Hyperbolic and cubical rigidities of Thompson’s group
In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson’s group is hyperbolically elementary, and we deduce that it satisfies Property , ie., every isometric action of on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.
A major theme in geometric group theory is to make a given group act on a metric space which belongs to a specific class in order to deduce some information about it. However, not every group is sensitive to a given class of spaces, meaning that every isometric action of a fixed group on any one of these spaces may turn out to be trivial in some sense. Nevertheless, although the machinery of group actions on spaces of cannot be applied, it turns out that the non-existence of good actions provides interesting information as well. Roughly speaking, it implies some rigidity phenomena.
The first occurrence of such an idea was Serre’s Property (FA). A group satisfies Property (FA) if every isometric action on a simplicial tree fixes a point. We refer to [Ser03, §6] for more information about this property. For instance, Property (FA) imposes restrictions on how to embed a given group into another (see for instance [Fuj99] in the context of 3-manifolds), and more generally on the possible homomorphisms between them (see for instance [DS08, Corollary 4.37] in the context of relatively hyperbolic groups). Also, such a rigidity has been applied in [Her88] to determine when the fundamental groups of two graph of groups whose vertex-groups satisfy Property (FA) are isomorphic.
Another famous fixed-point property is Kazhdan’s Property (T). Usually, Property (T) is defined using representation theory, but alternatively, one can say that a (discrete) group satisfies Property (T) if every affine isometric action on a Hilbert space has a global fixed, or equivalently if every isometric action on a median space has bounded orbits. See [BdlHV08, CDH10] for more information. Property (T) for a group imposes for instance strong restrictions on the possible homomorphisms starting from that group (for a geometric realisation of this idea, see for example [Pau91], whose main construction has been very inspiring in other contexts), and plays a fundamental role in several rigidity statements, including the famous Margulis’ superrigidity. We refer to [BdlHV08], and in particular to its introduction, for more information about Property (T).
In this article, we are mainly interested in the class of Gromov-hyperbolic spaces. We say that a group is hyperbolically elementary if every isometric action on a hyperbolic space either fixes a point at infinity or has bounded orbits. Once again, such a property imposes restrictions on the possible homomorphisms between two groups. For instance, it is proved in [Hae16] that higher rank lattices are hyperbolically elementary, from which it is deduced that any morphism from a higher rank lattice to the mapping class group of a closed surface with punctures must have finite image (a statement originally due to Farb, Kaimanovich and Masur).
We emphasize the fact that it is not reasonable to remove the possibility of fixing a point at infinity from the definition of hyperbolic elementarity. Indeed, any infinite group admits a proper and parabolic action on a hyperbolic space; see for instance the classical construction explained in [Hru10, Section 4]. However, being hyperbolically elementary does not mean that any isometric action on a hyperbolic space is completely trivial, since the definition does not rule out lineal actions (ie., actions on a quasi-line) nor quasi-parabolic actions (ie., actions with loxodromic isometries all sharing a point at infinity). And these actions may provide interesting information on a group. For instance, admitting lineal actions is related to the existence of quasimorphisms; and admitting a quasi-parabolic action implies the existence of free sub-semigroups, so that the group must have exponential growth.
The first main objective of our article is to prove a general criterion leading to some hyperbolic rigidity. More precisely:
Let be a group. Suppose that there exist two subsets satisfying the following conditions.
is boundedly generated by , ie., there exists some such that every element of is the product of at most elements of .
For every , there exists some such that .
For every , there exist some such that the following holds. For every , there exists some such that the elements , and all belong to .
Then any isometric action of on a hyperbolic space fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits.
Our main motivating in proving this criterion is to show that Thompson’s group is hyperbolically elementary.
Any isometric action of Thompson’s group on a Gromov-hyperbolic space either fixes a point at infinity or has bounded orbits.
