Rigidity of graphs of germs and homomorphisms between full groups
We study topological full groups of étale groupoids and show that they satisfy new rigidity phenomena of topological dynamical nature. If is a minimal groupoid of germs on the Cantor set, actions of the (alternating) full group of on compact spaces satisfy the following dichotomy: either there is a point such that no element has trivial germ at that point, or the action is induced from an action of a (reduced) power of the groupoid . This dichotomy is a simultaneous generalisation of the fact that isomorphisms of full groups are implemented by isomorphisms of the underlying groupoids, and of the simplicity of the alternating full group. Using this result we obtain that, for a vast class of groupoids (defined in terms of the geometry of their Cayley graphs), not only isomorphisms but all embeddings between the full groups are induced from the groupoids in a suitable sense. We also show that various quantitative invariants of étale groupoids, such as the orbital growth and the complexity function, can be used to produce obstructions to the existence of embeddings. A key tool in the proofs is a characterisation of the subgroups of the full group whose conjugacy class does not accumulate on the trivial subgroup in the Chabauty topology. As another application, we provide the first examples of finitely generated groups that do not admit infinite Schreier graphs that grow uniformly subexponentially, but do admit co-amenable subgroups of infinite index.
- 1 Introduction
- 2.1 Notations on stabilisers and graphs of group actions
- 2.2 The Chabauty topology and confined subgroups
- 2.3 Étale groupoids
- 2.4 Actions of groupoids
- 2.5 Standing assumptions on unit spaces of groupoids
- 2.6 Compactly generated and expansive groupoids.
- 2.7 The topological full group
- 2.8 The alternating full group
- 2.9 Graphs and growth functions of étale groupoids
- 2.10 Complexity of expansive groupoids
- 3 Confined subgroups and proximal actions
- 4 Confined subgroups of topological full groups
- 5 Actions by bounded displacement and growth of Schreier graphs
- 6 Dichotomy for actions and for homomorphisms
- 7 Homomorphisms and Cayley graphs of groupoids
- 8 Obstructions to homomorphisms
Let be an étale groupoid whose unit space, denoted , is homeomorphic to the Cantor set. A fundamental example is given by the groupoid of germs associated to an action of a countable group. We are interested in the topological full group of , denoted or in the case of the groupoid of germs of a group action on a compact space. When is a groupoid of germs (i.e. when it is an effective étale groupoid), the group is defined as the group of all homeomorphisms of whose germs belong to . See Definition 2.11 for a more general and more detailed definition, and Subsection 2.3 for preliminaries on étale groupoids.
The notion of full group was introduced by Dye in the context of orbit equivalence of measure preserving actions [Dye59]. Topological full groups were introduced by Giordano Putnam and Skau [GPS99] for minimal -actions on the Cantor set, and by Matui [Mat12, Mat15] for a wider class of étale groupoids. Matui [Mat06, Mat15] has shown that for some classes of groupoids the derived subgroup of is simple, and in some cases finitely generated. For more general groupoids, Nekrashevych [Nek15a] defines a subgroup , that we will call the alternating full group, which will play an important role in this paper (see Definition 2.13). It is shown in [Nek15a] that whenever is a minimal groupoid of germs on the Cantor set (i.e. every orbit is dense), the group is simple; see also [MB16, Th. 1.2] for an equivalent result with a different definition of . Moreover the group is finitely generated whenever is expansive [Nek15a] (for instance, if the groupoid of germs an expansive action of a finitely generated group). Topological full groups have been considered in recent year to produce various new interesting examples of groups [Nek16a, JM13, JNdlS16, Mat15].
A fundamental result on topological full groups is Matui’s isomorphism theorem: if are minimal groupoids of germs with Cantor set unit space, and if there exists an isomorphism , then there exists an isomorphism of the étale groupoids and that induces . This result was proven in [GPS99], for -actions, and in [Mat15] in general. The same result holds for isomorphisms between alternating full groups [Nek15a]. This result, analogous to the work of Dye on measurable full groups [Dye59], is an example of a reconstruction result. It can also be deduced from a more general reconstruction theory developed by Rubin (see [Rub89]), which provides widely applicable tools to study isomorphisms between “large” groups of homeomorphisms.
The main goal of this paper is to show that actions of topological full groups on compact spaces enjoy an interesting rigidity property that encompasses the isomorphism theorem, and to discuss various consequences of it. Our approach is different from the above-mentioned reconstruction methods, and involves the study the dynamics of the conjugation action on the space of subgroups endowed with the Chabauty topology, combined with the language provided by the setting of étale groupoids.
