant 0$ be a non-negative integer. A space $\Gamma$ is said to have $n$ ends, written $e(G)=n$, if for all compact subsets of $\Gamma$, there are at most $n$ non-compact connected components, and for some choice of compact subset $K$ there are exactly $n$ non-compact connected components. It has infinitely many $n$ if you can get arbitrarily many non-compact connected components. In the example we gave above, the hyperbolic plane and all fundamental groups of closed hyperbolic surfaces have 1 end. Quite a lot is known about this invariant. The first theorem proved was

Freundental-Hopf Theorem: A group $G$ has either 0,1,2, or infinitely many ends, and $e(G)=2$ if and only if $G$ is virtually cyclic.

Observe that a group has 0 ends if and only if it is finite. The next big breakthrough came in the form of

Stallings' theorem on ends: A group $G$ has infinitely many ends if and only if it splits over a finite subgroup; equivalently it acts without a global fixed point on a tree with finite edge stabilisers.

This means one-ended groups are the most mysterious class, and indeed very little in general is known here, hence we need a finer invariant, which we discuss for hyperbolic groups next. Before that, however, it is worth pointing out as a corollary of Stallings' theorem, every finitely generated group of cohomological dimension one is free and that every torsion-free virtually free group is free. Stallings' theorem also implies that the property of having a nontrivial splitting over a finite subgroup is a quasi-isometry invariant of a finitely generated group since the number of ends of a finitely generated group is easily seen to be a quasi-isometry invariant. The theory of ends of groups also proves a nice result about when one can remove points from $\mathbb{R}^n$ and still get a space that is a universal covering space of a manifold; see this cute application.

Hyperbolic groups

The theory of hyperbolic groups has long been a cornerstone of geometric group theory. Recall that a finitely generated group $G$ is called $\textit{hyperbolic}$ if its Cayley graph (with respect to some finite generating set) satisfies the $\textit{thin triangles condition}$ - there exists $\delta \geq 0$ such that for every geodesic triangle there exist two sides whose $\delta$-neighbourhood contains the third side.

One of the most remarkable features of hyperbolic groups is that they admit a natural $\textit{boundary at infinity}$ $\partial G$, which captures the asymptotic geometry of the group. This boundary is defined as the set of equivalence classes of geodesic rays, where two rays are equivalent if they remain within bounded distance from each other.

The boundary $\partial G$ possesses several important properties:

1. It is a compact metrizable space of finite topological dimension, which Bestvina and Mess prove is $vcd(G)-1$ whenever $G$ is virtually torsion-free, and $H^*(G, \mathbb{Z}G)=H^{*-1}(\partial G, \mathbb{Z})$.
2. The group $G$ acts on $\partial G$ by homeomorphisms.
3. A quasi-isometry between hyperbolic groups induces a homeomorphism between their boundaries, so the topological type of $\partial G$ is a quasi-isometry invariant. This lets one study quasi-isometries using topology.
4. For non-elementary hyperbolic groups, $\partial G$ is perfect and uncountable.

In fact one can define boundary for general hyperbolic metric spaces and check that this coincides with the notion above. However, the boundaries here are much wilder: any compact metrisable space can be realised as the boundary of a proper hyperbolic metric space. When one only has non-positive curvature the boundary is no longer well-defined as a group invariant, so this notion of boundary really does rely on the hyperbolicity.

Common examples include:

• Free groups: $\partial F_n$ is a Cantor set 
• Surface groups: $\partial \pi_1(S_g)$ is a circle for $g \geq 2$
• Fundamental groups of compact hyperbolic manifolds: $\partial G$ is a sphere $S^{n-1}$

The boundary construction provides powerful tools for studying hyperbolic groups, connecting geometric group theory with dynamics, analysis, and topology. It allows us to translate algebraic properties of the group into geometric and topological properties of this boundary space. For instance, Stallings' theorem on ends of groups says that a finitely generated infinite group is two-ended if and only if it is virtually cyclic, infinite-ended if and only if it splits over a finite subgroup, and one-ended otherwise.

Thus it is natural to concentrate on hyperbolic groups with connected non-empty boundary (that is to say one-ended hyperbolic groups). Suppose $A$ and $B$ are non-elementary hyperbolic groups and suppose $G = A \ast_C B$ where $C = \langle c \rangle$ is an infinite cyclic group. Put $c^+ = \lim_{n \to \infty} c^n \in \partial G$ and $c^- = \lim_{n \to \infty} c^{-n} \in \partial G$. It is easy to see that $c^+$ is a local cut point in $\partial G$.

