Two new families of finitely generated simple groups of homeomorphisms of the real line
aa r X i v : . [ m a t h . G R ] F e b TWO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OFHOMEOMORPHISMS OF THE REAL LINE
JAMES HYDE, YASH LODHA, AND CRISTÓBAL RIVASAbstract. The goal of this article is to exhibit two new families of finitely gener-ated simple groups of homeomorphisms of R . These families are strikingly dif-ferent from existing families owing to the nature of their actions on R , and ex-hibit surprising algebraic and dynamical features. In particular, one construc-tion provides the first examples of finitely generated simple groups of homeo-morphisms of the real line which also admit a minimal action by homeomor-phisms on the circle. This provides new examples of finitely generated simplegroups with infinite commutator width, and the first such left orderable ex-amples. Another construction provides the first examples of finitely generatedsimple left orderable groups that admit minimal actions by homeomorphismson the torus. Introduction
In [7] the first two authors constructed the first examples of finitely generatedinfinite simple groups of homeomorphisms of R . In a subsequent article [12],Matte Bon and Triestino demonstrated that certain groups of piecewise linearhomeomorphisms of flows are also examples of this phenomenon. Whethersuch groups exist had been a longstanding open question of Rhemtulla [11](also asked by Clay and Rolfsen in [5], by Navas in [14], and in the “KourkovaNotebook" [9].)The question as stated originally asks whether finitely generated simple left or-derable groups exist. However, note that left orderability for countable groupsis equivalent to requiring that they admit a faithful action by orientation pre-serving homeomorphisms of the real line. Such constructions are difficult sinceachieving the combination of finite generation and simplicity presents certaintechnical challenges owing to the lack of compactness of R . Moreover, there alsocertain natural obstructions to simplicity for various finitely generated groupsof homeomorphisms of R . If such a group is amenable, then it admits a homo-morphism onto Z (see [16]). The same holds if the group admits a nontrivial Yash Lodha is supported by the Samsung Science and Technology Foundation under ProjectNumber SSTF-BA1301-51 and by a KIAS Individual Grant at the Korea Institute for AdvancedStudy. action by C -diffeomorphisms on a closed interval (or even [
0, 1 ) , see [15]). Fora more detailed discussion around these issues, we refer the reader to [7].The goal of this article is to exhibit two new families of examples that exhibitnew, strikingly different dynamical and algebraic features, compared to existingfamilies. Groups in the first family are finitely generated by definition, howeverit is surprising that they are simple, and the proof of simplicity involves an in-tricate analysis of the group action. The groups in the second family emerge asthe derived subgroups of certain well known examples called fast n -ring groups (defined independently by Brin, Bleak, Kassabov, Moore and Zaremsky in [2]and by the second author with Kim and Koberda in [10].). The simplicity ofthese examples is less surprising, however it is surprising that they are finitelygenerated and left orderable, the proof of finite generation involves an intricateanalysis of the group action.We now present the first family. Recall that Thompson’s group T is the groupof piecewise linear orientation preserving homeomorphisms of the circle S = R / Z such that:(1) Each linear part is of the form n + d for n ∈ Z , d ∈ Z [ ] / Z .(2) There are finitely many points where the slopes do not exist, and theylie in Z [ ] .The group T <
Homeo + ( R ) is the “lift" of this action to the real line. In partic-ular, there is a short exact sequence → Z → T → T → Here the group Z is the group of integer translations of the real line, and it liesin the center of T . It is easily seen that T is finitely presented, since T is finitelypresented. The group T was first studied by Ghys and Sergiescu in [6], and ithas several remarkable features. For instance, it was observed recently by thefirst author with Belk and Matucci in [1] that it contains an isomorphic copyof Q , and is the first example of an “explicit" finitely presented group with thisproperty.One may modify the “lift", T , as follows. Let S be as above, and consider themap φ λ : R → S R → R /λ Z for each λ > 0 . The map φ λ provides the alternative lift T λ < Homeo + ( R ) ,which as an abstract group is isomorphic to T = T . Note that the center of T λ is the group h t → t + nλ | n ∈ Z i . In spite of the fact that T , T λ are not simple,we prove the following: Theorem 1.0.1.
Let λ > 1 be irrational. The group G λ = h T , T λ i is simple. WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 3
This provides a family of finitely generated simple subgroups of homeomor-phisms of the real line which are very elementary to define. Moreover, they areshown to admit minimal actions on the torus by homeomorphisms.
Corollary 1.0.2.
There exist finitely generated simple groups of homeomorphisms ofthe real line that admit a minimal action by homeomorphisms on the torus.
To describe the second family, we recall the notion of a fast n -ring group . Definition 1.0.3.