1.1. A dichotomy for actions on compact spaces
In order to state our results we need to fix some terminology regarding induction of actions of groupoids to actions of their topological full groups.
In analogy with the case of groups, an étale groupoid can act. The natural objects on which it can act are pairs consisting of a space together with a surjective continuous map ; such a pair will be called a fibre space over (see Subsection 2.4). Every continuous action on a compact fibre space over can be induced to an action of the full group on (as we shall explain in Section 6.1). The action obtained in this way factors onto the natural action via the map .
We need to consider a slightly more general way to induce actions of , as follows. Let be the compact space consisting of non-empty finite subsets of with cardinality at most . The group acts on and the germs of this action belong to a natural groupoid , that we call the -th reduced power of , that can be thought of as a relative of the usual cartesian power (we refer to Definition 6.5 for the detailed definition of ). Hence, the group is naturally a subgroup of . It follows that every continuous action of the reduced power also induces a continuous action of the group .
We will say for short that a continuous action is induced from if there exists a continuous surjective map and a continuous groupoid action which induces .
Our first result provides a necessary and sufficient condition ensuring that an action of the alternating full group is induced from for some . If is a group action on a compact space, for every point we denote by the germ-stabiliser, i.e. the subgroup consisting of elements that fix point-wise a neighbourhood of . The following is a consequence of Corollary 6.13
Theorem 1.1 (Germ-stabiliser rigidity for actions on compact spaces).
Let be a minimal groupoid of germs with Cantor set unit space . Let be a non-trivial action by homeomorphisms on a compact space. Then:
either there exists a point such that ; or
up to removing a clopen subset of consisting of global fixed points (possibly empty), the action is induced from for some ; in particular, it factors onto the natural action .
For minimal actions, this statement simplifies as follows:
Corollary 1.2 (Case of minimal actions).
A non-trivial minimal action of the group on a compact space is either topologically free or it is induced from .
Recall that an action of a countable group on a compact space is said to be topologically free if a dense set of points have trivial stabiliser. The first special case of Corollary 1.2 was established in [LBMB16, Th. 1.8] for the Thompson’s group (and for the Thompson’s group ).
An action on a compact space can be equivalently thought of as a homomorphism , where is a groupoid of germs (namely the groupoid of germs of the action). Taking this point of view on Theorem 1.1, it turns out that the assumption that is a groupoid of germs is irrelevant: we shall in fact prove a more general version of Theorem 1.1 where we allow to be an arbitrary étale groupoid, see Theorem 6.9.
Formally, Theorem 1.1 is a generalisation of the simplicity of the group . To see the connection, consider a proper normal subgroup , and apply Theorem 1.1 to the action on the one-point compactification of the quotient. It is easy to see that for every . Since is countable, no subset of it can admit a surjective map to , hence we are in case a, and . In fact, the proof of Theorem 1.1 is based on a generalisation of the simplicity of the group . See also Example 6.12.
In addition, Theorem 1.1 recovers the isomorphism theorem. To explain the connection, let be a minimal groupoid of germs on unit spaces homeomorphic to the Cantor set. If there is an isomorphism , we can apply Corollary 1.2 to the induced actions and . These actions are not topologically free and the isomorphism theorem readily follows (see Examples 6.15 for details and for the case of isomorphisms between groups ).
Therefore Theorem 1.1 unifies to a single more general phenomenon these two facts. In addition, it has various new consequences that we outline in the rest of the introduction.
1.2. Embeddings between full groups
It is natural to ask whether some sort of generalisation of Matui’s isomorphism theorem holds for homomorphisms between topological full groups. In other words, is it true that all homomorphisms have to come from the underlying dynamics, in some sense to be made precise?
A moment of reflection shows that one cannot hope for a result of this type to hold in full generality: every action of the group on the Cantor set gives rise to a homomorphisms where is any étale groupoid that contains the groupoid of germs of the action. As every countable group the group admits a wild multitude of actions on the Cantor set, not all of which are related to the groupoid in a clear way. However the question becomes interesting if we require some restriction on the étale groupoids we are interested into. For example, when are both groupoids of germs of minimal -actions on the Cantor set, this problem is raised by Cornulier in [Cor14, Question (2f)].
Theorem 1.1 (along with the more general Theorem 6.9) provides a convenient tool to study homomorphisms. Roughly speaking, the reason is that if we are given a homomorphism , case a in Theorem 1.1 can be often be ruled out using geometric information on the groupoid . The possibility of studying homomorphisms beyond the case of isomorphisms is an interesting difference with the respect to the reconstruction methods previously used in the context of full groups or in similar contexts.
