On dense totipotent free subgroups in full groups

We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated IRS’s. The perfect kernel of a countable group Γ is the largest closed subspace of the space of subgroups of Γ without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of Γ : those for which almost every point-stabilizer has dense conjugacy class in the perfect kernel. Equivalently, the support of the associated IRS is as large as possible, namely it is equal to the whole perfect kernel. We prove that every ergodic p.m.p. equivalence relation R of cost < r can be realized by the orbits of an action of the free group F r on r generators that is totipotent and such that the image in the full group [ R ] is dense. We explain why these actions have no minimal models. This also provides a continuum of pairwise orbit inequivalent invariant random subgroups of F r , all of whose supports are equal to the whole space of inﬁnite index subgroups. We are led to introduce a property of topologically generating pairs for full groups (we call evanescence) and establish a genericity result about their existence. We show that their existence characterizes cost 1 .


Geometry & Topology msp
Volume 27 (2023) In this context, clarifying precisely what is meant by "totipotency" and how it is experimentally determined will both avoid unnecessary controversy and potentially reduce inappropriate barriers to research.
-M Condic [5] Let be a countable discrete group.Denote by Sub./ the space of subgroups of .It is equipped with the compact totally disconnected topology of pointwise convergence and with the continuous -action by conjugation.Let ˇbe a Borel -action on the standard Borel space X ' OE0; 1.Its stabilizer map Stab ˇW X !Sub./; x 7 !f 2 W ˇ. /x D xg; is -equivariant.If is a probability measure on X which is preserved by ˇ, then the pushforward measure Stab ˇ is invariant under conjugation.It is the prototype of an invariant random subgroup (IRS).When is atomless and the stabilizer map is essentially injective (a.k.a. the action ˇis totally nonfree), the support of the associated IRS Stab ˇ ./ has no isolated points: it is a perfect set.The largest closed subspace of Sub./ with no isolated points is called the perfect kernel of Sub./.We say that an ergodic probability measure preserving (p.m.p.) action is totipotent when the support of its IRS is equal to the perfect kernel of Sub./.By ergodicity, the following stronger property holds: almost every element of the associated IRS has dense orbit in the perfect kernel; see Proposition 2.3.We call such an IRS totipotent.
Given a p.m.p. action Õ ˇ.X; /, we consider the associated p.m.p. equivalence relation R ˇWD f.x; y/ 2 X X W ˇ./x D ˇ./yg; and its full group OER ˇ as the group of all measure-preserving transformations whose graph is contained in R ˇ.The (bi-invariant) uniform distance between two measurepreserving transformations S and T is defined by d u .T; S/ WD .fx 2 X W S.x/ ¤ T .x/g/: It endows the full group OER ˇ with a Polish group structure.The cost is a numerical invariant attached to the equivalence relation R ˇ.If ˇis a p.m.p. action of the free group F r on r generators, then the cost of R ˇis exactly r when ˇis free, and the cost of R ˇis < r when ˇis nonfree; see Gaboriau [8].
The main result of Le Maître [15] is that for any ergodic p.m.p. equivalence relation R, if R has cost < r for some integer r 2, then there exists a homomorphism W F r !OER with dense image.
This result has been sharpened in order to ensure that the homomorphism is injective.
Actually, the associated (almost everywhere defined) p.m.p. action ˛ can be made to satisfy the following two opposite conditions: high faithfulness and amenability on -almost every orbit; see Le Maître [18].
These two conditions can be phrased in terms of the support of the IRS associated to the action: the first one means that the support contains the trivial subgroup, and one can show that the second one is equivalent to the support containing a coamenable subgroup (which in the construction of [18] is the kernel of a certain surjective homomorphism F r !Z).
The purpose of the present paper is to show that the homomorphism can be chosen so that the support of the associated IRS is actually the largest perfect subspace of Sub.F r /, which consists of all its infinite-index subgroups; see Proposition 2.1.
Theorem Let R be an ergodic p.m.p. equivalence relation whose cost is < r for some integer r 2. Then there exists a homomorphism W F r !OER whose image is dense and whose associated p.m.p. action ˛ is totipotent.
The density in OER of the image of implies that R ˛ ' R and that the stabilizer map Stab ˛ is essentially injective [18,Proposition 2.4].In particular, the actions F r Õ .Sub.F r /; Stab ˛ / and F r Õ ˛ .X; / are conjugate (thus produce the same equivalence relation) and almost every subgroup for the IRS Stab ˛ equals its own normalizer.It follows that, up to isomorphism, every p.m.p. ergodic equivalence relation of cost < r comes from a totipotent IRS of F r (actually, from continuum many different totipotent IRSs of F r ; see Remark 5.1).Such a statement is optimal since p.m.p. equivalence relations of cost r cannot come from a nonfree F r action.To our knowledge, it was not even clear until now whether F r admits ergodic totipotent IRSs.Since there are continuum many pairwise nonisomorphic ergodic p.m.p. equivalence relations of cost < r , our approach provides continuum many pairwise distinct ergodic totipotent IRSs of the free group on r generators, whose associated equivalence relations are even nonisomorphic.