Let { J , ..., J n } be a set of connected open intervals in S thatcover S , and homeomorphisms { f , ..., f n } that satisfy:(1) J i ∩ J j = ∅ if | i − j | > 1 ( mod n ) and is a nonempty, proper, connectedsubinterval of both J i , J j if | i − j | = ( mod n ) .(2) J i = Supp ( f i ) = { x ∈ S | x · f = x } for each ≤ i ≤ n .The aforementioned configuration is called an n -ring of intervals and homeomor-phisms . The group G n = h f , ..., f n i is said to be a fast n -ring group if the followingholds. In what appears below, we interpret the subscripts as modulo n . For each ≤ i ≤ n , let x i be the endpoint of J i + that lies in J i . Then we have the followingdynamical condition which we refer to throughout the article as ( ∗ ) : x i · f i f i + ...f i + l ∈ J i + l + ∀ ≤ l ≤ n It was demonstrated in [2] that the isomorphism type of G n does not dependon the choice of homeomorphisms f , ..., f n , provided the dynamical condition ( ∗ ) is satisfied. Note that an elementary application of the classical ping ponglemma demonstrates that the group G is the nonabelian free group of rank . The isomorphism type of G n for n ≥ remains mysterious, however. Oursecond family emerges from the derived subgroups of these examples. Theorem 1.0.4.
For each n ≥ , the group H n = G ′ n is finitely generated, simple andleft orderable. In fact, the prescribed action of H n on S lifts to a faithful action of H n on R . To provide another dynamical motivation for this second family, we recall thefollowing dynamical trichotomy for groups actions on the real line. For every ac-tion of a finitely generated group G by orientation preserving homeomorphismsof the real line without global fixed points, there are one of three possibilities:(i) There is a σ -finite measure µ that is invariant under the action.(ii) The action is semiconjugate to a minimal action for which every smallenough interval is sent into a sequence of intervals that converge to apoint under well chosen group elements, however, this property doesnot hold for every bounded interval. JAMES HYDE, YASH LODHA, AND CRISTÓBAL RIVAS (iii) The action is globally contracting; more precisely, it is semiconjugate toa minimal one for which the contraction property above holds for allbounded intervals.(For details, we refer the reader to [13]). Note that if a group admits a faithfulaction of type ( i ) , then it is indicable : it admits a homomorphism onto Z . There-fore, finitely generated simple groups of homeomorphisms of the real line mayonly admit actions of type ( ii ) or ( iii ) . It was shown in [8] that the groups G ρ constructed by the first two authors in [7] have the property that every actionon the real line by homeomorphisms without global fixed points is of type ( iii ) .The same was shown by Matte Bon and Triestino for their examples in [12].The following is a corollary of Theorem 1.0.4, which illustrates a striking newphenomenon associated with the groups H n , n ≥ . Corollary 1.0.5.
There exist finitely generated simple left orderable groups which admitactions by orientation preserving homeomorphisms on the real line which are of type ( ii ) . Given a group that admits an action of type ( ii ) on R , it is easy to see that theaction of the group on the orbit of provides an unbounded homogenous quasi-morphism. As a consequence, we have the following. Corollary 1.0.6.
There exist finitely generated simple left orderable groups that admitnon-trivial (unbounded) homogeneous quasimorphisms into the reals.
This also has a nice algebraic consequence.
Corollary 1.0.7.
There exist finitely generated simple left orderable groups that haveinfinite commutator width: for each n ∈ N there is an element that cannot be expressedas a product of fewer than n commutators. Note that the homeomorphisms that generate the group G n can be realised aselements of Thompson’s group T . It follows that the groups H n are subgroupsof T . It was shown in [6] that the given action of T on S is conjugate to an actionon S by C ∞ -diffeomorphisms.We conclude the following. Corollary 1.0.8.
There exists a finitely generated simple left orderable group that ad-mits: (1) a faithful action by C ∞ -diffeomorphisms of the circle. (2) a faithful action by piecewise linear homeomorphisms of the circle (with finitelymany allowable breakpoints for each element). (3) an embedding into Thompson’s group T . WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 5
Note that one may show ( ) directly without appealing to [6], since we cansimply choose the homeomorphisms that generate G n to be smooth. Convention 1.0.9.
In this article, all group actions will be right actions. We willuse the notation [ f, g ] = fgf − g − and f g = g − fg . For f ∈ Homeo + ( R ) , wedefine Supp ( f ) = { x ∈ R | x · f = x } .2. The first family
The goal of this section is to prove Theorem 1.0.1. We first state and discuss afew preliminaries.2.1.
Preliminaries.