We need again to fix some terminology in order to state this result. Let be an étale groupoid with compact unit space , that we assume to be compactly generated in the sense of Haefliger [Hae02] and Nekrashevych [Nek15b] (see Definition 2.8). This assumption allows to endow every fibre of the source map of with the structure of a graph , called the Cayley graph of based at . This graph depends on the choice of a generating set of , but its bi-Lipschitz equivalence type is independent of it (but may depend on the base-point ). A tightly related graph is the orbital graph , whose vertex set is the -orbit of . We refer to Subsection 2.9 for more details, but for now the reader can have in mind the following example: if is the groupoid of germs of an action of a finitely generated group, then it is compactly generated, and for every the Cayley graph is bi-Lipschitz equivalent to the Schreier graph of with respect to the subgroup (also called the graph of germs, while the orbital graph is bi-Lipschitz equivalent to the Schreier graph with respect to the usual stabiliser . In particular, if the action is free, both graphs are quasi-isometric the Cayley graph of of the group .
Definition 1.4 (The class ).
Let be a compactly generated étale groupoid with compact unit space . We say that belongs to the class if at least one of the following conditions is satisfied for some finite generating set of bisections of .
For every the graph has finite asymptotic dimension.
For every the graph has polynomially bounded growth.
The following classes of groupoids belong to the class .
The groupoid of germs of actions of and of , and more generally of any topologically free action of a finitely generated group with finite asymptotic dimension. A wide class of groups are known to have finite asymptotic dimension. See [BD08] for a survey.
The groupoid of germs of associated to the fragmentations of dihedral group actions, introduced by Nekrashevych, whose topological full group are torsion and have intermediate growth [Nek16a].
To state the next result, recall that every continuous action of an étale groupoid on a compact space gives rise to another étale groupoid, called the semi-direct product groupoid , whose unit space is (its definition is recalled in Subsection 2.4). We say that a cocycle (i.e. a groupoid homomorphism) between étale groupoids is a spatial inclusion if it is injective in restriction to the unit space of .
The following theorem is a consequence of Theorem 7.5.
Theorem 1.6 (Rigidity of embeddings).
Retain the same assumptions on as above and further assume that it is compactly generated. Let be an étale groupoid in the class . The following are equivalent.
There exists a non-trivial homomorphism .
There exists an embedding .
There exists an integer , an action on a compact fibre space over , and a continuous open spatial inclusion
Moreover, for every non-trivial homomorphism there exists a spatial inclusion as in c which naturally induces .
For a description of the homomorphisms “naturally induced by” a spatial inclusion , see Subsection 6.1.
See Corollary 7.8 for a more precise statement in the special case of topological full groups of minimal -actions on the Cantor set, which gives an answer to [Cor14, Question (2f)]. See also Corollary 7.12 for a statement in the special case of groupoids associated to one-sided SFT’s, which applies in particular to homomorphisms between groups in the family of Higman–Thompson groups .
1.3. Quantitative invariants of groupoids
Étale groupoids bear rich structure of geometric and dynamical nature, and various quantitive invariants can be associated to them. It is natural to study how these invariants are related to the behaviour of the topological full group.
The isomorphism theorem implies that every groupoid invariant can be used to distinguish topological full groups up to isomorphism. As a motivating application of Theorems 1.1 and 1.6 we strengthen this by showing that various quantitative invariants associated to étale groupoids produce obstructions to embeddings between the corresponding topological full groups. This confirms the natural intuition that a “more complicated” dynamical system produces a “larger” topological full group.
The first invariant that we consider is the orbital growth function, which measures the maximal growth of the orbital graphs . Given a graph and a vertex , we denote the ball of radius around , and . Let be a compactly generated étale groupoid over with finite generating set of bisections . The orbital growth of is defined as the function . Given two functions we write if for some , and if . The growth rate of does not depend on the choice of up to the equivalence , and is denoted .
Another invariant that we consider is the complexity function. Let be a finitely generated group and let be a subshift over a finite alphabet, i.e. the translation action on a closed invariant subset of the shift. Let be a finite generating set of , and let be the ball of radius in the Cayley graph of . The complexity function is defined as the number of configurations that appear as restrictions of elements . The complexity function of subshifts over the group is a well-studied invariant in symbolic dynamics, see [CN10]. In this case, it is related to the topological entropy of the subshift via the formula .
The complexity function of subshift admits a very natural generalisation that can be defined for every expansive étale groupoid on the Cantor set endowed with an expansive generating set . A definition was given by Nekrashevych in [Nek16b]; we propose a slightly different one given in Section 2.10. Its growth type doesn’t depend on (up to ), and it agrees with the previous definition if is the groupoid of germs of a subshift over a finitely generated groups (these facts are checked in Section 2.10).