Another interesting fact about totipotent p.m.p.F r -actions is that they have no minimal model, ie they cannot be realized as minimal actions on a compact space.Indeed, it follows from a result of Glasner and Weiss [10,Corollary 4.3] that as soon as the support of the IRS of a given p.m.p. action contains two distinct minimal subsets (eg when it contains two distinct fixed points), the action does not admit a minimal model; see Theorem 2.5.In our case the perfect kernel of Sub.F r / contains a continuum of fixed points (namely, all infinite-index normal subgroups), so that totipotent p.m.p. actions of F r are actually very far from admitting a minimal model.
Let us now recall the context around our construction.The term IRS was coined by Abert, Glasner and Virag [1] and has become an important subject on its own at the intersection of group theory, probability theory and dynamical systems.The notion of IRS is a natural generalization of a normal subgroup, especially in the direction of superrigidity type results.It has thus been present implicitly in the work of many authors, a famous landmark being the Stuck-Zimmer theorem [21], which gives examples of groups admitting very few IRSs.On the contrary, some groups admit a "zoo" of IRSs, starting with free groups; see Bowen [2] and, for other examples, Bowen, Grigorchuk and Kravchenko [3; 4] and Kechris and Quorning [13].
In particular, Bowen proved that every p.m.p. ergodic equivalence relation of cost < r comes from some IRS of F r .He obtained this result through a Baire category argument which required that the first generator act freely.In particular, such IRSs can never be totipotent.
Eisenmann and Glasner [7] then used homomorphisms F r !OER with dense image so as to obtain interesting IRSs of F r .They proved that given a homomorphism !OER with dense image, the associated IRS is always cohighly transitive almost surely, which means that for almost every ƒ Ä , the -action on =ƒ is n-transitive for every n 2 N.They also showed that the IRSs of F r obtained by Bowen for cost 1 equivalence relations are faithful and, moreover, almost surely coamenable.
The third author [18] then used a modified version of his result on the topological rank of full groups to show that every p.m.p. ergodic equivalence relation of cost < r comes from a coamenable, cohighly transitive and faithful IRS of F r .Also in this construction, the first generator continues to act freely, thus preventing totipotency.Let us now briefly explain how our new construction (Section 5) allows us to circumvent this.
The main idea is to use a smaller set Y ¨X such that the restriction of R to Y still has cost < r , so that we can find some homomorphism F r !OER Y with dense image.
This provides us with some extra space in order to obtain totipotency via a well-chosen perturbation of the above homomorphism.This perturbation is obtained by mimicking all Schreier balls on X nY and then merging these amplifications with the action on Y so as to obtain both density in OER and totipotency.The use of evanescent pairs of topological generators (see Definition 4.1) with Theorem 4.5 and Proposition 3.8 will grant us that this perturbation maintains the density.We establish in Theorem 4.6 that the existence of an evanescent pair of topological generators is equivalent to R having cost 1.
Finally, let us mention the case of the free group on infinitely many generators F 1 .
Here, the space of subgroups is already perfect (see Proposition 2.1), and one can easily adapt our arguments to show that: For every ergodic p.m.p. equivalence relation R, there exists a homomorphism W F 1 !OER whose image is dense and whose associated p.m.p. action ˛ is totipotent.
This result could, however, also be obtained by a purely Baire-categorical argument: it is not hard to see that the space of such homomorphisms is dense G ı in the Polish space of all homomorphisms W F 1 !OER.
Going back to the case of finite rank, it is not even true that a generic homomorphism W F r !OER generates the equivalence relation R. In order to hope for a similar genericity statement, one should first answer the following question.
Question Consider a p.m.p. ergodic equivalence relation R of cost < r .Is it true that, in the space of homomorphisms W F r !OER whose image generates R, those with dense image are dense?
The fact that Bowen and then Eisenmann and Glasner had to work in the even smaller space where the first generator acts freely, indicates that a Baire-categorical approach to our main result is out of reach at the moment, if not impossible.