Indeed, suppose $x \in \partial A$, $y \in \partial B$ are points close to $c^+$. Let $p$ be a path from $x$ to $y$ in $\partial G$ which stays close to $c^+$. We claim that $p$ must in fact pass through $c^+$. Indeed, $p$ is approximated by paths $p'$ from $a \in A$ to $b \in B$ in the Cayley graph of $G$. Since $G = A \ast_C B$, every such $p'$ must pass through some power $c^i$ of $c$. Moreover we must have $i \to \infty$ since $p'$ approximates $p$. Hence $c^i \to c^+$ and so $p$ passes through $c^+$. Thus we see that $c^+$ is indeed a local cut point in the boundary. A similar argument applies to hyperbolic groups which split essentially over two-ended (i.e. virtually infinite cyclic) subgroups as an amalgamated product or an HNN-extension. (An amalgamated free product over a two ended subgroup is called essential if the associated two-ended subgroup has infinite index in both factors. Similarly, an HNN-extension over a two-ended subgroup is essential if the associated two-ended subgroup has infinite index in the base group). A remarkable result of B. Bowditch shows that the converse is also true.

$\textbf{Theorem 1:}$ Let $G$ be a one-ended word-hyperbolic group such that $\partial G \neq S^1$. Then

1. The boundary $\partial G$ is locally connected and has no global cut points. 
2. The group $G$ essentially splits over a two-ended subgroup if and only if $\partial G$ has a local cut point.

Bowditch studies the combinatorics of local cut-points in the boundary of a one-ended hyperbolic group. It turns out that one can use this combinatorics (and that of connected components of the boundary with several local cut-points removed from it) to construct a discrete ``pre-tree'', that is a structure with a tree-like ``betweenness'' relation. This pre-tree is used to produce an actual simplicial tree which naturally inherits a $G$-action. The action is then shown to have two-ended edge-stabilizers, which implies Theorem 1. Moreover, a detailed analysis of the simplicial tree action of $G$ described above can be used to read-off the JSJ-decomposition of $G$. (There are notes from the reading group on JSJ decompositions here) Roughly speaking, the JSJ decomposition is a canonical splitting of $G$ with two-ended edge stabilizers such that any essential splitting of $G$ over a two-ended subgroup can be obtained from the JSJ using some simple operations. Since Bowditch's construction uses only the topological properties of the boundary, his proof not only provides the uniqueness and existence of the JSJ-decomposition for hyperbolic groups (with or without torsion), but also demonstrates that certain basic properties of the JSJ-decomposition are preserved by quasi-isometries of $G$. Thus, for example, the proof implies that two quasi-isometric (e.g. commensurable) hyperbolic groups have the same number of ``quadratically hanging'' subgroups in their JSJ-decompositions.

The techniques also yielded the following theorems which have since been generalised to all finitely presented groups (by Dunwoody-Swenson and Papasoglu respectively):

$\textbf{Theorem 2:}$ Let $G$ be a one-ended word-hyperbolic group which is not quasi-fuchsian. Let $C$ be a two-ended subgroup of $G$. Then the number of relative ends $e(G, C) > 1$ if and only if $G$ splits essentially over a two-ended group.

$\textbf{Theorem 3:}$ Let $G_1$ and $G_2$ be quasi-isometric non-elementary one-ended word-hyperbolic groups with the boundaries different from the circle. Then $G_1$ splits over a two-ended subgroup if and only if $G_2$ splits over a two-ended subgroup.

There are also many intriguing open questions about the boundary. Most prominent among these is the Cannon conjecture: if a hyperbolic group has boundary the sphere $S^2$ must it be virtually Kleinian? This is related to a theorem of Kapovich-Kleiner which classifies the possible one-dimensional boundaries of groups, via the fact that hyperbolic 3-manifolds with boundary are 2-dimensional. They conjecture that the Sierpinski carpet case occurs exactly when the group is commensurable to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. Let us first recall some facts about 1-dimensional fractals.

