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An easy consequence of 5.1 above is that the type of the co-induced action only depends on the type of the action. Corollary 5.3 |
Let [MATH] and [MATH] . If [MATH] then [MATH] Remark 5.4 If [MATH] and [MATH] , we may also view [MATH] as an element of [MATH] . In BGK17 the notion of a characteristic random subgroup is defined to be an IRS which moreover is invariant under the action of the full automorphism group. So, as any group [MATH] is contai... |
Continuity of co-induction We will consider here continuity properties of the co-induction operation. First, on the level of invariant random subgroups we have the following result. |
Proposition 6.1 Let [MATH] . The map [MATH] is continuous if and only if either [MATH] or [MATH] Proof. It is easily seen that if [MATH] , then [MATH] for any [MATH] . Thus in this case the co-induction operation is constant. If [MATH] , then the operation is continuous because the product in the definition is finite. |
Conversely, assume [MATH] and [MATH] For each [MATH] let [MATH] Then [MATH] as [MATH] in [MATH] . But we have [EQUATION] for any [MATH] finite, while |
[EQUATION] for all [MATH] finite. Since [MATH] is a homeomorphism between the space of weak equivalence classes of an amenable group and the space of invariant random subgroups on the group, we now have the following corollary. |
Corollary 6.2 Let [MATH] and assume [MATH] amenable. Then we have that the map [MATH] is continuous if and only if either [MATH] or [MATH] |
One implication holds in general. To prove this we will use a sequence of weak equivalence classes, which converges to the weak equivalence class of the trivial action in [MATH] . We denote the trivial action by [MATH] |
Proposition 6.3 Let [MATH] and assume [MATH] and [MATH] . Then [MATH] is not continuous. Proof. Consider a sequence of actions [MATH] for which there exists [MATH] Borel satisfying that [MATH] [MATH] is free and [MATH] is trivial for all [MATH] . Then, since |
[EQUATION] for all [MATH] and all Borel [MATH] , we have [MATH] as [MATH] in [MATH] . However, since [MATH] for all [MATH] , it follows as in the proof of 6.1 , that [MATH] , when [MATH] , in [MATH] . Thus [MATH] cannot be continuous. |
By similar arguments as those above, we can also say something about the continuity of countable powers of an action. In general for an action [MATH] and [MATH] the action [MATH] is defined by |
[EQUATION] for all [MATH] and [MATH] . We then have the following result. Proposition 6.4 The map [MATH] from [MATH] to [MATH] is not continuous if [MATH] |
Proof. Let [MATH] and [MATH] be as in the proof of 6.3 and assume towards a contradiction that the map is continuous. Then we would have [MATH] as [MATH] in [MATH] . But, |
[EQUATION] for all [MATH] , while [MATH] Remark 6.5 In B18 , Theorem 1.2] it is shown that for a class of groups, containing the non-abelian free groups, the operation [MATH] is not continuous, not even when restricted to the space of free weak equivalence classes. As a corollary, for any group [MATH] in this class, th... |
[EQUATION] is not continuous, again, not even when restricted to the space of free weak equivalence classes. So, while co-induction on weak equivalence classes is continuous in the finite index case, when the big group is amenable, this is not the case in general. |
Properties of the co-induced invariant random subgroups We will study here different properties of the co-induced invariant random subgroups such as mixing properties and non-atomicity. We also obtain a characterization of when the co-induced action is free. |
First note that it is clear that if [MATH] is ergodic (resp., weakly mixing), then [MATH] is ergodic (resp., weakly mixing). The converse does not hold in general. For example, [MATH] can be a free non-ergodic action, but [MATH] is ergodic. However, for each [MATH] , if [MATH] is ergodic (resp., weakly mixing), the act... |