A result relating the complexity to the topological full group was proven in [MB14]: if the complexity function of a subshift grows slower than for some , then the simple random walk on every finitely generated subgroups of the topological full group has trivial Poisson boundary. In [Nek15a], dynamical systems with linear growth of the repetitivity (which implies linear growth of the complexity) are used to construct étale groupoids whose topological full group has intermediate growth. These results motivate the question whether for more general dynamical systems, the growth and the complexity (or the positivity/ vanishing of entropy) constraints the behaviour of the full group, especially in relation to its subgroup structure, see [Nek16b].
The following theorem is proven in Section 8.
Theorem 1.7 (Obstructions to embeddings).
Retain the same assumptions on and further assume that it is compactly generated. Let be a compactly generated étale groupoid. Then the group cannot embed into the group provided any of the following holds.
The groupoid is a groupoid of germs, and the orbital growth functions of and satisfy .
The groupoid belongs to the class , both are expansive with Cantor set unit space, and their complexity functions satisfy .
The asymptotic dimension of every Cayley graph of is strictly larger than the asymptotic dimension of every Cayley graph of .
It is worth pointing out the following special case of the non-embedding criterion b.
Let and be minimal subshifts whose topological entropies satisfy and . Then every homomorphism has abelian image.
1.4. Actions by bounded displacement
Let us now describe a more combinatorial rigidity property of the group . Let be a graph of bounded degree. Its group of permutations of bounded displacement, or wobbling group is the group of permutations of the set of vertices of , having the property that
For example, let be a finitely generated group, and be a subgroup, and let be its Schreier graph with respect to a finite symmetric generating set of . Then the action defines a homomorphism (studying actions by bounded displacement of finitely generated groups is essentially equivalent to the study of their Schreier graphs, but the wobbling group point of view has the advantage to make sense also for non-finitely generated groups).
It is easy to see that the natural action of the group on every -orbit defines a homomorphism , which is injective if is minimal. In many interesting cases the graphs are very explicit and this allows to take a rather concrete point of view on the group (this point of view is used e.g. in [JM13, Nek16a]).
The following theorem says that the group cannot act by permutation of bounded displacements on any graph for which the growth function grows slower than . Recall that the group is finitely generated if and only if is expansive [Nek15a].
Let be a compactly generated minimal groupoid of germs with Cantor set unit space. Let be a bounded degree graph such that . Then there is no non-trivial homomorphism .
When is expansive, (so that is finitely generated), it follows that every non-trivial Schreier graph of (with respect to any finite generating set of ) satisfies .
Similar properties for a given group were previously known only as a consequence of stronger properties of analytic nature. It is a well-known observation that a group with property does not admit any Schreier graph of subexponential growth [Gro93, Rem. 0.5.F]; see also [JdlS15] for a similar statement in terms of wobbling groups. More generally, the same conclusion holds as soon as does not admit co-amenable subgroups of infinite index, a property which is in some sense a discrete weakening of property (T) (see [Cor15, GM07] for an account of the examples of groups that are known to have this property but do not have (T), and [Cor15] for a similar statement in terms of wobbling groups). Note also that many groups admit Schreier graphs whose growth functions is a slow as possible, i.e. linear in (for instance any group that virtually has infinite abelianization; another famous example of group with this property is given by the Grigorchuk group [BG00]). Theorem 1.9 has the following corollaries; the first can be compared with a question asked by Cornulier in [Cor15, Question 1.19 (3)].
There exists a finitely generated group with the following properties.
the group has a co-amenable subgroup of infinite index;
the group does not admit any infinite Schreier graph which has uniformly subexponential growth, i.e. .
For every function which is the orbital growth function of a minimal expansive groupoid of germs, there exists a finitely group with the following properties.
For every non-trivial Schreier graph of we have .
There exists a Schreier graph of such that .
Note that the function in Corollary 1.11 can take a wide range of behaviours, and can be arbitrarily close to exponential but still be subexponential. It is a natural question to characterise exactly the growth types of functions that are realisable as growth functions of expansive groupoids. It seems plausible that rather arbitrary subexponential functions are realisable.
1.5. Chabauty space and confined subgroups
Given a countable group , its space of subgroup is naturally a compact space, endowed with the Chabauty topology. We say that a subgroup is confined if the closure of its conjugacy class in the space does not contain the trivial subgroup . A homonymous notion of confined subgroups was introduced by Hartley and Zalesskii [HZ97] in the special case of simple locally finite groups, and has been further studied in [LP03, LP02]. The original definition is different, but it is equivalent for locally finite groups (as recently pointed out by Thomas [Tho17]). Therefore we chose to keep the same terminology.