The groups , and were defined by Richard Thompson in 1965. Historically, Thompson’s groups and are the first explicit examples of finitely presented simple groups. Thompson’s groups were also used in [MT73] to construct finitely presented groups with unsolvable word problems, and in [Tho80] to shows that a finitely generated group has a solvable word problem if and only if it can be embedded into a finitely generated simple subgroup of a finitely presented group. We refer to [CFP96] for a general introduction to these three groups. Since then, plenty of articles have been dedicated to Thompson’s groups, and they have been the source of inspiration for the introduction of many classes of groups, now referred to as Thompson-like groups; see for instance [Hig74, Bro87, Ste92, Bri07, FK08, Bri04, BF15]. Nevertheless, Thompson’s groups remain mysterious, and many questions are still open. For instance, it is a major open question to know whether is amenable, and the structure of subgroups of is still essentially unknown [BCR17].
Our initial motivation in proving Theorem 1.2 came from another fixed-point property, in the class of CAT(0) cube complexes. A group
satisfies Property , for some , if every isometric action of on an -dimensional CAT(0) cube complex has a global fixed point;
satisfies Property if it satisfies Property for every ;
satisfies Property if every isometric action of on a CAT(0) cube complex has a global fixed point.
Property was introduced by Barnhill and Chatterji in [BC08]111In [BC08], Barnhill and Chatterji named Property as Property . However, since then, Property refers to the fixed-point property in CAT(0) cube complexes of arbitrary dimensions. So we changed the terminology., asking the difference between Kazhdan’s Property (T) and Property . It turns out that in general Property (T) is quite different from Property or Property . For instance, it is conjectured in [Cor15] that higher rank lattices satisfy Property , but such groups may be far from satisfying Property (T) since some of them are a-T-menable. For positive results in this direction, see [CFI16, Corollary 1.7], [Cor13, Example 6.A.7], [Cor15, Theorem 6.14]. For some (very) recent developments related to Property , see [CC17, LMBT18, Cor18].
The second main result of our article shows how to deduce Property from some hyperbolic rigidity. More explicitly:
A finitely generated group all of whose finite-index subgroups
are hyperbolically elementary,
and do not surject onto ,
satisfies Property .
We emphasize the fact that we do not know if the property of being hyperbolically elementary is stable under taking finite-index subgroups. Since Thompson’s group is a simple group, the combination of our two main theorems immediately implies that satisfies Property .
Any isometric action of Thompson’s group on a finite-dimensional CAT(0) cube complex fixes a point.
We emphasize that it was previously known that (as well as and ) does not act properly on a finite-dimensional CAT(0). In fact, since contains a free abelian group of arbitrarily large rank, it follows that cannot act properly on any contractible finite-dimensional complex.
Corollary 1.4 contrasts with the known fact that acts properly on a locally finite infinite-dimensional CAT(0) cube complexes. (Indeed, Guba and Sapir showed in [GS97, Example 16.6] that coincides with the braided diagram group where is the semigroup presentation ; and Farley constructed in [Far05] CAT(0) cube complexes on which such groups act.) As a consequence, provides another negative answer to [BC08, Question 5.3], ie., is a new example of a group satisfying Property but not Property (T). Indeed, as a consequence of [NR98], a group acting properly on a CAT(0) cube complex does not satisfy Property (T); in fact such a group must be a-T-menable, according to [NR97], which is a strong negation of Kazhdan’s Property (T).
So provides an example of a tough transition between finite and infinite dimensions, since on the one hand, has the best possible cubical geometry in infinite dimension: it acts properly on a locally finite CAT(0) cube complex; and on the other hand, it has the worst possible cubical geometry in finite dimension: every isometric action of on a finite-dimensional CAT(0) cube complex has a global fixed point. Using the vocabulary of [Cor13], Thompson’s group satisfies Property PW and Property for every . It seems to be the first such example in the literature.
We would like to emphasize the fact that, although our article it dedicated to Thompson’s group , we expect that Theorem 1.1 applies to most of the generalisations of . For instance, without major modifications, our arguments apply to Higman-Thompson groups (, ) and to the group of interval exchange transformations . However, since there does not exist a common formalism to deal with all the generalisations of , we decided to illustrate our strategy by considering only . Therefore, our paper should not be regarded as proving a specific statement about , but as proposing a general method to prove hyperbolic and cubical rigidities of groups looking like . In particular, we expect that our strategy works for higher dimensional Thompson’s groups.