Let be a countable discrete group.The topology on its space of subgroups Sub./ admits V .I; O/ WD fƒ 2 Sub./ W I Â ƒ and O \ ƒ D ∅g as a basis of open sets, where I and O are finite subsets of .By the Cantor-Bendixson theorem, Sub./ decomposes in a unique way as the disjoint union of a perfect set, called the perfect kernel K./ of Sub./, and of a countable set.We indicate some isolation properties of subgroups: (1) If ƒ 2 Sub./ is not finitely generated, then writing ƒ D .j / j 2N we obtain ƒ as the nontrivial limit of the infinite index (both in ƒ and in ) of the finitely generated subgroups ƒ n WD h 0 ; 1 ; : : : ; n i.
(2) If is finitely generated, then its finite-index subgroups are isolated.Indeed, a finite-index subgroup ƒ is finitely generated as well and it is alone in the open subset defined by a finite family I of generators and a finite family O of representatives of its cosets =ƒ except fƒg.
(3) If is not finitely generated, then its finite-index subgroups are also not finitely generated and thus are not isolated by property (1).
Let us denote by Sub 1i ./ the subspace of infinite-index subgroups of .The following is probably well known, but we were not able to locate a proof in the literature.
Proposition 2.1 For the free group F r on r generators, with 2 Ä r Ä 1: Proof We first show that if ƒ 2 Sub 1i .F r /, with 2 Ä r Ä 1, then it is a nontrivial limit of finitely generated infinite-index subgroups of F r .If ƒ is not finitely generated, then property (1) above applies.Thus, assume ƒ is finitely generated.If r is infinite, then ƒ has infinite index in some finitely generated noncyclic free subgroup ƒ Ä ƒ F 2 Ä F 1 .We can thus assume that the rank r 2 is finite.By the Hall theorem, ƒ is a free factor of a finite-index subgroup ƒ of the free group F r (we include the case ƒ D f1g).Since ƒ has infinite index, is nontrivial.If g 2 is a nontrivial element, then ƒ is the nontrivial limit of the sequence of finitely generated infinite-index subgroups .ƒhg n i/ n 2 of F r .
Remark 2.2 This also shows that the Cantor-Bendixson rank of Sub.F r / equals 1 when r is finite and equals 0 when r D 1.
Computations of the perfect kernel for some other groups have been performed in [3; 20].
The following is a classical result: Assume acts by homeomorphisms on a Polish space Z and is an ergodic -invariant probability measure on Z. Then the orbit of -almost every point z 2 Z is dense in the support of .
In particular: Thus, our main theorem produces IRSs on Sub.F r / for which almost every F r -orbit (under conjugation) is dense in K.F r / D Sub 1i .F r /.In other words, for almost every subgroup ƒ, the Schreier graph of the action F r Õ F r =ƒ contains arbitrarily large copies of Schreier balls of every infinite transitive F r -action.

Remark 2.4
In the introduction, we defined an IRS to be totipotent when almost every subgroup has dense orbit in the perfect kernel.But an IRS can also be considered as a p.m.p. dynamical system whose associated IRS can be different.The connections between the two notions of totipotency are unclear to us.However, since the actions that we construct are totally nonfree, this situation does not happen and our IRSs are totipotent in both senses.
Moreover, this proposition can be combined with [10,Corollary 4.3] to give the following result.
Theorem 2.5 Let Õ .X; / be a p.m.p. ergodic action on a standard probability space.Suppose that the support of the associated IRS contains at least two distinct minimal subsets.Then the action has no minimal model.This is in wide contrast with free actions of countable groups: they always admit minimal models [22].
Proof By the previous proposition, the orbit closure of the stabilizer of -almost every point is equal to the support of the IRS, and hence contains two distinct minimal subsets.Admitting a minimal model would thus be incompatible with [10, Corollary 4.3].

Full groups and density
We fix once and for all a standard probability space .X; / and denote by Aut.X; / the group of all its measure-preserving transformations, two such transformations being identified if they coincide on a full measure set.In order to ease notation, we will always neglect what happens on null sets.Given an element T 2 Aut.X; /, its set of fixed points is denoted by A partial isomorphism of .X; / is a partially defined Borel bijection ' W dom ' !rng ', with dom ' and rng ' Borel subsets of X , such that ' is measure-preserving for the measures induced by on its domain dom ' and its range rng '.In particular, we have .dom'/ D .rng'/.The support of ' is the set Given two partial isomorphisms with '; disjoint domains and ranges, one can form their union, which is the partial isomorphism A graphing is a countable set of partial isomorphisms ˆ.Its cost C.ˆ/ is the sum of the measures of the domains of its elements, which is also equal to the sum of the measures of their ranges since they preserve the measure.
Given a graphing ˆ, the smallest equivalence relation which contains all the graphs of the elements of ˆis denoted by R ˆand called the equivalence relation generated by ˆ.When ˆD f'g, we also write it as R ' and call it the equivalence relation generated by '.
