Connectedness properties and splittings of groups with isolated flats

A major theme of geometric group theory over the last few decades has been the study of a group using various boundaries at infinity attached to spaces on which the group acts. There has been a fruitful connection between topological properties of these boundaries and algebraic properties of the group. Our main theorem provides another example of such a connection for the groups that act geometrically on $\operatorname{CAT}(0)$ spaces with isolated flats.

This article is concerned with the following problem: describe the topological spaces that can arise as the boundary of a $\operatorname{CAT}(0)$ space with isolated flats that admits a geometric group action. Kapovich–Kleiner posed an analogous question for word hyperbolic groups in [KK00]. In order to describe boundaries, one first needs to know which topological properties boundaries can have.

If one can prove that the boundary is connected and locally connected, then it is a Peano continuum. This is a class of compact metric spaces with a rich structure theory that is substantially more powerful than general continuum theory. Much of this paper focuses on determining which groups can have such a boundary.

In the hyperbolic setting, all connected boundaries are locally connected, by a deep, nontrivial theorem (see [BM91, Lev98, Bow99, Swa96]) with far reaching consequences such as [Bow98, KK00]. Groups acting on $\operatorname{CAT}(0)$ spaces with isolated flats [KL95, HK05] are, in a sense, the simplest generalization of hyperbolicity in the $\operatorname{CAT}(0)$ setting. By Hruska–Kleiner, a $\operatorname{CAT}(0)$ group with isolated flats is relatively hyperbolic with respect to the collection $\mathbb{P}$ of maximal virtually abelian subgroups of higher rank [HK05]. A virtually abelian group has higher rank if its rank over $\mathbb{Q}$ is at least two.

By a theorem of Mihalik–Ruane [MR99, MR01], many one-ended $\operatorname{CAT}(0)$ groups with isolated flats have connected but non–locally connected boundary. These examples occur when the group exhibits a particular kind of splitting as a graph of groups. A simple example of the non–locally connected type is the fundamental group of the space obtained from a closed genus two surface and a torus by gluing together an essential simple closed curve from each surface (see Example 2.1).

In the general $\operatorname{CAT}(0)$ setting, determining local connectedness seems to be quite a delicate issue. One-ended hyperbolic groups and fundamental groups of closed Hadamard manifolds all have locally connected boundary, while all $\operatorname{CAT}(0)$ boundaries of one-ended nonabelian right-angled Artin groups are non–locally connected. Very few natural classes of $\operatorname{CAT}(0)$ groups are known to admit boundaries of both types, and among those none had been completely characterized in terms of local connectivity. Indeed this question is not even completely understood for right-angled Coxeter groups, a family that includes many groups with both types of boundary.

Let $G$ act geometrically on a one-ended $\operatorname{CAT}(0)$ space $X$ with isolated flats. Our main theorem determines exactly when $G$ has locally connected boundary.

The reverse implication of Theorem 1.1 follows immediately from the Mihalik–Ruane splitting theorem. This article concerns the forward implication of Theorem 1.1, which was not known previously.

A compactum is semistable if it is shape-theoretically equivalent to a locally connected continuum [GS]. A well-known conjecture of Geoghegan–Mihalik would imply that every boundary of every one-ended $\operatorname{CAT}(0)$ group is semistable [Mih83, GS]. This conjecture has been proven in the $\operatorname{CAT}(0)$ with isolated flats setting by combining work of Mihalik–Swenson and the authors [MS, HR]. In other words, all torsion-free one-ended $\operatorname{CAT}(0)$ groups with isolated flats have semistable boundary. Thus the conclusion of Theorem 1.1 gives finer information than one could derive from semistability alone. When considered up to shape equivalence, all isolated flats boundaries are equivalent to Peano continua, but when considered up to homeomorphism it turns out that many are not Peano continua. In a sense, semistability is too weak to see the precise homeomorphism type of a given boundary.