Finally, we would like to mention that Thompson’s group is also hyperbolically elementary, since it does not contain any non-abelian free subgroup, but it does not satisfy Property since its abelianisation is infinite. About Thompson’s group , the situation is less clear, and our strategy does not work. So we leave it as an open question:
Is Thompson’s group hyperbolically elementary? Does it satisfy Property ?
The paper is organised as follows. First, Section 2 is dedicated to basic definitions and preliminary lemmas about hyperbolic spaces and CAT(0) cube complexes. In Section 3, we introduce and study a family of particular elements of , named reducible elements. Finally, in Sections 4 and 5 respectively, we prove our general criteria, namely Theorems 1.1 and 1.3, and we prove Theorem 1.2 and Corollary 1.4 by appling them to .
I am grateful to Yves Cornulier, for his comments on an earlier version of this paper, which lead to a great improvement of the presentation. I also would like to thank the university of Vienna for its hospitality during the elaboration of this work. I was supported by the Ernst Mach Grant ICM-2017-06478, under the supervision of Goulnara Arzhantseva.
2.1 Hyperbolic spaces
In this section, we recall some basic definitions about Gromov-hyperbolic spaces, we fix the notations which will be used in the paper, and we prove a few preliminary lemmas which will be useful later on. For more general information about hyperbolic spaces, we refer to [Gro87, GdlH90, CDP06, BH99].
Let be a metric space. For every , the Gromov product is defined as
Fixing some , the space is -hyperbolic if the inequality
is satisfied for every .
Usually, it is easier to work with geodesic metric spaces instead of general metric spaces. The following lemma explains a classical trick which allows us to restrict our study to hyperbolic graphs.
Let be metric space. If denote the graph whose vertices are the points of and whose edges link two points at distance at most one, then the inclusion is a -quasi-isometry such that any isometry of extends uniquely to an isometry of . As a consequence, if is hyperbolic, then so is .
From now on, all our (hyperbolic) metric spaces will be graphs.
Fixing a graph , three vertices and a geodesic triangle , there exists a unique tripod and a unique map such that:
are the endpoints of ;
restricts to an isometry on each , , .
The data is the comparison tripod of , and the three (not necessarily distinct) points of sending to the center of define the intriple of .
The following statement is an alternative definition of hyperbolic spaces (among geodesic metric spaces). We refer to the proof of [GdlH90, Proposition 2.21] for more information.
Let be a -hyperbolic graph. For every vertices and every geodesic triangle , if denotes the comparison tripod of then for every satisfying .
The next statement is a fundamental property satisfied by hyperbolic spaces, often referred to as Morse Property. See for instance [BH99, Theorem III.H.1.7].
Let be a -hyperbolic graph. For every and every , there exists some , called the Morse constant, such that: for every , any two -quasigeodesics between and stay at Hausdorff distance at most .
Now, let us prove two preliminary lemmas which will be useful in the next sections.
Let be a -hyperbolic space and two lines which are -quasiconvex for some . For every and , any geodesic between and intersects the -neighborhood of the nearest-point projection of onto .
Fix two points and , and a geodesic between them. Let be a nearest-point projection of onto . Fixing some geodesics and , we claim that is a -quasigeodesic.
The only point to verify is that, given two points and , the inequality
holds. Let us consider a geodesic triangle , and let denote its intriple where , and . Notice that, since is -quasiconvex, there exists some satisfying . One has
On the other hand,
hence . Our claim follows. We register our conclusion for future use.
Let be a -hyperbolic graph, a -quasiconvex line, and , two vertices. If denotes a nearest-point projection of onto , then any concatenation defines a -quasigeodesic.
Now, we conclude from the Morse property that the Hausdorff distance between and is at most . The desired conclusion follows. ∎
Let be a -hyperbolic space, two vertices and a -quasiconvex line. Fix two nearest-point projections respectively of onto , and suppose that . Then
The right-hand side of our inequality is a consequence of the triangle inequality, so we only have to prove its left-hand side.
Fix some geodesics , , , and . Let be the intriple of the geodesic triangle where , , ; and similarly let be the intriple of the geodesic triangle where , , . Notice that, since is -quasiconvex, there exists some satisfying . The configuration is summarised by Figure 1. Notice that
hence . Now, we distinguish two cases.