The equivalence relations that can be generated by graphings are called p.m.p. equivalence relations; they are Borel as subsets of X X and have countable classes.
The cost C.R/ of a p.m.p. equivalence relation R is the infimum of the costs of the graphings which generate it.
Whenever ˛W !Aut.X; / is a p.m.p. action, we denote by R ˛the equivalence relation generated by ˛./.Given a p.m.p. equivalence relation R, the set of partial isomorphisms whose graph is contained in R is denoted by OEOER and called the pseudo full group of R.Here is a useful way of obtaining elements of the pseudo full group that we will use implicitly.Say that R is ergodic when every Borel R-saturated set has measure 0 or 1.Under this assumption, given any two Borel subsets A; B Â X of equal measure, there is ' 2 OEOER such that dom ' D A and rng ' D B [12, Lemma 7.10].
The full group of R is the subgroup OER of Aut.X; / consisting of almost everywhere defined elements of the pseudo full group.Endowed with the uniform metric given by d u .S; T / D .fx 2 X W S.x/ ¤ T .x/g/, it becomes a Polish group.Observe that d u .T; id X / D .suppT /.
For more material about this section, we refer for instance to [12; 9].

Around a theorem of Kittrell and Tsankov
In this paper, we will be interested in p.m.p. actions W F r !OER with dense image in OER.To that end, the following result of Kittrell and Tsankov is very useful.Given a family .R i / of equivalence relations on the same set X , we define W i2I R i as the smallest equivalence relations which contains each R i .Theorem 3.1 [14, Theorem 4.7] Let R be a p.m.p. equivalence relation on .X; /, and suppose that .R i / i2I is a family of Borel subequivalence relations such that We will also use two easy corollaries of their result, which require us to set up a bit of notation.
Definition 3.2 Given an equivalence relation R on a set X and Y Â X , we define the equivalence relation R Y restricted to Y and the equivalence relation Observe that given a p.m.p. equivalence relation R, we have a natural way of identifying the full group of the restriction R Y with the full group of the induced equivalence relation R lY by making its elements act trivially outside of Y .In particular, if .Y / > 1 2 , then k D 1 and hence ˛./ D OER.
Proof Let B Y be a subset of maximal measure such that ˛./ OER lB .Then by the above corollary, for every 2 such that ˛. /B ¤ B we must have .B \˛./B/ D 0; hence B is an atom of a finite partition preserved by the -action ˛.

From graphings to density
The following is a slight variation of [15,Definition 8].
Definition 3.5 Let n 2. A precycle of length n is a partial isomorphism ' such that if we set B WD dom ' n rng ' (the basis of the precycle), then f' i .B/g iD0;:::;n 2 is a partition 0 of dom ', and f' i .B/g iD1;:::;n 1 is a partition of rng '.
We say that T 2 Aut.X; / extends ' if T x D 'x for every x 2 dom.'/.
Observe that a precycle of length The following proposition is obtained by a slight modification of the proof of the main theorem of [15].Proposition 3.8 Let R be a p.m.p. ergodic equivalence relation on X and let Y Â X be a positive measure subset.Let R 0 Ä R lY be a hyperfinite equivalence relation whose restriction to Y is ergodic (and trivial on X n Y ).Suppose that C.R lY / < r .Y / for some integer r 2. Then there are r 1 precycles ' 2 ; ' 3 ; : : : ; ' r 2 OEOER lY such that .supp.' i // < .Y / and such that whenever U 2 ; U 3 ; : : : ; U r 2 OER extend ' 2 ; ' 3 ; : : : ; ' r , we have hOER 0 ; U 2 ; U 3 ; : : : ; U r i OER lY .
Proof Let T 2 OER 0 be such that its restriction to Y is ergodic.Our assumption C.R lY / < r .Y / means that the normalized cost of the restriction R Y is less than r .Lemma III.5 from [8] then provides a graphing ˆon Y of normalized cost < .r1/ such that fT Y g [ ˆgenerates the restriction R Y .We now view ˆas a graphing on X , so that fT g [ ˆgenerates R lY , and C.ˆ/ < .r1/ .Y /.Let c WD C.ˆ/=.r 1/ < .Y /.We take p 2 N so large that c.p C 2/=p < .Y /.