Sierpinski carpets and Menger curves

The classical construction of a Sierpinski carpet is analogous to the construction of a Cantor set: start with the unit square in the plane, subdivide it into nine equal subsquares, remove the middle open square, and then repeat this procedure inductively on the remaining squares. If we take a sequence \(D_{i}\subset S^{2}\) of disjoint closed 2-disks whose union is dense in \(S^{2}\) so that \(\text{Diam}(D_{i})\to 0\) as \(i\to\infty\), then \(S^{2}-\cup_{i}\text{Interior}(D_{i})\) is a Sierpinski carpet; moreover one can show that any Sierpinski carpet embedded in \(S^{2}\) is obtained in this way. Sierpinski carpets can also be characterized as follows: a compact, 1-dimensional, planar, connected, locally connected space with no local cut points is a Sierpinski carpet.

We will use a few topological properties of Sierpinski carpets \(\mathcal{S}\):

• There is a unique embedding of \(\mathcal{S}\) in \(S^{2}\) up to post-composition with a homeomorphism of \(S^{2}\).
•  There is a countable collection \(\mathcal{C}\) of ``peripheral circles'' in \(\mathcal{S}\), which are precisely the nonseparating topological circles in \(\mathcal{S}\).
• Given any metric \(d\) on \(\mathcal{S}\) and any number \(D>0\), there are only finitely many peripheral circles in \(\mathcal{S}\) of diameter \(>D\).

The Menger curve may be constructed as follows. Start with the unit cube \(I^{3}\) in \(\mathbb{R}^{3}\). Consider the orthogonal projections \(\pi_{ij}:I^{3}\to F_{ij}\) of the unit cube onto the \(ij\) coordinate square, and let \(\mathcal{S}_{ij}\subset F_{ij}\) be the Sierpinski carpet as constructed above. The Menger curve is the intersection \(\cap_{i<j}\pi^{-1}_{ij}(\mathcal{S}_{ij})\). The Menger curve is universal among all compact metrizable 1-dimensional spaces: any such space can topologically embedded in the Menger curve. A compact, metrizable, connected, locally connected, 1-dimensional space is a Menger curve provided it has no local cut points, and no nonempty open subset is planar.

We can now state the Kapovich-Kleiner classification:

$\textbf{Theorem 4:}$ Let $G$ be a hyperbolic group which does not split over a finite or virtually cyclic subgroup, and suppose $\partial^\infty G$ is $1$-dimensional. Then one of the following holds:

1. $\partial^\infty G$ is a Menger curve;
2. $\partial^\infty G$ is a Sierpinski carpet; 
3. $\partial^\infty G$ is homeomorphic to $S^1$ and $G$ maps onto a Schwartz triangle group with finite kernel.

In higher dimensions such results will be much harder to obtain if at all possible since there are so many compacta.

$\textbf{Proof:}$ The fact that $G$ does not split over a finite group implies that $G$ is one-ended, and $\partial^\infty G$ is connected. By theorem 1, the boundary of a one-ended hyperbolic group is locally connected and has no global cut points; furthermore, if $\partial^\infty G$ has local cut points then $G$ splits over a virtually infinite cyclic subgroup unless $\partial^\infty G \simeq S^1$ and $G$ maps onto a Schwarz triangle group with finite kernel (the classification of groups with $S^1$ boundary is a landmark result due to many people, including Gabai, Casson-Jungreis, Tukia...). Therefore from now on we will assume that $\partial^\infty G$ has no local cut points.

A $1$-dimensional, compact, metrizable, connected, locally connected space $Z$ with no local cut points is a Menger curve provided no point $z \in Z$ has a neighborhood which embeds in the plane. Hence either $\partial^\infty G$ is a Menger curve or some $\xi \in \partial^\infty G$ has a planar neighborhood $U$; therefore we assume the latter holds.

$\textbf{Claim:}$ Let $\Gamma \subset \partial^\infty G$ be a subset homeomorphic to a finite graph. Then $\Gamma$ is a planar graph.

$\textbf{Proof:}$ Since the action of $G$ on $\partial^\infty G$ is minimal, every $G$-orbit intersects the planar neighborhood $U$, and so every point of $\partial^\infty G$ has a planar neighborhood. Because $\partial^\infty G$ has no local cut points, we have $\partial^\infty G \setminus \Gamma \neq \emptyset$. So we can find a hyperbolic element $g \in G$ whose fixed point set $\{\eta_1, \eta_2\} \subset \partial^\infty G$ is disjoint from $\Gamma$. Hence for sufficiently large $n$, $g^n(\Gamma)$ is contained in a planar neighborhood of $\eta_1$ or $\eta_2$. $\blacksquare$