The conclusion of Theorem 1.1 has already been used as an essential ingredient in constructing examples of non-hyperbolic $\operatorname{CAT}(0)$ groups with boundary homeomorphic to the Menger curve [Hau18, HHS, DHW], which were not previously known to exist. Although groups with Menger boundary in the hyperbolic setting are well-known [Ben92, Cha95, Bou97, KK00, DGP11], substantially different techniques are needed in the non-hyperbolic setting.

In the locally connected case we also obtain a detailed description of the boundary of $G$ as the tree of metric compacta in the sense of Świątkowski [Świ]. This structure explicitly determines the boundary as a tree of spaces built in the same fractal manner as the trees of manifolds of Jakobsche and Ancel–Siebenmann [Jak80, AS85, Jak91], which generalize the classical Pontryagin mod-$2$ surface [Pon30].

The building blocks in this construction are boundaries of vertex groups in a natural graph of groups splitting of $G$. These vertex groups are atomic in the following sense. Suppose $G$ is a one-ended $\operatorname{CAT}(0)$ group with isolated flats. Let $\mathcal{A}$ be the family of subgroups of $G$ that are contained in higher rank virtually abelian subgroups. We say that $G$ is atomic if $G$ does not split over any subgroup in $\mathcal{A}$. The methods used in the proof of Theorem 1.1 also give the following theorem explicitly describing the topology of the boundary.

As suggested by Świątkowski, the structure of a tree of metric compacta may prove useful in classifying which topological spaces arise as visual boundaries of $\operatorname{CAT}(0)$ groups.

In a general setting, attempts to understand a group by splitting into indecomposable pieces often require a study of hierarchies of splittings (as in the settings of one-relator groups, $3$–manifold groups, cubulated groups, etc.). For example in [HR], the authors prove semistability in the isolated flats setting using a nontrivial theorem of Louder–Touikan [LT17] on the termination of slender hierarchies of relatively hyperbolic groups. The study of hierarchies is technically much more elaborate than the study of a single splitting.

In contrast with [HR], in the locally connected case of Theorem 1.2 it is notable that we reach atomic vertex groups after only a single splitting. We show that the hierarchy terminates after one splitting in Theorem 6.4. In Corollary 7.8 we show that the resulting vertex groups are $\operatorname{CAT}(0)$ with isolated flats.

In this section we illustrate both directions of Theorem 1.1 with examples. The first example shows a group whose visual boundary is not locally connected, but whose Bowditch boundary is locally connected. The remaining examples illustrate various constructions of non-hyperbolic groups with locally connected visual boundaries, some illustrating the “indecomposable” case of Theorem 1.4 and others constructed using “locally finite” amalgams of indecomposable groups.

In order to illustrate the difference between the non–locally connected $\operatorname{CAT}(0)$ boundary of $G$ and the locally connected Bowditch boundary $\partial(G,\mathbb{P})$ in Example 2.1, we briefly sketch a significant non–locally connected subset of the boundary. We then describe its locally connected image in the Bowditch boundary.

Each flat $F$ in $\tilde{X}$ is a lift of the torus $T^{2}$. Inside $F$ are infinitely many lifts of the geodesic $\gamma$, along each of which there is a copy of $\mathbb{H}^{2}$ attached. The lifts of $\gamma$ in the flat $F$ are all parallel and thus all share the same pair of endpoints $a,b$ in $\partial F$. Let $Y$ be the convex subcomplex of $\tilde{X}$ consisting of the flat $F$ along with the countably many hyperbolic planes glued to $F$ along the lifts of $\gamma$. Then $\partial Y$ is a suspension of a set $K$ that is countably infinite with two limit points. The suspension points are $a$ and $b$. Note that this suspension itself is not locally connected. Furthermore, each of the points in $\partial F-\{a,b\}$ turns out to be a point of non–local connectivity in $\partial\tilde{X}$.