Pick 2 OEOER 0 a precycle of length 2 whose domain B has measure c=p.By cutting and pasting the elements of ˆand by conjugating them by elements of OER 0 , we may as well assume that ˆD f' 2 ; : : : ; ' r g, where each ' i is a precycle of length p C 2 extending of basis B, whose support is a strict subset of Y .Assume that U i 2 OER extends ' i for every i D 2; 3; : : : ; r .Since 2 OEOER 0 , then OER Ä OER 0 .We can apply Lemma 3.7 and obtain that the closure of the group generated by OER 0 and U i contains we can conclude the proof of the theorem using Theorem 3.1.Remark 3.9 We have a lot of freedom in constructing the precycles ' 2 ; ' 3 ; : : : ; ' r of Proposition 3.8.To start with, their length can be chosen to be any integer n D p C 2 large enough that c= .Y / < .n2/=n.Actually, they could even have been chosen with any (possibly different) lengths n 2 ; n 3 ; : : : ; n r , integers large enough that c= .Y / < .nj 2/=n j : simply pick r 1 precycles j 2 OEOER 0 of length 2 whose domain B j has measure c=.n j 2/, and proceed as in the proof above.
In particular, the periodic closing cycles U 2 ; U 3 ; : : : ; U r can be assumed to have any large enough period n 2 ; n 3 ; : : : ; n r and domains contained in Y of measure < .Y /.Up to conjugating by elements of OER 0 , one can further assume that the closing cycles have a nonnull common subset of fixed points in Y :

Evanescent pairs and topological generators
In this section our main goal is to obtain two topological generators of the full group of a hyperfinite ergodic equivalence relation with new flexibility properties relying on the following definition.(2) for every n 2 N, the full group OER is topologically generated by the conjugates of V n by the powers of T , ie hT j V n T j W j 2 Zi D OER.
Geometry & Topology, Volume 27 (2023) In particular, if .T; V / is an evanescent pair of topological generators, then the pair .T; V / topologically generates OER, .T; V n / is an evanescent pair of topological generators for any n 2 N, d u .V n! ; id X / tends to 0 when n tends to 1.
We will show in Theorem 4.5 that the odometer T 0 can be completed to form an evanescent pair .T 0 ; V / of topological generators for R T 0 , and that the set of possible V is actually a dense G ı .
We now define a sequence of involutions U n 2 OER T 0 with disjoint supports as in [17, Section 4.2]: U n WD ˛n.n /, where n 2 Sym.f0; 1g n / is the 2-point support transposition that exchanges 0 n 1 1 and 1 n 1 0. Observe that U n is the involution with support N 1 n 1 0 t N 0 n 1 1 (of measure 2 nC1 ) which is equal to T 0 on N 1 n 1 0 and T 1 0 on N 0 n 1 1 .
Recall that if n 2 Sym.f0; 1g n / is 2 n -cycle and w n is a transposition which exchanges two n -consecutive elements, then the group Sym.f0; 1g n / is generated by the conjugates of w n by powers of n (actually 2 n 1 of them are enough).A straightforward modification gives the following lemma; see [17,Lemma 4.3] for a detailed proof.
Lemma 4.2 For every n 2 N, the group ˛n.Sym.f0;1g n // is contained in the group generated by the conjugates of U n by powers of T 0 .
Given a periodic p.m.p. transformation U and k 2 N, we say that V is a k th root of U when supp U D supp V and V k D U .The following lemma is well-known.
Lemma 4.3 Whenever R is an ergodic equivalence relation, every periodic element in OER admits a k th root in OER.
Proof Let us first prove that every n-cycle U 2 OER admits a k th root.To this end, pick a fundamental domain A for the restriction of U to its support.Since R is ergodic, we can pick a k-cycle V 2 OER supported on A. Let B be a fundamental domain for V , and put C WD A n B. Then it is straightforward to check that W 2 OER, defined as follows, is a k th root of U : x otherwise.
In the general case, one glues together the k th roots obtained for every n 2 N by considering the restrictions of U to U -orbits of cardinality n.

Remark 4.4
The same proof works more generally for aperiodic p.m.p. equivalence relations.
Theorem 4.5 The set of V 2 OER T 0 such that .T 0 ; V / is an evanescent pair of topological generators of R T 0 is a dense G ı subset of OER T 0 .
Proof Denote by P the set of periodic elements of OER T 0 .It is a direct consequence of Rokhlin's lemma that P is dense in OER T 0 .And similarly the subset P 0 Â P of V 2 OER T 0 with finite order (or equivalently, with bounded orbit size) is dense in OER T 0 .
Writing P as the intersection (over the positive integers q) of the open sets Denote by E the set of V 2 OER T 0 such that for every n, the group OER T 0 is topologically generated by conjugates of V n by powers of T 0 .We want to show that P \ E is dense G ı , and since P is dense G ı it suffices (by the Baire category theorem in the Polish group OER T 0 ) to show that E is dense G ı .