We now quote the classification that a compact, metrizable, connected, locally connected space $X$ with no global cut points is planar as long as no nonplanar graph embeds in $X$. Therefore $\partial^\infty G$ is planar. Finally, by known results, $\partial^\infty G$ is Sierpinski carpet. $\blacksquare$

Recall that a hyperbolic group $G$ is said to be topologically rigid if every homeomorphism $f : \partial G \to \partial G$ is induced by an element of $G$. Kapovich-Kleiner also construct topologically rigid groups with 2-dimensional boundary and contrast this with a sketch of a proof that non-elementary hyperbolic groups with 1-dimensional boundary aren't topologically rigid. Topologically rigid groups have the following remarkable property: If $G'$ is a hyperbolic group whose boundary is homeomorphic to the boundary of a topologically rigid hyperbolic group $G$, then $G'$ embeds in $G$ as a finite index subgroup.

North-South dynamics

A quasi-isometry between two hyperbolic spaces extends to a homeomorphism between their boundaries. In particular, every isometry of a hyperbolic space is a quasi-isometry and thus induces a homeomorphism of the boundary (which is bi-Lipschitz with respect to any visual metric). It turns out that any isometry belongs to one of three types: elliptic, parabolic or loxodromic (hyperbolic):

$\textbf{Proposition 5:}$ Let $(X, d)$ be a proper hyperbolic space and let $\gamma : X \to X$ be an isometry of $X$. Then exactly one of the following occurs:

1. For any $x \in X$ the orbit of $x$ under the cyclic group $\langle \gamma \rangle$ is bounded. In this case $\gamma$ is said to be $\textit{elliptic}$.
2. The homeomorphism $\gamma : \partial X \to \partial X$ has exactly two distinct fixed points $\gamma^+, \gamma^- \in \partial X$. For any $x \in X$ the $\langle \gamma \rangle$-orbit map $\mathbb{Z} \to X$, $n \mapsto \gamma^n x$ is a quasi-isometric embedding and $\lim_{n \to \infty} \gamma^n x = \gamma^+$, $\lim_{n \to \infty} \gamma^{-n} x = \gamma^-$. In this case $\gamma$ is said to be $\textit{hyperbolic}$ or $\textit{loxodromic}$.
3. The homeomorphism $\gamma : \partial X \to \partial X$ has exactly one fixed point $\gamma^+ \in \partial X$. For any $x \in X$ $\lim_{n \to \infty} \gamma^n x = \gamma^+$, $\lim_{n \to \infty} \gamma^{-n} x = \gamma^+$. In this case $\gamma$ is said to be $\textit{parabolic}$.

Notice that if $\gamma$ has finite order in the isometry group of $X$ then $\gamma$ has to be elliptic.

A word-hyperbolic group $G$ acts by isometries on its Cayley graph and therefore this action extends to the action of $G$ on $\partial G$ by homeomorphisms. We will summarize the basic topological properties of the action of $G$ on $\partial G$ in the following statement:

$\textbf{Proposition 6:}$ Let $G$ be a word-hyperbolic group. Then:

1. Any element $g \in G$ of infinite order in $G$ acts as a loxodromic isometry of the Cayley graph of $G$. There are exactly two points in $\partial G$ fixed by $g$: the point $g^+ = \lim_{n \to \infty} g^n$ and the point $g^- = \lim_{n \to \infty} g^{-n}$. The points $g^+$ and $g^-$ are referred to as $\textit{poles}$ or $\textit{rational points}$ corresponding to the infinite cyclic subgroup $\langle g \rangle \leq G$.
2. The action of $G$ on $\partial G$ is $\textit{minimal}$, that is for any $p \in \partial G$ the orbit $Gp$ is dense in $\partial G$.
3. The set of rational points $Q(G) := \{g^+ \mid g \in G \text{ is of infinite order} \}$ is dense in $\partial G$.
4. The set of pole-pairs $Q'(G) := \{(g^-, g^+) \mid g \in G \text{ is of infinite order} \}$ is dense in $\partial G \times \partial G$.

It turns out that the action by an element of infinite order on the boundary of a hyperbolic group has very simple “North-South” dynamics. This was first observed for Fuchsian groups but turns out to be a much more fundamental fact.

$\textbf{Theorem 7:}$ Let $G$ be a word-hyperbolic group and let $g \in G$ be an element of infinite order. Then for any open sets $U, V \subseteq \partial G$ with $g^+ \in U$, $g^- \in V$ there is $n > 1$ such that $g^n (\partial G - V ) \subseteq U$.