The following result provides a characterisation of all confined subgroups of the group and, is a key tool used throughout the paper.
Theorem 1.12 (Characterisation of the confined subgroups).
Let be a minimal groupoid of germs with unit space homeomorphic to the Cantor set. Let be any subgroup of such that . A subgroup is confined if and only if there exists a unique finite subset (possibly empty) such that .
Here denotes the subgroup of consisting of elements that fix point-wise a neighbourhood of , and denotes the set-wise stabiliser of in .
When is an AF-groupoid (see Subsection 4.1), Theorem 1.12 provides a classification of the confined subgroup of a class of simple locally finite groups, namely block-diagonal limits of products of finite alternating groups (or LDA-groups), completing previous work of Leinen-Puglisi [LP03] in the case of diagonal limits of product of alternating groups.
A classification of the confined subgroups of a group provides a more precise information than a classification of the uniformly recurrent subgroups (URS’s) in the sense of Glasner and Weiss [GW15]. Recall that a URS of a countable group is a closed minimal conjugation-invariant subset. A URS can be associated to every minimal action of on a compact space, and is called the stabiliser URS of the action, see [GW15, Proposition 1.2] (conversely, every URS arises in this way [MBT17, Ele17]). In particular, Theorem 1.12 implies that the group admits a unique non-trivial URS, namely the stabiliser URS of its action (see Corollary 4.6). In the case of the LDA groups, the classification of URS’s was recently obtained by Thomas [Tho17].
The first result of uniqueness of the URS’s of a group given by a minimal action on a compact space was provided in [LBMB16], and applies to the Thompson’s groups and and to some related groups. The groups and are examples of topological full groups of minimal groupoids of germs on the circle and the Cantor set, respectively. Two main novelties arise in Theorem 1.12 with respect to this result. First, it is valid for a much wider class of topological full groups (with Cantor set unit space). In particular, its proof no longer relies on the extreme proximality condition of the natural actions of and used in [LBMB16]. As a consequence, it can be applied to groupoids whose Cayley graphs are amenable or have subexponential growth, which is important for Theorem 1.9 and its corollaries. Second, we no longer classify only URS’s but rather all the confined subgroups. This is important to study non-minimal actions, and opens the way to the applications to homomorphisms between topological full groups (given a homomorphism , we are certainly not allowed to assume that the image acts minimally on the unit space of ).
One of the purposes of this paper is to illustrate that Chabauty methods and confined subgroups can be used as tools to study embeddings between groups. In a different direction, A. Le Boudec [LB18] shows that, for a class of discrete groups, URS’s can be used as a tool to study lattice embeddings into locally compact groups.
The reader may have noticed a similarity between Theorem 1.1 and a theorem of Nekrashevych [Nek10], stating that if the free group acts faithfully on a rooted tree , then there is a non-cyclic subgroup and a point such that . Although Theorem 1.1 is not directly related to this result, its formulation draws inspiration from it.
Structure of the paper
Section 2 fixes the notations used throughout the paper and contains elementary preliminaries. In Section 3 we prove a refinement of a result from [LBMB16], which provides a tool to study confined subgroups of groups given by a proximal minimal action on a Hausdorff space, and is perhaps of independent interest. In Section 4 we characterise the confined subgroups and the uniformly recurrent subgroups of the groups and . In Section 5 we use this characterisation to establish Theorem 1.9 and its application to growth and amenability of Schreier graphs. In Section 6 we state and prove the main dichotomy theorem in a more general form (Theorem 6.9) and discuss its first consequences. In Section 7 we show Theorem 1.6. We discuss in more detail how it specialises to the case of topological full groups of Cantor minimal systems and of one-sided shifts of finite type. Section 8 is devoted to obstruction to embeddings; we prove Theorem 1.7 and discuss applications of it in some special cases.
The derivation of Theorem 1.6 from Theorem 6.9 uses a property of asymptotic dimension whose proof was explained to me by Alessandro Sisto (Proposition 7.3); I thank him for this suggestion that allowed to improve the statement of Theorem 1.6. I am grateful to Adrien Le Boudec and to Volodia Nekrashevych for many conversations related to the topics of this paper. I also thank Laurent Bartholdi, Yves Cornulier and Adrien Le Boudec for useful remarks on a first draft.