For every m; n 2 N, set The density of the union of the ˛n.Sym.f0;1g n // in OER T 0 recalled above implies that Let us first check that each E m;n is G ı .Denote by W the subgroup of F 2 D ha 1 ; a 2 i generated by the conjugates of a 2 by powers of a 1 .So for w D w.a 1 ; a 2 / 2 W and V 2 OER T 0 , the element w.T 0 ; V m / is a product of conjugates of V m by powers of T 0 .By the definition of the closure we can write E m;n as Since the map V 7 !w.T 0 ; V / is continuous, each of the above right-hand sets is open, so their union over w 2 W is also open, and we conclude that E m;n is G ı .
To check the density, it suffices to show that, for each m; n, one can approximate arbitrary elements of P 0 by elements of E m;n .So let U 2 P 0 and let > 0. Denote by K the order of U .Pick p n such that 2 p K < 1 2 .Let A be the U -saturation of the support of U p D ˛p. p / (defined at the beginning of the section).The measure of A is at most .Finally, let V be a .Km/ th root of U p and define Let us make a few comments on the above result.First, one can check that the pair .T 0 ; V / produced in the construction of [17,Theorem 4.2] provides an explicit example of an evanescent pair of topological generators of R T 0 .Also, the above proof can be adapted to show that any rank one p.m.p. ergodic transformation [19,Section 8] can be completed to form an evanescent pair of topological generators; see [16,Theorem 5.28] for an explicit example of a pair which is evanescent.Proving these results is beyond the scope of this paper, so we leave it as an exercise for the interested reader.
It is unclear whether every p.m.p. ergodic transformations can be completed to form an evanescent pair of topological generators for its full group.Nevertheless, we can characterize the existence of an evanescent pair as follows.
Theorem 4.6 Let R be an ergodic p.m.p. equivalence relation.Then R admits an evanescent pair of topological generators if and only if R has cost 1.
Proof If R admits an evanescent pair .T; V /, then since V is periodic we have .suppV n! / !0. Since any set of topological generators for OER generates the equivalence relation R, we conclude that R has cost 1.
As for the converse, Theorems 4 and 5 from [6] provide an ergodic hyperfinite subequivalence relation which is isomorphic to that of the odometer.So we can pick a conjugate of the odometer T 2 OER.Repeating the proof of Theorem 4.5, we see that the set E T of V 2 OER such that for every n 2 N, OER T is contained in hT j V n T j W j 2 Zi, is dense G ı in OER.
Let us now consider the set E R of V 2 OER such that .T; V / is an evanescent pair of topological generators of R, and for n 2 N the set E n of V 2 OER such that V is periodic and hT j V n T j W j 2 Zi D OER.Each E n is G ı by the same argument as in the proof of Theorem 4.5.Since E R D T n E n , it suffices to show that each E n is dense in order to apply the Baire category theorem and finish the proof.
Let us fix n 2 N. Since E T is dense in OER, we only need to approximate elements of E T by elements of E n .Moreover, the set of V 2 OER such that .suppV / < 1 is open and dense, so we only need to approximate every V 2 E T with .suppV / < 1 by elements of E n .So let V 2 E T with .suppV / < 1, and take > 0.
Lemma III.5 from [8] yields a graphing ˆof cost < 1 3 min.; .X n supp V // such that fT g [ ˆgenerates R, since R has cost 1.Conjugating by elements of OER T and pasting the elements of ˆ, we may as well assume that ˆD f'g, where .dom'/ < 1 3 , and ' is a precycle of length 2 whose support is disjoint from supp V .We then pick 2 OEOER T such that ' t is a precycle of length 3 of support disjoint from supp V , and denote by U 1 the associated 3-cycle.Now let U 2 be an n th root of U 1 and let V 2 WD V U 2 .Then d u .V 2 ; V / < , and we claim that V 2 belongs to E n .In order to prove this, let us denote by G the closed group generated by the conjugates of V n 2 by powers of T .
Since U 2 and V have disjoint support, they commute, and so , and since V 2 E T , we have that OER T Ä G.In particular OER Ä G, and conjugating by V n 2 (which acts as U 1 on supp U 1 ), we get that OER ' Ä G; see also Lemma 3.7.Since R D R T _ R , we conclude by Theorem 3.1 that G contains OER, as wanted.

Proof of the main theorem
As shown in Proposition 2.1, the perfect kernel of Sub.F r / for 1 < r < 1 is the space of infinite-index subgroups.We will construct a p.m.p. action of F r for which almost every Schreier graph contains all possible balls of Schreier graphs of transitive F r -actions on infinite sets.
Step 1 (using a smaller subset) We start with a p.m.p. ergodic equivalence relation R on .X; / of cost < r .By the induction formula [8, Proposition II.6], there is a subset Y Â X such that 1  2 < .Y / < 1 and such that the (normalized) cost of the restriction R Y is still < r .Thus the cost of the induced equivalence relation R lY is < r .Y /.