Manifolds as boundaries

We now finally get to the question which prompted me to write this blog post.

$\textbf{Theorem 8:}$ Let $G$ be an infinite hyperbolic group. Suppose $\partial G$ contains an open subset homeomorphic to $\mathbb{R}^n$, where $n \geq 2$. Then $\partial G$ is homeomorphic to $S^n$.

$\textbf{Proof:}$ Since pole-pairs $Q'(G)$ are dense in $\partial G$, there is $g \in G$ of infinite order such that $g^+, g^-$ are both contained in the open subset of $\partial G$ homeomorphic to $\mathbb{R}^n$. Let $U', V'$ with $g^+ \in U'$, $g^- \in V'$ be disjoint open neighborhoods of $g^+$ and $g^-$ which are both homeomorphic to $\mathbb{R}^n$. Thus there are homeomorphisms $f^+ : \mathbb{R}^n \to U'$ and $f^- : \mathbb{R}^n \to V'$ such that $f^+(0) = g^+$ and $f^-(0) = g^-$. Let $B_2$ and $B_1$ be open balls of radius $2$ and $1$ centered at the origin in $\mathbb{R}^n$. Put $U = f^+(B_2)$ and $V = f^-(B_1)$. Thus $U$ is an open neighborhood of $g^+$ and $V$ is an open neighborhood of $g^-$ in $\partial G$. Therefore there is $m > 1$ such that $g^m (\partial G - V ) \subseteq U$. Consider the sphere $S$ of radius $2$ in $\mathbb{R}^n$ centered at the origin. Thus $f^-(S) \subseteq \partial G - V$ is an embedded sphere in $\partial G$. Moreover, this is a collared sphere since we clearly can extend $f^-|_S$ to a topological embedding $S \times [-\varepsilon, \varepsilon] \to \partial G - V$. Consider now the sets $S' = g^m [f^-(S)] \subseteq U \subseteq \partial G$ and $S'' = (f^+)^{-1}(S') \subseteq B_1 \subseteq \mathbb{R}^n$. Then $S''$ is a topologically embedded sphere in $\mathbb{R}^n$ which possesses a bi-collar.

Therefore by the Generalized Shoenflies Theorem $S''$ separates $\mathbb{R}^n$ into two open connected pieces: one bounded and homeomorphic to an open ball and the other unbounded and homeomorphic to the exterior of a closed ball. Let $C$ be the bounded piece. Since $\partial G - V$ is compact (as a closed subset of $\partial G$), the set $g^m (\partial G - V)$ is compact as well. Hence the set $(f^+)^{-1}(g^m (\partial G - V))$ is a bounded subset of $\mathbb{R}^n$. It is clear that $\partial G - V$ is a closed subset of $\partial G$ with the topological boundary of $\partial G - V$ equal to $f^-(S)$. Since $g^m$ and $f^+$ are homeomorphisms, the interior of $(f^+)^{-1}(g^m (\partial G - V))$ is an open connected subset of $\mathbb{R}^n$ with topological boundary $S''$. We have already seen that $(f^+)^{-1}(g^m (\partial G - V))$ is bounded and therefore it is equal to the closure of $C$. Thus $(f^+)^{-1}(g^m (\partial G - V))$ is homeomorphic to a closed $n$-ball. Hence $\partial G - V$ is also homeomorphic to a closed $n$-ball. The closure of $V$ is equal to $f^-(\overline{B}_1)$ and so it is also a closed $n$-ball by construction. Thus $\partial G$ can be obtained by gluing two closed $n$-balls $\overline{V}$ and $\partial G - V$ along the $(n - 1)$-sphere $f^-(S)$. Therefore $\partial G$ is homeomorphic to an $n$-sphere as required. $\blacksquare$

Cannon-Thurston maps

I would like to end by pointing out that it is far from obvious that for any hyperbolic subgroup $H$ of a hyperbolic group $G$ that the maps on boundaries should extend - this is only immediate if $H$ is quasi-convex. Nevertheless, in many circumstances this is possible and gives rise to strange maps. For example, Cannon and Thurston were first interested in the inclusion map from the fibre (a surface subgroup) of a fibred hyperbolic 3-manifold. They managed to show that there is an induced map $S^1 \to S^2$ which is a space filling curve. I leave it to the reader to explore this brief account of the theory.