2.1. Notations on stabilisers and graphs of group actions
Throughout the paper, we use the following notations. If is a set and is a subset, we denote its complement. Assume that is a countable group acting on .
the stabiliser of a point will be denoted , and the set-wise stabiliser of will be denoted ,
the point-wise stabiliser of will be denoted ,
we call the subgroup the rigid stabiliser of and denote it .
If, moreover, has a topology we denote and the subgroups consisting of elements that fix point-wise a neighbourhood of (respectively, ), and call it the germ-stabiliser of (respectively, ). Thus .
Note that . Assume that is a compact space. Then a simple Baire argument shows that the set of points such that is a dense -subset of . A point satisfying this condition will be said to be a regular point.
Let be a finitely generated group with finite symmetric generating set . Assume that is an action of on a set. For every point , the orbital graph is the graph whose vertex set is the orbit of and it has as a distinguished vertex, where are connected by an oriented edge labelled by if . It coincides with the Schreier graph of the stabiliser , where the Schreier graph of a subgroup is the orbital graph for the action of on the coset space . If has a topology, the Schreier graph of the germ stabiliser is called the graph of germs at and will be denoted . We omit when they are clear from the context.
2.2. The Chabauty topology and confined subgroups
Let be a countable discrete group. The set of subgroups is a compact space endowed with the topology induced by the product topology on the set of all subsets of , which is called the Chabauty topology. The conjugation action of on is by homeomorphisms.
Given a finite subset , we denote
These sets are open in , and form a pre-basis of the topology as varies over finite subsets of . In particular, the sets of the form form a fundamental system of neighbourhoods of the trivial subgroup .
If is finitely generated, an equivalent description of the Chabauty topology can be given in terms of the space of marked graphs. Recall that given an integer and a finite set , the space of oriented graphs with a distinguished base-point , degree bounded by and edges labelled by is naturally a compact metrisable space, where a sequence converges to if for every the ball is eventually isomorphic to as a rooted labelled graph. The correspondence embeds homeomorphically into the space of marked graphs with degree bounded by and edges labelled by .
A subgroup is said to be confined if the Chabauty-closure of does not contain . If is another subgroup, the subgroup is said to be confined by A if the closure of does not contain .
This property can be thought of as a weak notion of normality. Note that being confined is equivalent to the existence of a finite set such that the conjugacy class of avoids , i.e. .
The terminology is due to Hartley and Zalesskii [HZ97], who introduced an equivalent property for simple locally finite groups
Another concept that will play an important role in this paper is the lower and upper semicontinuity of various maps taking values in . Given a a (Hausdorff) space , a map is said to be upper (respectively lower) semicontinuous if for every net in converging to , every cluster point of in verifies (respectively ). A characterisation of these properties is given by the following lemma, which readily follows from the definition of the Chabauty topology.
Let be a map from a Hausdorff space to .
The map is upper semicontinuous if and only if for every , the set is closed.
The map is lower semicontinuous if and only if for every , the set is open.
In particular, is continuous if and only if both conditions hold.
Two basic example of upper and lower semicontinuous maps are the following.
Let act by homeomorphisms on a compact space . Then it follows from the lemma above that:
the map is upper semicontinuous;
the map is lower semicontinuous.
We will encounter other examples of semicontinuous maps later.
We record for later use the following fact.
Assume that acts continuously on a compact space . Let be a lower semicontinuous map such that for all . Then the closure of the image of does not contain . In particular is confined for every .
Let be in the closure of the image of , and let be a net such that tends to . Upon extracting, tends to a limit and by lower semicontinuity we have . It follows that . ∎
Recall also that a uniformly recurrent subgroup, or URS, is a closed minimal invariant subset [GW15]. Whenever is a minimal action on a compact space, the closure of contains a unique URS, called the stabiliser URS of the action, see [GW15] (this notion will not play an essential role in this paper).
2.3. Étale groupoids
We briefly recall the basic definitions regarding groupoids and étale groupoids. A groupoid is a small category in which every morphism is an isomorphism. In other words it consists of a set of objects (called the unit space), a set of morphisms , two maps called the source and the range that indicate the initial and terminal object of a morphism. The product of two elements is defined if and only if and in this case . For every there is a unique element (called the unit at ) which satisfies and for every . There is an inversion so that and .
The map allows to identify the unit space with a subset of . We will systematically use this identification, and write instead of .
It is a common use to denote the unit space of a groupoid by ; we will deviate from this convention to avoid conflicts with other notations used in this paper (, ..).
Given a subset of the unit space, the restriction of to is the subgroupoid .
A cocycle is a groupoid homomorphism, i.e. a map between groupoids such that and for ever such that , were denote the source and range maps of .