By use of the analogous result for actions, we obtain the following result. Proposition 7.1 Let [MATH] with [MATH] . Then [MATH] is weakly mixing for any [MATH] |
Proof. Let [MATH] and let [MATH] satisfy [MATH] . Then by 3.3 we have [MATH] is weakly mixing and hence, by 5.1 , so is [MATH] In general, weakly mixing is the strongest mixing property one can hope for a non-atomic IRS. Indeed, by the result of T-D15a , if an IRS [MATH] is totally ergodic (i.e., the restriction of the... |
Let [MATH] be any group and [MATH] . We let [MATH] Note that [MATH] is a subgroup of [MATH] Proposition 7.2 Let [MATH] with [MATH] and [MATH] . If [MATH] is atomic, then [MATH] |
Proof. Assume [MATH] satisfies [MATH] . Then the orbit of [MATH] must be finite and [MATH] restricted to this orbit is a uniform measure. The diagonal of the orbit is then a fixed positive measured subset of [MATH] , which is invariant under the diagonal action of [MATH] . Thus, as [MATH] is weakly mixing, the orbit mu... |
[EQUATION] Thus [MATH] , as wanted. From the proposition above, it follows that in order to obtain a non-atomic IRS via the co-induction operation, we just need to ensure that we do not obtain a Dirac measure. Thus we have the following criterion. |
Corollary 7.3 Let [MATH] with [MATH] and [MATH] . Moreover, let [MATH] be a transversal for the left cosets in [MATH] . Then [MATH] is non-atomic if and only if there is [MATH] such that |
[EQUATION] and [MATH] for all [MATH] Proof. First note that [MATH] is weakly mixing by 7.1 . Hence it follows by 7.2 that [MATH] is non-atomic if and only if [MATH] , and the latter is equivalent to the statement in the corollary. |
Note that if [MATH] for some [MATH] , then we have [MATH] . Thus [EQUATION] where [MATH] denotes the automorphism in [MATH] induced by [MATH] |
It is clear that if [MATH] and [MATH] is normal with [MATH] , we have that [MATH] satisfies [MATH] . Thus all possible Dirac measures in [MATH] are contained in the image of the co-induction operation. In some cases these are the only ones. |
Proposition 7.4 If [MATH] with [MATH] , then any [MATH] satisfies [MATH] Proof. Let [MATH] satisfy [MATH] . The for each [MATH] and [MATH] we have [MATH] . Thus, since [MATH] we have |
[MATH] if [MATH] and [MATH] , if [MATH] , as wanted. In such cases, this means that the co-induced action has almost everywhere fixed stabilizers and the co-induced action of a faithful action is free. In general it is easily seen that if an action is free, then so are all its co-induced actions. By use of the descript... |
Proposition 7.5 Let [MATH] [MATH] a transversal for the left cosets in [MATH] and [MATH] . Then [MATH] is not free if and only if for some [MATH] we have |
[EQUATION] and [MATH] for all [MATH] Proof. It follows directly from 5.1 that [MATH] is not free if and only if for some [MATH] we have |
[EQUATION] Since the latter is equivalent to the statement in the proposition, the conclusion follows. Note that for any [MATH] we have |
[EQUATION] So for the co-induced action to be non-free, in the case [MATH] , the conjugates of some [MATH] under the transversal [MATH] must uniformly converge very fast to the identity in [MATH] |
Remark 7.6 For any group [MATH] there exists a group [MATH] such that [MATH] and [MATH] . Thus for such groups any action [MATH] will satisfy that [MATH] is free. |
New constructions of non-atomic, weakly mixing invariant random subgroups In this section we will apply the co-induction operation on invariant random subgroups to construct new examples of continuum size families consisting of non-atomic, weakly mixing invariant random subgroups on several classes of groups. |
8.A A sufficient criterion We will provide in this subsection a sufficient criterion for an infinite index subgroup to generate continuum many non-atomic, weakly mixing co-induced invariant random subgroups on the bigger group. |