Case 1: Suppose that . Then there exists some satisfying , and because is -quasiconvex, there exists some satisfying . One has
hence . Next, notice that
We conclude that
Case 2: Suppose that . As a consequence, there exists some satisfying . Notice that
hence . Therefore,
ie., . This contradicts our assumptions, so our second case cannot happen. ∎
Let be a -hyperbolic space, two vertices, a geodesic between and , and a -quasiconvex line. Fix two nearest-point projections respectively of onto , and suppose that . Then , so that there exists a unique point satisfying , and moreover .
First of all, notice that, as a consequence of Lemma 2.7, one has
which proves the first assertion of our statement.
Let be a -hyperbolic space, a -quasiconvex line and two points. If are nearest-point projections onto of respectively, then
If there is nothing to prove, so suppose that . As a consequence of Lemma 2.7,
which concludes the proof of our corollary. ∎
Now, let be a -hyperbolic graph and an isometry. The translation length of is
and the minimal set of is
It is worth noticing that, because is a graph, the infinimum in the definition of turns out to be a minimum, so that is non-empty.
Let be a hyperbolic graph and a loxodromic isometry. An axis of is a concatenation for some .
Noticing that an axis of is a -local geodesic, the following lemma follows from [BH99, Theorem III..1.13].
Let be a -hyperbolic graph and a loxodromic isometry satisfying . Any axis of is -quasiconvex.
We conclude this section with a last preliminary lemma, which will be fundamental in the proof of the hyperbolic rigidity of Thompson’s group .
Let be a hyperbolic space and two isometries. Suppose that is loxodromic of translation length at least and that is elliptic. Fix an axis of . If is elliptic, then there exists a point such that
For convenience, fix a -equivariant map sending every point of to one of its nearest-point projections, and set . Because is elliptic, we know from [BH99, Lemma III..3.3] that there exists some such that has diameter at most .
First, suppose that there exists some such that the distances and are both greater than . Fix a geodesic . We know from Corollary 2.8 that there exists a point satisfying such that . Similarly, we know from Corollary 2.8 that there exists a point satisfying such that . Notice that
where the last inequality is justified by Corollary 2.9. Consequently,
We conclude that is a point satisfying the conclusion of our lemma, since
Next, suppose that for every satisfying one has . Consequently, since
it follows that . So
where the first inequality of the second line is justified by Corollary 2.9. Next, since
we deduce from Lemma 2.7 that
By noticing that
the previous inequality becomes
Since by assumption, it follows that
According to [GdlH90, Corollaire 8.22], this inequality implies that is loxodromic, contradiction our hypotheses. ∎
2.2 CAT(0) cube complexes
In this paper, we suppose that the reader is familiar with the basic definitions and properties of CAT(0) cube complexes. For details, we refer to [Sag14, Wis12]. Nevertheless, we recall the following fundamental property of cubical complexes, which will be used several times in Section 5 without mentioning it. We refer to [Rol98, Theorem 11.9] for a proof.
Let be a group acting on some CAT(0) cube complex . If has a bounded orbit, then stabilises a cube. As a consequence, the action has a global fixed point.
Let be a CAT(0) cube complex. An ultrafilter is a collection of halfspaces of such that
contains exactly one of the two halfspaces delimited by a given hyperplanes;
if and are two halfspaces satisfying , then implies .
For every vertex , the collection of all the halfspaces of containing is the principal ultrafilter defined by .
The Roller compactification of is the graph whose vertices are the ultrafilters of and whose edges link two ultrafilters whenever their symmetric difference has cardinality two. The Roller compactification is usually not connected, but each connected component turns out to be a median graph (which we identify canonically with a CAT(0) cube complex; see [Che00]). Moreover, the map defines an embedding whose image is a connected component of . We refer to the connected components of as its cubical components, and we identify with the cubical component of the principal ultrafilters. The Roller boundary of is .
Finally, we define a topology on , and a fortiori on , as follows. By labelling the two halfspaces delimited by a given hyperplane with and , we can naturally thought of as a subset of , where denotes the set of all the hyperplanes of . The topology of is the topology induced by the product topology on . Since is closed in , it follows that is compact. More details about Roller boundary can be found in [Sag14, Rol98].