Using results of Dye [6,Theorems 4 and 5] as in the proof of Theorem 4.6, one can pick a conjugate of the odometer T 2 OER Y .We view T as an element of OER lY .Now we apply Proposition 3.8 (where R 0 D R T ) to obtain precycles ' 2 ; : : : ; ' r 2 OEOER Y whose supports have measure < .Y /.For i Ä r , we let U i be the closing cycle of ' i as defined after Definition 3.5.Set Á WD .Y n supp U 2 / > 0. Let m 0 be a positive integer such that .X n Y /=m 0 < 1 2 Á.
Step 2 (preparing the finite actions) Let .G n / n 1 be an enumeration of the (finite radius) balls of the Schreier graphs of all the transitive F r -actions over an infinite set, up to labeled graph isomorphism, and for which the number of vertices satisfies jG n j m 0 .Since G n comes from a transitive action over an infinite set, we can choose some `2 f1; : : : ; r g and some n 2 G n so that there is no a `-labeled edge whose source is equal to n .
Pick ı n ; n 6 2 G n , set G 0 n WD G n tfı n ; n g and add an a `-edge from n to ı n , an a 1 -edge from ı n to n and an a 2 -edge from n to itself.
In this way we obtain a finite partial Schreier graph, and this can be extended to a genuine Schreier graph of an F r D ha 1 ; : : : ; a r i-action on the same set as follows: for each i 2 f1; 2; : : : ; r g, we consider the connected components of the subgraph obtained by keeping only the edges labeled a i .These are either cycles (we don't modify them) or oriented segments (possibly reduced to a single vertex), in which case we add one edge labeled a i from the end to the beginning of the segment.
Therefore we obtain an action n of F r on the finite set G 0 n and a special point n 2 G 0 n n G n such that n .a 2 / n D n .
Step Gluing all the ˛n together, we obtain an action ˛1 of F r on X with the properties that ˛1.F r / Ä OER lC and ˛1 restricted to C n is ˛n.
Let T 2 OER lY be the conjugate of the odometer introduced in Step 1. Theorem 4.5 states that the set of V 2 OER T such that .T; V / is an evanescent pair of generators for R T is dense so we can choose such a V with .suppV / <  ˛.a i / WD U i ˛1.a i / for i 3: See Figure 1 for the action of ˛.a 2 /.Note that a 2 is the only generator of F r which does not leave the set Y invariant, because of the presence of the involution I in the definition of its action.
X The crucial assumption that .T; V / is an evanescent pair of generators now comes into play: there is a word w.T; V k /, which is a product of conjugates of V k by powers of T , such that d u .w.T; V k /; S / < 1 2 .We remark that ˛.a 1 / acts on Y the same way as T , and that ˛.a 2 / k acts on Y the same way as V k .Also note that ˛.a 1 / preserves each C j while ˛.a 2 / k is the identity on each C j for j D 1; 2; : : : ; n 0 , so that for all m 2 Z, the transformation ˛.
is a p.m.p. ergodic action on a standard probability space, then the stabilizer Stab.x/ of almost every point x 2 X has dense -orbit in the support of the associated IRS D Stab of Sub./.

Corollary 3 . 3
Let R be an ergodic p.m.p. equivalence relation on .X; /.Let T 2 OER and Y Â X be measurable and such that .Y \T Y / > 0. Put Y T WD S n2Z T n Y .Then hT; OER lY i OER lY T .Proof Since .Y \T Y / > 0 and R is ergodic, we have that R lY [T Y D R lY _R lT Y .Therefore Theorem 3.1 implies that hOER lY ; T OER lY T 1 i D OER Y [T Y : Now observe that .Y [ T Y / \ T .Y [ T Y / Ã T Yhas positive measure.Therefore Theorem 3.1 implies that hT; OER lY i contains OER l.Y [T Y [T 2 Y / , and the corollary follows by induction.Corollary 3.4 Consider an ergodic p.m.p. equivalence relation R on .X; / and let Y Â X be a positive-measure subset.Let ˛be a p.m.p. action of on .X; / such that ˛./ Ä OER, .˛./Y / D 1 and OER lY Ä ˛./.Then either ˛./ D OER, or preserves a finite partition fY i g k iD1 of X , with Y Â Y 1 and OER lY i Ä ˛./ for each i Ä k.