To understand what the Bowditch boundary is in this example, note that there are two types of circles in the visual boundary $\partial\tilde{X}$—those that occur as the boundary of a hyperbolic plane and those that occur as the boundary of a flat. By [Tra13], the Bowditch boundary is obtained from the visual boundary by collapsing the circles arising as boundaries of flats. In the quotient, each such circle becomes a global cut point of the Bowditch boundary. Each of these cut points is incident to a countable family of circles whose union forms a Hawaiian earring.

We refer the reader to [Bal95, BH99] for introductions to the theory of $\operatorname{CAT}(0)$ spaces. Throughout this section $X$ is assumed to be a proper $\operatorname{CAT}(0)$ space, a condition that holds whenever $X$ admits a proper, cocompact, isometric group action.

The $\operatorname{CAT}(0)$ geometry $X$ gives rise to the visual boundary $\partial X$, which is a compact metrizable space. We first, define the boundary $\partial X$ as a set as follows:

It is a well-known result that for any proper $\operatorname{CAT}(0)$ space, both $X$ and $\overline{X}$ are ANR’s (Absolute Neighborhood Retracts). We refer the reader to [Ont05] for a proof that $X$ is an ANR, using a theorem of [Hu65]. See also [Gui14, §2.9] for a proof that $\overline{X}$ is an ANR using work of [Han51].

The main consequence of being an ANR that we will use is that $\overline{X}$ is locally connected. This consequence is much easier to prove directly, which we do below.

Let us pause for a moment to warn the reader that the notation $\overline{N}$ does not refer to a closed set of $\overline{X}$, but rather refers to the fact that each point of $\overline{N}$ is a limit point of $N$.

Throughout this paper we will often need to compare the sizes of different subsets of the visual boundary. To make such a comparison, it is useful to consider explicit metrics on the visual boundary. The following definition due to Osajda introduces a family of such metrics.

In this section we define the notions of relative hyperbolicity and the Bowditch boundary. The definitions we use are due to Yaman [Yam04] and are given in terms of dynamical properties of an action on a compact space, which turns out to be the Bowditch boundary.

A convergence group action is an action of a finitely generated group $G$ on a compact, metrizable space $M$ satisfying the following conditions, depending on the cardinality of $M$:

• •

  If $M$ is the empty set, then $G$ is finite.

• •

  If $M$ has exactly one point, then $G$ is infinite.

• •

  If $M$ has exactly two points, then $G$ is virtually cyclic.

• •

  If $M$ has at least three points, then the action of $G$ on the space of distinct (unordered) triples of points of $M$ is properly discontinuous.

In the first three cases the convergence group action is elementary, and in the final case the action is nonelementary.

Suppose $G$ has a convergence group action on $M$. An element $g\in G$ is loxodromic if it has infinite order and fixes exactly two points of $M$. A subgroup $P\leq G$ is a parabolic subgroup if it is infinite and contains no loxodromic element. A parabolic subgroup $P$ has a unique fixed point in $M$, called a parabolic point. The stabilizer of a parabolic point is always a maximal parabolic group. A parabolic point $p$ with stabilizer $P:=\operatorname{Stab}_{G}(p)$ is bounded parabolic if $P$ acts cocompactly on $M-\{p\}$. A point $\xi\in M$ is a conical limit point if there exists a sequence $(g_{i})$ in $G$ and distinct points $\zeta_{0},\zeta_{1}\in M$ such that $g_{i}(\xi)\to\zeta_{0}$, while for all $\eta\in M-\{\xi\}$ we have $g_{i}(\eta)\to\zeta_{1}$.

We remark that by work of Yaman, the Bowditch boundary can be obtained as the Gromov boundary of a certain $\delta$–hyperbolic space on which $G$ acts. Although we will not use this point of view in the present article, the construction of this $\delta$–hyperbolic space is the basis of Yaman’s proof that Definition 4.1 is equivalent to other definitions appearing in the literature (such as those in [Bow12]).