A topological groupoid is a groupoid endowed with a topology such that the source and range maps, the composition and the inversion are continuous. Here is the set of composable pairs, endowed with the topology induced from the product topology on . The unit space is endowed with the topology induced by the inclusion .
An étale groupoid is a locally compact topological groupoid such that the source and range maps are open and are local homeomorphisms. We do not require the topology on to be Hausdorff (as many interesting examples are not). However, we do require the unit space to be Hausdorff.
A bisection of an étale groupoid is an open subset such that and are homeomorphisms onto their image. By definition of an étale groupoid, bisections form a basis of the topology. Bisections are multiplied and inverted by the rules
Every open bisection defines a homeomorphism between and , given by . Note that is the unique element such that . An étale groupoid is said to be effective, or a groupoid of germs, if every bisection is uniquely determined by the associated homeomorphism, or equivalently if the associated homeomorphism is non-trivial unless .
Given , the orbit of is the set of such that there exists with and . The groupoid is said to be minimal if every orbit is dense in .
For every the set such that forms a group, called the isotropy group at . The groupoid is said to be principal of is trivial for every . It is said to be essentially principal if the set of points with trivial isotropy group is dense in .
Every countable group can be seen as an étale groupoid with one-point unit space and with the discrete topology. A groupoid of this form is never effective (unless is trivial).
Let be a countable group acting on a compact space. The associated action groupoid is . Its unit space is (with the obvious identification), and source and range map are given by and . The product and inversion are defined by the rules and . It is an étale groupoid if is endowed with the product topology, where has the discrete topology. A groupoid of this form is always Hausdorff. It is effective if and only if the action is topologically free, i.e. the set of points with trivial stabiliser is dense.
Let again be a countable group action on a compact space. For every denote by the germ of at , i.e. the equivalence class of the pair under the equivalence relation that identifies with if and coincide in restriction to a neighbourhood of Then the set of germs is naturally a groupoid , called the groupoid of germs of the action, with unit space identified with the set of germs . Source, range, composition and inversion are given exactly as in the case of the action groupoid by replacing with . The groupoid has a natural topology for which a basis of open sets is given by sets of the form where and is open, and is étale with this topology. The groupoid of germs of a group action is always effective. It is not necessarily Hausdorff.
Let be a countable group and let be a closed -invariant subset in the Chabauty space. The associated coset étale groupoid is the disjoint union of all coset spaces of subgroups in . Its unit space is , and source and range maps are given by . Composition and inversion are given by . A basis of open sets of its topology is given by sets of the form , where and is open. This topology makes it an étale groupoid. It is effective if and only if there is a dense subset of which has trivial normaliser in .
2.4. Actions of groupoids
Let be a set together with a surjective map . We will call for short the pair a fibre space over . It is said to be continuous if is a topological space and the map is continuous, and compact if is further compact. A continuous action is a continuous map
where denotes the fibre product of and over , with the properties that and for every such that and . Note that it follows that every defines a homeomorphism between and .
Let be a continuous action on a compact space and let be its action groupoid. Then it is not difficult to see that actions on compact fibre spaces are in one-to-one correspondence with extensions of the action , namely actions on compact spaces such that there exists a continuous, surjective, -equivariant map . If the action is topologically free, the same holds for the groupoid of germs of . (However, this is not true if the action is not topologically free.)
If is a continuous -action, the set is naturally an étale groupoid, called the action groupoid and denoted . Its unit space is , identified with the subset . Range and source map are given by and . The product is defined by .
2.5. Standing assumptions on unit spaces of groupoids
We will be mostly interested in studying topological full groups of étale groupoids whose unit space is homeomorphic to the Cantor set. However unit spaces that are locally homeomorphic to the Cantor set, but non-compact appear naturally in the proofs, for example when considering the restriction of to an open set . Moreover we will also consider groupoids with a more general unit space, that will play the role of the “target” groupoid in Theorem 1.6. Therefore, we fix the following convention.
For the rest of the paper, unless differently specified, will denote a second countable étale groupoid whose unit space is isomorphic either to the Cantor set, or to the locally compact non-compact Cantor set.
We denote an étale groupoid whose unit space is arbitrary.
Recall that the locally compact non-compact Cantor set is described as the unique space up to homeomorphism which is locally compact, metrisable, second countable, totally disconnected and without isolated points. Any open, non-closed subset of the Cantor set is homeomorphic to it.
2.6. Compactly generated and expansive groupoids.
We recall the definition of a compactly generated étale groupoid, due to Haefliger [Hae02] (in the language of pseudogroups), see also [Nek15b, Sec. 2.3]. We only recall it in the special case of groupoids with a compact unit space. Note that the definition for groupoids with non-compact unit space is different, see [Hae02].