Definition 4 . 1
A pair .T; V / of elements of the full group OER of the p.m.p. equivalence relation R is called an evanescent pair of topological generators of R if (1) V is periodic, and Therefore there exists a subset A Â Y n D of measure .A/ D .B/.Let I 2 OER be an involution with support A [ B and which exchanges A and B. We finally define the desired action ˛of F r by setting ˛.a 1 / WD T ˛1.a 1 /; ˛.a 2 / WD V .W U 2 W 1 /.I˛1.a 2 //;
a 1 / m ˛.a 2 / k ˛.a 1 / m acts on C 1 [ C 2 [ [ C n 0 as the identity.It now follows from the fact that w is a product of conjugates of V k by powers of T that w.T; V k / and w.˛.a 1 /; ˛.a 2 / k / coincide on Y and can only differ on C >n 0 , whichThe subscription price for 2023 is US $740/year for the electronic version, and $1030/year ( C $70, if shipping outside the US) for print and electronic.Subscriptions, requests for back issues and changes of subscriber address should be sent to MSP.Geometry & Topology is indexed by Mathematical Reviews, Zentralblatt MATH, Current Mathematical Publications and the Science Citation Index.Geometry & Topology (ISSN 1465-3060 printed, 1364-0380 electronic) is published 9 times per year and continuously online, by Mathematical Sciences Publishers, c/o Department of Mathematics, University of California, 798 Evans Hall #3840, Berkeley, CA 94720-3840.Periodical rate postage paid at Oakland, CA 94615-9651, and additional mailing offices.POSTMASTER: send address changes to Mathematical Sciences Publishers, c/o Department of Mathematics, University of California, 798 Evans Hall #3840, Berkeley, CA 94720-3840.GT peer review and production are managed by EditFLOW ® from MSP. PUBLISHED BY and Legendrian invariants TOBIAS EKHOLM and YANKI LEKILI 2181 Filtering the Heegaard Floer contact invariant ÇA GATAY KUTLUHAN, GORDANA MATI Ć, JEREMY VAN HORN-MORRIS and ANDY WAND 2237 Large-scale geometry of big mapping class groups KATHRYN MANN and KASRA RAFI 2297 On dense totipotent free subgroups in full groups ALESSANDRO CARDERI, DAMIEN GABORIAU and FRANÇOIS LE MAÎTRE 2319 The infimum of the dual volume of convex cocompact hyperbolic 3-manifolds FILIPPO MAZZOLI 2347 Discrete subgroups of small critical exponent BEIBEI LIU and SHI WANG 2383 Stable cubulations, bicombings, and barycenters MATTHEW G DURHAM, YAIR N MINSKY and ALESSANDRO SISTO 2479 Smallest noncyclic quotients of braid and mapping class groups SUDIPTA KOLAY An n-cycle is a measure-preserving transformation all of whose orbits have cardinality either 1 or n.Given a precycle ' of length n, we can extend it to an n-cycle U ' 2 OER ' as follows: .n 1/ .x/ifx 2 rng ' n dom ';x otherwise.This n-cycle U ' is called the closing cycle of ' and supp U ' D supp '.If f' 1 ; : : : ; ' n 1 g is a pre-n-cycle in the sense of [15, Definition 8], then ' 1 t t ' n 1 is a precycle of length n in our sense; and if ' is a precycle of length n in our sense, then f' ' i .B/ W i D 0; : : : ; n 2g is a pre-n-cycle in the sense of [15, Definition 8].The reason for this change of terminology will become apparent in the statement of the next lemma, which was proved for U D U ' in [15, Proposition 10].
2 is an element ' 2 OEOER such that dom.'/\rng.'/D ∅.If ' is a precycle of length n, then .supp'/ D n .B/ and .dom'/ D .n1/ .B/.Geometry & Topology, Volume 27 (2023) Lemma 3.7 Suppose that ' is a precycle of basis B, let WD ' B , and suppose that U 2 Aut.X; / extends '.Then OER ' is contained in the closure of the group generated by OER [ fU g.Proof Let n be the length of '.For i D 0; : : : ; n 2, let i D ' ' i .B/ .Then we have R ' D W n 2 iD0 R i .Since U extends ', we have U i U 1 D iC1 for all i D 0; : : : ; n 3, and hence U OER i U 1 D OER i C1 .Since 0 D , the group generated by U [ OER contains OER i for all i D 0; : : : ; n 2. Theorem 3.1 finishes the proof.
3 (defining the action) Set C WD X n Y .Consider a partition C D F n 1 C n , where .C n / > 0 for every n.We are going to define an amplified version of the action n on C n as follows.For each n 1, we take a measurable partitionC n D F / for every g 2 G 0 n .Set B n WD B nn .Using ergodicity of R, for every g 2 G 0 n nf n g we choose g W B n !B g n in the pseudo full group OEOER of R. In this way we obtain an action ˛n of F r , defined on C n by the formula if x 2 B g 0 n and n ./g 0 D g 1 ; then ˛n./x WD g 1 1 g 0 .x/;andtrivial on X n C n .Thus, ˛n.F r / Ä OER lC n .