The following statement provides a useful trick when arguing by induction on the dimension.
Let be a finite-dimensional CAT(0) cube complex. For every cubical component , the inequality holds.
A proof can be found for instance in [Fio17, Proposition 4.29], in the more general context of median spaces.
Hyperbolic model of cube complexes.
In [Gen17], we introduced a hyperbolic model (depending on a parameter) of CAT(0) cube complexes. Below, we recall the first definitions and properties, and we prove a proposition related to its Gromov-boundary.
Let be a CAT(0) cube complex and an integer. A facing triple is the data of three pairwise disjoint hyperplanes such that no one separates the other two. Two hyperplanes are -well-separated if they are not transverse and if every collection of hyperplanes transverse to both and which does not contain any facing triple has cardinality at most . An isometry is -contracting if it skewers a pair of -well-separated hyperplanes, ie., if there exist two -well-separated hyperplanes delimiting two halfspaces respectively such that .
Recall that an isometry of some metric space is contracting if the map , for some , defines a quasi-isometric embedding , and if the nearest-point projection of any ball disjoint from onto has diameter uniformly bounded. In [Gen16, Theorem 3.13], we proved that the two previous definitions of contraction coincide:
Let be a CAT(0) cube complex. An isometry is contracting if and only if there exists some such that is -contracting.
Given a CAT(0) cube complex and an integer , one next defines a new metric on (the set of vertices of) by:
We showed in [Gen17] that is indeed a metric, and we proved the following statement:
Let be a CAT(0) cube complex and some integer. The metric space is hyperbolic, and an isometry of defines a loxodromic isometry of if and only if it -contracting.
In the rest of the section, we would like link the Gromov-boundary of with the Roller boundary of . Notice that it is not clear whether is a geodesic metric space, so, given a basepoint , the boundary will be defined as the quotient of the collection of sequences satisfying modulo the equivalence relation: if . (Nevertheless, it follows from [Gen17, Lemma 6.55] that is a quasigeodesic metric space, so the boundary can also be defined as the asymptotic classes of quasigeodesic rays.) Our main statement is:
Let be a CAT(0) cube complex and an integer. There exists an -equivariant map sending a point of to a subset of diameter at most in a cubical component of .
First, we recall [Gen17, Lemma 6.55], which essentially states that the quasigeodesics in fellow-travel the geodesics in .
Let be a CAT(0) cube complex and three vertices such that belongs to a geodesic between and in . Then
As a consequence of the previous lemma, we are able to estimate the Gromov product in . (In the following, Gromov products will always refer to the distance .)
Let be a CAT(0) cube complex, an integer and three vertices. Then
where denotes the median point of .
For convenience, set . By applying Lemma 2.19, we get
which concludes the proof. ∎
Proof of Proposition 2.18..
For every , we denote by the set of all the accumulation points in of all the sequence of vertices representing . We want to prove that is the map we are looking for. First of all, notice that is non-empty for every , as a consequence of the compactness of , and that our map is clearly -equivariant.
Next, we claim that for every . Indeed, let be a sequence representing and one of its accumulation points. For convenience, suppose that converges to in . Because
where the last inequality is justified by Lemma 2.20, it is clear that cannot belong to , so it must belong to .
Finally, we need to verify that, given some , if and are two sequences representing and converging respectively to and in , then and belong to the same cubical component and there the distance between them is at most .
Let be hyperplanes such that, for every , the ultrafilters and does not contain the same halfspace delimited by . Set . By definition of the topology of , there exists some such that, for every and every halfspace delimited by one the ’s, belongs to the principal ultrafilter defined by if and only if and similarly belongs to the principal ultrafilter defined by if and only if . It follows that the ’s separate and for every . We also want to choose sufficiently large so that for every . Now, fix some . As a consequence of Lemma 2.20, we have
Consequently, there exist pairwise -well-separated hyperplanes separating and . Without loss of generality, suppose that separates and for every and that separates from . For every , notice that intersects the halfspace delimited by which contains and since it separates these two vertices; on the other hand, cannot be included into the halfspace delimited by which contains since the distance between and is at most , so we conclude that