Let be an étale groupoid with compact unit space . It is said to be compactly generated if there exists a finite set of open bisections such that every element of is a product of elements of and their inverses, and there exists open bisections such that and the closure of in is contained in . The set is called a finite generating set (of bisections) of .
If is a finitely generated group action on a compact space, its groupoid of germs is compactly generated, a finite generating set being given by (the sets of germs of) a finite generating set of .
For groupoids with Cantor set unit space the existence of the bisections in the definition can be replaced by the requirement that are compact open bisections (the resulting definition is equivalent). In this case, we will always consider generating sets consisting of compact open bisections.
Recall that a group action on the Cantor set is said to be expansive if there exists a finite partition of into clopen sets whose -translates separate points, equivalently if it is conjugate to a subshift over a finite alphabet. The following definition is due to Nekrashevych [Nek15a]
Let be an étale groupoid with Cantor set unit space. It is said to be expansive if it is compactly generated and there exists a generating set of compact open bisections such that is a basis of the topology of . Such a set is called an expansive generating set.
Expansivity of an action of a finitely generated group on the Cantor set is equivalent to the expansivity of its groupoid of germs and to the expansivity of its action groupoid, see [Nek15a, Prop. 5.5].
2.7. The topological full group
Let be an étale groupoid with unit space . Its topological full group is the group of all continuous maps , with the following properties:
for every we have ;
the set is a bisection of such that ;.
the set is compact, or equivalently for outside a compact set.
It is a group with the composition defined by .
There is natural action via the homeomorphism associated to bisections , or equivalently given by for every and . This action is faithful if is a groupoid of germs, but not necessarily otherwise. The map
is a continuous open cocycle from the action groupoid to .
The correspondence defines a bijection between and the set of bisections of verifying b and c, and we have where bisections are are multiplied according to (1). Therefore, one can simply define the group as the group of all bisections with these properties, as in [Nek15a]. However, we will keep the distinction between and the associated bisection , to avoid conflict between well-established notations regarding group actions and groupoids. In particular, if , the notation will refer to the image of under the natural action , while denotes the set-wise product of and in , i.e. .
2.8. The alternating full group
In this subsection, let be as in Convention 2.7. We recall the definition of multisections and of the group , following the point of view of Nekrashevych [Nek15a] (with some minor differences in the notations and the terminology).
A multisection of degree of is a map from to the set of compact of compact open bisections of , such that the following conditions are satisfied:
we have for every , and if ;
for every we have .
The set is called the domain of the multisection and the subsets are called the components of the domain.
There is a natural action of the group on the set of multisections given by .
A multisection of degree induces an embedding from the symmetric group into , still denoted . The bisection associated to is given by
Informally, the associated homeomorphism permutes the components in a way prescribed by , and acts within each component according to the bisection . We denote the image of the alternating group under .
The alternating full group of is the subgroup of generated by , where varies over multisections of degree .
It follows from the definitions that when the unit space is the locally compact non compact Cantor set, the group is the direct limit of the groups , were varies among compact open subsets of (and hence is isomorphic to the Cantor set).
Theorem 2.15 ([Nek15a]).
Assume that is a minimal groupoid of germs. Then every non-trivial subgroup of normalised by contains . In particular, is simple and contained in every non-trivial normal subgroup of .
Theorem 2.16 ([Nek15a]).
Assume that has Cantor set unit space and is expansive, and that every -orbit contains at least 5 points. Then the alternating full group is finitely generated.
We now list some useful general facts about multisections and the alternating group, that will be used in the sequel. It is convenient to introduce the following terminology.
A multielement of degree of is an embedding of the complete equivalence relation on into . Explicitly, an injective map such that for every and for every we have .
Note that it follows that and , and that all lie in the same orbit. The group acts on the set of multielements by .
If is a multielement and a multisection of the same degree, we will write to mean that for every .
Let be a degree multielement.
Assume for are units in the same orbit of such that the points are pairwise distinct. Then there exists a degree multielement such that and for .
Assume that is a neighbourhood of for every . Then there exists a degree multisection such that and for every .
Part b allows as to talk about “sufficiently small” multisections containing , and we will sometimes do so without mention.
To prove (i), set set for . Choose arbitrarily for every an element such that and , and set . In the remaining cases, let .
Let us prove (ii). First, for every we can assume that , upon replacing all the by by . Choose a decreasing basis of compact open neighbourhoods of in . If is large enough, we can assume , that for every , and that are pairwise disjoint. Define a sequence of multisections by setting and for and in the remaining cases. Then for every the sequence of bisections