Suppose $G$ acts properly, cocompactly, and isometrically on a $\operatorname{CAT}(0)$ space $X$. Suppose also that $G$ has a family of subgroups $\mathbb{P}$ such that $(G,\mathbb{P})$ is relatively hyperbolic. In this case, we have introduced two different types of boundary that one may associate with $G$: the visual boundary $\partial X$ of the $\operatorname{CAT}(0)$ space $X$ and the Bowditch boundary $\partial(G,\mathbb{P})$. These boundaries are closely related by a theorem of Hung Cong Tran, as was mentioned in the introduction. We give a precise statement here.

A $k$–flat in a $\operatorname{CAT}(0)$ space $X$ is an isometrically embedded copy of Euclidean space $\mathbb{E}^{k}$ for some $k\geq 2$. In particular, note that a geodesic line is not considered to be a flat.

The previous theorem also has the following converse.

A group $G$ that admits an action on a $\operatorname{CAT}(0)$ space with isolated flats has a “well-defined” visual boundary, often denoted by $\partial G$, by the following theorem.

Finally, we point out a key geometric fact about $\operatorname{CAT}(0)$ spaces with isolated flats that will be used several times throughout this paper.

In this section, we give the definition of a peripheral splitting of a relatively hyperbolic group and examine some of their basic properties. We introduce the notion of a locally finite peripheral splitting, which plays a key role in the proof of Theorem 1.1. The goal of this section is to prove Theorem 6.4, which roughly states that in the locally finite case many features of a relatively hyperbolic group are inherited by the vertex groups of its maximal peripheral splitting.

Such a splitting is called trivial if one of the vertex groups is equal to $G$ and nontrivial otherwise. One peripheral splitting $\mathcal{H}$ is a refinement of another $\mathcal{G}$ if $\mathcal{G}$ can be obtained from $\mathcal{H}$ by a finite sequence of foldings of edges that preserve the colors of vertices. A refinement is trivial if it involves no folds, i.e., the refinement is an isomorphism of graphs. A peripheral splitting of $(G,\mathbb{P})$ is maximal if it admits only trivial refinements.

Let $\mathcal{A}$ be the collection of all subgroups of the peripheral subgroups of $G$. If $(G,\mathbb{P})$ admits only a trivial peripheral splitting, then $G$ does not split over $\mathcal{A}$ relative to $\mathbb{P}$ in the following sense: let $G$ act without inversions on a simplicial tree $T$ such that each peripheral subgroup has a fixed point in $T$. Then the action of $G$ on $T$ has a global fixed point.

The notion of an atomic group, mentioned in the introduction, is a much stronger condition that does not include the phrase “relative to $\mathbb{P}$.”

If $(G,\mathbb{P})$ is atomic then the last conclusion of this theorem holds. In general the component vertex groups of the maximal peripheral splitting inherit a natural relatively hyperbolic structure, but they do not need to be atomic. See [HR] for examples of relatively hyperbolic groups whose component groups are not atomic, and which require a nontrivial hierarchy of splittings over parabolic subgroups in order to reach atomic groups.

In order to ensure that the maximal peripheral splitting has atomic component vertex groups, we need to focus on the special case of locally finite splittings, which are better behaved than the general case. A peripheral splitting $\mathcal{G}$ is locally finite if for each peripheral vertex group $P$ the adjacent edge groups include as finite index subgroups of $P$. Equivalently, in the Bass–Serre tree $T$ for $\mathcal{G}$, the vertex $v$ stabilized by $P$ has finite valence. The following theorem summarizes key properties of the component vertex groups in the unique maximal peripheral splitting in the special case that this splitting is locally finite. For simplicity we have stated this result only in the isolated flats setting, which is the only case needed in this paper.

In this section we prove Theorem 1.3, which is a convex splitting theorem for $\operatorname{CAT}(0)$ groups. This splitting theorem does not use the notion of isolated flats, and could potentially be a useful tool in many other $\operatorname{CAT}(0)$ settings.