In the following, for a group [MATH] and a subset [MATH] , we let [MATH] denote the subgroup [MATH] in [MATH] and [MATH] denote the normal subgroup [MATH] in [MATH] |
Proposition 8.1 Let [MATH] with [MATH] . Consider the statements: (1) There exists a transversal [MATH] for the left cosets in [MATH] and [MATH] such that the chain of normal subgroups [MATH] , given by |
[EQUATION] is not constant. (2) There exists a continuum size family [MATH] such that we have [MATH] are all non-atomic, weakly mixing and satisfy [MATH] for all [MATH] with [MATH] |
(3) There exists [MATH] such that [MATH] is non-atomic. (4) There exists [MATH] such that [MATH] is not a Dirac measure. (5) For any transversal [MATH] for the left cosets in [MATH] there is [MATH] such that the chain of subgroups [MATH] , given by |
[EQUATION] is not constant. Then [MATH] Proof. It is clear that [MATH] . Thus it suffices to prove [MATH] and [MATH] For the implication [MATH] , assume [MATH] holds for [MATH] and [MATH] We will first construct one such invariant random subgroup. Afterwards we will argue how to obtain uncountably many. |
Let [MATH] . Then the non-constant assumption on the sequence [MATH] ensures that [MATH] , as for some [MATH] with [MATH] we have [MATH] . Moreover, it follows directly by the construction of [MATH] that [MATH] for all [MATH] and that |
[EQUATION] So the assumptions of 7.3 are satisfied and thus the co-induced measure must be non-atomic. Since [MATH] , it will also be weakly mixing. |
Now to construct uncountably many of these, let [MATH] be least such that [MATH] and let [MATH] . Next, fix [MATH] such that for [MATH] we have [MATH] if and only if [MATH] For each [MATH] put |
[EQUATION] Then we have [EQUATION] for all [MATH] , while [EQUATION] for all [MATH] . Thus, by the description of the co-induction operation given in 5.1 , we obtain that |
[EQUATION] So [MATH] is a continuum size family of non-atomic, weakly mixing invariant random subgroups of [MATH] , as wanted. For the implication [MATH] , assume that [MATH] holds. Let [MATH] be a transversal for [MATH] . We have that |
[EQUATION] for all [MATH] finite. Since [MATH] is not a Dirac measure, there exists [MATH] such that [MATH] . Thus, if we let [MATH] satisfy that [MATH] , we have [MATH] for some [MATH] and |
[EQUATION] By convergence of the series, it follows that for some [MATH] we have [EQUATION] Thus we must have [MATH] , as wanted. |
Remark 8.2 In general, if [MATH] is a normal subgroup, then [MATH] is a transversal for the left cosets in [MATH] if and only if [MATH] is a transversal, as well. Thus in this case, the statement |
(1’) There exists a transversal [MATH] for the left cosets in [MATH] and [MATH] such that the chain of normal subgroups [MATH] given by |
[EQUATION] is not constant. is equivalent to condition [MATH] in 8.1 Remark 8.3 If [MATH] in 8.1 is abelian, then all the statements are equivalent. We also point out that the invariant random subgroups constructed in the proof of [MATH] above, are not weakly mixing when restricted to [MATH] |
8.B Wreath products and HNN-extensions We will here apply the criterion in 8.1 to wreath products and HNN-extensions. (1) Let [MATH] be countable groups and consider the action [MATH] given by [MATH] . The wreath product of [MATH] by [MATH] is then the semidirect product [MATH] and is denoted by [MATH] |
Construction of continuum many non-atomic, weakly mixing invariant random subgroups on [MATH] , for [MATH] countable groups such that [MATH] is infinite and [MATH] is not trivial. |
Let [MATH] and [MATH] . Then [MATH] and [MATH] is a transversal for the left cosets [MATH] . Fix an enumeration [MATH] and let [MATH] . Define [MATH] by |
[EQUATION] Then, as [MATH] is not constant, following 8.1 we construct continuum many non-atomic, weakly mixing invariant random subgroups on [MATH] |
If [MATH] is a countable set and we have an action [MATH] , we may form a wreath product by letting [MATH] be given by [MATH] and then consider the semidirect product [MATH] . We denote such a wreath product by [MATH] . Arguments similar to those in the preceding paragraph work as well for [MATH] , if the action [MATH]... |
(2) Next we will consider HNN-extensions over “small” subgroups. Construction of continuum many non-atomic, weakly mixing invariant normal subgroups for the HNN extension [MATH] , where [MATH] is a countable group, [MATH] and [MATH] is an embedding with |
[MATH] Let [MATH] for each [MATH] be a copy of [MATH] and put [EQUATION] Then [MATH] , where [EQUATION] for all [MATH] and [MATH] (see B08 , Theorem 17.1] ). Now let [MATH] and consider the homomorphism [MATH] induced by the homomorphisms [MATH] for [MATH] given by [MATH] if [MATH] and [MATH] for all [MATH] For a fixed... |
[EQUATION] and hence using 8.1 we construct continuum many non-atomic, weakly mixing invariant random subgroups in [MATH] Note that this covers the case where [MATH] is an automorphism of a non-trivial normal subgroup of [MATH] . We also have the following application. |
Corollary 8.4 If [MATH] are not relatively prime, then there are continuum many non-atomic, weakly mixing invariant random subgroups on [MATH] |
Proof. We have that [MATH] is the HNN-extension of [MATH] with respect to the isomorphism [MATH] and [MATH] 8.C Non-abelian free groups |
We will now turn our attention towards the non-abelian free groups. It follows already from the results in BGK17 that these groups admit continuum many non-atomic, weakly mixing invariant random subgroups. In this subsection we show how the co-induction can be used to give alternative constructions of invariant random ... |
(1) First we will use the co-induction operation from [MATH] to various semi-direct products of the form [MATH] , where [MATH] is induced by a permutation of the generators of [MATH] , to construct new invariant random subgroups on [MATH] |
Construction of continuum many non-atomic, weakly mixing invariant random subgroups on [MATH] Fix [MATH] and let [MATH] be a Borel partition with [MATH] . For each [MATH] , put [MATH] and fix a free action [MATH] . Then define [MATH] by [MATH] if [MATH] and [MATH] if [MATH] for some [MATH] . Finally, let [MATH] act as ... |
Next, let [MATH] be infinite and let [MATH] be a permutation which is transitive on [MATH] and fixes every element of [MATH] . We then define [MATH] by [MATH] and [MATH] for all [MATH] and [MATH] . Consider [MATH] and let |
[EQUATION] Note that by 3.3 we have [MATH] is weakly mixing and for [MATH] we have [EQUATION] So for [MATH] it holds that [EQUATION] |
and hence [MATH] is non-atomic. Moreover, this implies that whenever [MATH] are infinite with [MATH] we have [MATH] (2) Next consider another construction using co-induction, which allows us to construct continuum many non-atomic, weakly mixing invariant random subgroups on every non-abelian free group. |
Construction of continuum many non-atomic, weakly mixing invariant random subgroups on [MATH] for [MATH] with [MATH] Fix [MATH] with [MATH] and some free generators [MATH] . Consider the surjective group homomorphism [MATH] given by [MATH] for [MATH] and [MATH] . Then let [MATH] and note that |
[EQUATION] This set freely generates [MATH] as a copy of [MATH] inside [MATH] Moreover, the set [MATH] constitutes a transversal for [MATH] |
Now for each [MATH] let [MATH] satisfy that the action induced by [MATH] is weakly mixing and for each [MATH] we have [EQUATION] |
One way to choose [MATH] is to decompose [MATH] such that [MATH] and then let [MATH] be weakly mixing when restricted to [MATH] and trivial on [MATH] for [MATH] Next, define an action [MATH] by letting [MATH] and [MATH] for all [MATH] and [MATH] . Then put [MATH] . Note that all conditions of 7.3 are satisfied with res... |