We apply this splitting theorem to peripheral splittings of $\operatorname{CAT}(0)$ groups with isolated flats, in order to prove Theorem 7.7, which states that the vertex groups of such a splitting are also $\operatorname{CAT}(0)$ groups with isolated flats. Although this article is mainly focused on locally finite splittings, Theorem 7.7 has no such restriction and applies to all $\operatorname{CAT}(0)$ groups with isolated flats. Ben-Zvi uses Theorem 7.7 to show that all one-ended $\operatorname{CAT}(0)$ groups with isolated flats have globally path connected boundary [BZ].

The next two results are key tools used in the proof of Theorem 1.3.

The following folk result has been used implicitly in many places throughout the literature (see for example [HK05]). We have decided to include the (short) proof for the benefit of the reader. Dani Wise has described the proof as a “pigeonhole principle” for cocompact group actions.

The proof of Theorem 1.3 will be developed over the course of the next several lemmas and definitions.

Let $\pi\colon X\to T$ be the $G$–equivariant continuous map given by Lemma 7.1. For each edge $e$ of $T$, let $m_{e}$ denote the midpoint of $e$. For each vertex $v$ of $T$, let $S(v)$ be the union of all segments of the form $[v,m_{e}]$ where $e$ is an edge adjacent to $v$; in other words, $S(v)$ is the union of all half-edges emanating from $v$. Let $Q(e)\subset X$ be the preimage $\pi^{-1}(m_{e})$, and let $Q(v)\subset X$ be the preimage $\pi^{-1}\bigl{(}S(v)\bigr{)}$.

In the proof of Theorem 1.3 we assume that we have a splitting whose edge groups are convex, in other words each edge group $G_{e}$ stabilizes a closed convex subspace $Y_{e}$ on which it acts cocompactly. Although $G_{e}$ also acts cocompactly on $Q(e)$, it is unlikely that $Q(e)$ itself is a convex subspace of $X$. In the next lemma, we show that we can enlarge $Q(e)$ to its convex hull and preserve cocompactness.

In the next proposition, we deal with the vertex groups. We replace $Q(v)$ with a larger set $C(v)$. We first show that $G_{v}$ acts cocompactly on $C(v)$. Afterwards we will complete the proof of Theorem 1.3 by showing that $C(v)$ is a convex subspace of $X$.

We now complete the proof of Theorem 1.3 by showing that the subspace $C(v)$ constructed above is convex, which is the most delicate part of the argument.

We obtain the following corollary by combining Theorem 7.7 with Theorem 6.4.

In this section, we begin the proof of Theorem 1.1. Our goal is to study the boundary of a one-ended $\operatorname{CAT}(0)$ group with isolated flats. We need to show that if the maximal peripheral splitting is locally finite, then the boundary of the $\operatorname{CAT}(0)$ space is locally connected. The first step of the proof, which is the main goal of the current section, is to show that the visual boundary $\partial X$ is locally connected at any point not in the boundary of a flat, i.e., the rank one points.

We use decomposition theory to prove Corollary 8.13, which states that the map $\partial X\to\partial(G,\mathbb{P})$ given by Theorem 4.3 from the visual boundary to the Bowditch boundary is upper semicontinuous. Using this structure we are able to pull back local connectivity from the Bowditch boundary to the $\operatorname{CAT}(0)$ boundary, but only at rank one points (see Corollary 8.14).

Many of our techniques in the proof of the main theorem depend on elementary results from decomposition theory, which we summarize below. We refer the reader to [Dav86] for more details. Recall that a decomposition $\mathcal{D}$ of a topological space $M$ is a partition of $M$. Decompositions are equivalent to quotient maps in the following sense. A decomposition $\mathcal{D}$ of a space $M$ has a natural quotient map $\pi\colon M\to M/\mathcal{D}$ obtained by collapsing each equivalence class of $\mathcal{D}$ to a point and endowing the result with the quotient topology. Conversely, any quotient map $M\to N$ gives rise to an associated decomposition of $M$ consisting of the family of point preimages.

The proof of the following result is left as a routine exercise to the reader (cf. Proposition I.1.1 of [Dav86]).