[EQUATION] Thus [MATH] for all [MATH] with [MATH] Remark 8.5 Using an action similar to the one in the construction above one can give a proof of the following algebraic fact: Let [MATH] satisfy [MATH] if [MATH] . Then the sequence |
[EQUATION] does not extend to a basis of [MATH] . Indeed, assume towards a contradiction, that we may extend the sequence to a basis of [MATH] . Then we would have that [MATH] generates a copy, [MATH] , of [MATH] as a subgroup of [MATH] . So let [MATH] be an action such that [MATH] for all [MATH] and |
[EQUATION] as [MATH] in [MATH] Now, since the sequence extends to a basis, we may extend this action to an action [MATH] . Therefore we would have |
[EQUATION] which contradicts the convergence above. 8.D Free products with normal amalgamation Here we will use co-induction to give constructions of non-atomic, weakly mixing invariant random subgroups on certain free products of groups with normal amalgamation. Other constructions can be found in BGK17 but our proofs... |
Construction of continuum many non-atomic, weakly mixing invariant random subgroups on [MATH] , where [MATH] [MATH] are non-trivial countable groups with [MATH] infinite, with support in |
[EQUATION] Moreover, we can ensure that these invariant random subgroups are weakly mixing, when restricted to [MATH] Let [MATH] , consider the homomorphism [MATH] induced by the homomorphisms [MATH] , and put [MATH] . Then [MATH] is freely |
[EQUATION] where [MATH] and [MATH] is a transversal for the left cosets [MATH] . Now fix [MATH] and [MATH] . For each [MATH] , let [MATH] be an action satisfying [MATH] and [MATH] for all [MATH] and [MATH] . Let [MATH] and note that |
[EQUATION] We have [EQUATION] for all [MATH] [MATH] . Thus [MATH] or its inverse is in the reduced word over the alphabet of non-trivial commutators of |
[EQUATION] if and only if [EQUATION] Therefore [MATH] and so [MATH] constitute a continuum size family of non-atomic, weakly mixing invariant random subgroups of [MATH] |
To ensure weakly mixing when restricted to [MATH] , let [MATH] satisfy that [MATH] are distinct and that [MATH] [MATH] for all [MATH] . Then modify [MATH] such that the action of [MATH] is weakly mixing and each satisfies [MATH] for [MATH] . Note that the relation constrains on [MATH] ensure that at most one of [MATH] ... |
[EQUATION] when [MATH] and [MATH] . Moreover, as with [MATH] , each will appear exactly four times. So we then have [EQUATION] Again, [MATH] constitute a continuum size family of non-atomic, weakly mixing invariant random subgroups of [MATH] . These will now also be weakly mixing, when restricted to [MATH] by 3.3 |
Remark 8.6 Gaboriau pointed out that in the paper D. Gaboriau and N. Bergeron, Asymptotique des nombres de Betti, invariants [MATH] et laminations, Comment. Math. Helv. 79(2) (2004), 362–395, 2004, the following result is proved: Let [MATH] and [MATH] be residually finite, infinite groups such that either |
[MATH] or [MATH] for [MATH] Then the free product [MATH] admits continuum many IRS. These IRS are distinguished by their [MATH] -Betti numbers. (Here, [MATH] is the [MATH] -th Betti number of [MATH] while [MATH] is its [MATH] -th [MATH] -Betti number.) |
Next, note the following well-known simple fact. Proposition 8.7 Let [MATH] be countable groups and [MATH] a surjective group homomorphism. Then there is an embedding [MATH] such that if [MATH] is ergodic, weakly mixing or non-atomic, so is [MATH] |
Proof. Note that the map [MATH] given by [MATH] is a homeomorphism with image [EQUATION] Moreover, we have [EQUATION] for all [MATH] and [MATH] . So let [MATH] be given by [MATH] . It is then clear that [MATH] is ergodic, weakly mixing or non-atomic if [MATH] is. Since |
[EQUATION] it follows by 4.2 that [MATH] is continuous. Now by use of the previous construction for free products and 8.7 , we can construct continuum many non-atomic, weakly mixing invariant random subgroups for the groups [MATH] , where [MATH] and [MATH] are countable groups satisfying that [MATH] with [MATH] non-tri... |
The same applies to all the groups [MATH] , where [MATH] is a countable family of countable groups with [MATH] infinite and [MATH] non-trivial, by looking at the natural surjective group homomorphism [MATH] |
8.E Automorphism invariant random subgroups of the free group of rank two In this part we will use the co-induction operation to construct non-atomic invariant random subgroups on [MATH] which are invariant under the action of the full automorphism group, as well. Moreover, these invariant random subgroups will be weak... |
Fix a basis [MATH] . We think of an element of [MATH] as represented by the induced reduced word in the letters [MATH] . Consider the automorphisms [MATH] given by |
[EQUATION] [EQUATION] Let [MATH] denote the set of automorphisms [MATH] and [MATH] (and not [MATH] ). Then [EQUATION] is a set of representatives for the left cosets in [MATH] , where [MATH] is identified with the subgroup of inner automorphisms (see CMZ81 , Section 3] ). Note that [MATH] denotes the identity map. |
Consider the word [MATH] for some [MATH] . The first goal is to prove that the family [EQUATION] satisfies the [MATH] cancellation property. Recall that a subset of words [MATH] has the [MATH] cancellation property if the set [MATH] of all cyclically reduced cyclic conjugates of the words in [MATH] and their inverses s... |
[EQUATION] Here [MATH] denotes the length of a word in [MATH] . In case [MATH] satisfies the [MATH] cancellation property and [MATH] is a cyclically reduced word, then there is [MATH] such that [MATH] For a proof of this see for example LS77 , Theorem 4.5 in Chapter V] We will use this fact to ensure that condition [MA... |
Put [EQUATION] and let [MATH] for all [MATH] Below we will use the following terminology. For two words [MATH] cancellation of [MATH] and [MATH] is a string [MATH] which appears in the reduced cycles of both [MATH] We say that [MATH] is a bad cancellation of [MATH] and [MATH] if |
[EQUATION] Here [MATH] denote the length of the induced cyclically reduced word. We call a cancellation for maximal if it cannot be extended. The goal is then to prove that there is no bad cancellation between any pair of words in the set |
[EQUATION] Let [EQUATION] Then it suffices to prove that there is no bad cancellation among the words in [MATH] and then prove that there cannot be any bad cancellation between a word from [MATH] and a word from [MATH] . We will begin with the former. Most of our arguments for this are based on the following two lemmas... |
For a word [MATH] , we let [MATH] denote the word obtained from [MATH] by switching every negative power of [MATH] and [MATH] to be positive. |
Lemma 8.8 Let [MATH] [MATH] and let [MATH] be a cancellation of [MATH] . Assume [MATH] satisfies that for any cancellation [MATH] of [MATH] the total number of [MATH] ’s and the total number of [MATH] ’s in [MATH] are both less than [MATH] . Then |
[EQUATION] Proof. First let [MATH] and [MATH] be such that [MATH] . Assume [MATH] is a maximal cancellation of [MATH] Then there is a maximal cancellation [MATH] of [MATH] such that [MATH] is equal to one of the following strings: |
[EQUATION] This can be seen by considering the pre-images through [MATH] of all possible strings of the form [EQUATION] where [MATH] and [MATH] are such that [MATH] and [MATH] is a legal string in the reduced cycles induced by [MATH] and [MATH] , together with the cases where [MATH] [MATH] and [MATH] . The latter cases... |
Next, for [MATH] , we let [MATH] and [MATH] be such that [MATH] and [MATH] , for [MATH] . Now let [MATH] be a maximal cancellation of [MATH] . Then, by repeating the argument above, there is a maximal cancellation [MATH] of [MATH] such that |
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