Throughout the section we denote by F the standard piece-wise linear action of Thompson’s group F ≤ Homeo + [
0, 1 ] . Also, we fix λ ∈ R \ Q , λ > 1 . Recall that G λ = h T , T λ i where T λ < Homeo + ( R ) is the lift of theaction of T on S with the identification R → R /λ Z . We denote by Z ( T ) , Z ( T λ ) as the center of T , T λ respectively. Note that Z ( T ) = h t → t + n | n ∈ Z i Z ( T λ ) = h t → t + nλ | n ∈ Z i Recall that the pointwise stabilizer of Z in T , which we denote by F , is naturallyisomorphic to Thompson’s group F . Indeed the restriction of this action of F to each interval [ n, n + ] for each n ∈ Z is conjugate to the standard piecewiselinear action of F on [
0, 1 ] by the translation t → t + n . It is easy to see that forany element f ∈ T \ Z ( T ) , hh f ii = T . Note that the analogous statement holdsfor T λ .We observe the following. Lemma 2.1.1.
Every g ∈ G λ is Lipshitz. In particular, g is uniformly continuous.Proof. It is straightforward to see that the elements of
T , T λ are Lipshitz. Sincethis property for homeomorphisms is closed under composition and inverses,we are done. (cid:3) The proof.
The key idea in the proof of Theorem 1.0.1 is the following.
Proposition 2.2.1.
Let f ∈ G λ \ { id } . For each c ∈ {
1, λ } , there is an element g ∈ hh f ii that satisfies the following. (1) x · g = x for all x ∈ c · Z . (2) There exists a pair x, y ∈ [
0, c ] , x < y such that ( x + c · n ) · g > y + c · n ∀ n ∈ Z JAMES HYDE, YASH LODHA, AND CRISTÓBAL RIVAS
Using Proposition 2.2.1, we can finish the proof of Theorem 1.0.1 as follows.
Proof of Theorem 1.0.1.
Let g , g ∈ G λ \ { id } be elements that satisfy the conclu-sion of Proposition 2.2.1 for c =
1, c = λ , respectively. We will show that:(1) hh g ii G λ ∩ ( T \ Z ( T )) = ∅ .(2) hh g ii G λ ∩ ( T λ \ Z ( T λ )) = ∅ .We know that the normal closure of any element in ( T \ Z ( T )) is all of T . Similarly,the normal closure of any element in ( T λ \ Z ( T λ )) is all of T λ . So after showingthe above we can conclude the proof. Indeed, since the proofs for ( ) , ( ) areanalogous, we shall just prove ( ) .Thanks to Lemma 2.1.1, g is Lipshitz. Combining this with the fact that thereexists a pair x, y ∈ [
0, 1 ] , x < y such that ( x + n ) · g > y + n ∀ n ∈ Z we obtain the following. There is an interval I ⊂ (
0, 1 ) such that x ∈ I and X · g ∩ X = ∅ where X = ( [ n ∈ Z ( I + n )) We can find a nontrivial element h = [ h , h ] h, h , h ∈ F ≤ T such that Supp ( h ) , Supp ( h ) , Supp ( h ) ⊂ X It follows that [ h , [ h , g − ]] = [ h , h ] = h ∈ F ≤ ( T \ Z ( T )) finishing the proof. (cid:3) The rest of this section shall be devoted to proving Proposition 2.2.1. We willprove it for the case c = , the other case is completely analogous. Definition 2.2.2.
Let
G <
Homeo + ( R ) be a given group action. Given compactintervals I, J such that | I | = | J | , we denote by T J,I : R → R the unique translationso that T J,I ( J ) = I . Given g, g , g ∈ G and I, J as above, we define d g ( I, J ) = sup { ( | x · g − x · h | | x ∈ I } where h = T I,J ◦ g ◦ T J,I and d g ,g ( I ) = sup { ( | x · g − x · g | | x ∈ I } WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 7
Lemma 2.2.3.
Consider an element g = u v ...u n v n ∈ G λ u i ∈ T , v i ∈ T λ for each ≤ i ≤ n For each ǫ > 0 , there is a δ > 0 such that for each δ ∈ (− δ , δ ) , the element g = u ( f − v f δ ) ...u n ( f − v n f δ ) where x · f δ = x + δ satisfies that d g ,g ([
0, 1 ]) < ǫ .Proof. This follows from an elementary inductive argument on n , using conti-nuity. (cid:3) The following is a basic dynamical fact about irrational translations, and weleave the proof for the reader.
Lemma 2.2.4.
Fix λ ∈ R \ Q , λ > 1 . For each ǫ > 0 , there is an N ∈ N such that forany interval I such that | I | > N , there are m, k ∈ Z such that [ m, m + ] ⊂ I | m − kλ | < ǫ Definition 2.2.5.
An element g ∈ Homeo + ( R ) is repetitive if for each ε > 0 thereis an N ∈ N such that for each interval I such that | I | > N , there is a subinterval [ m, m + ] ⊂ I, m ∈ Z such that d g ([
0, 1 ] , [ m, m + ]) < ǫ We say that a group action G ≤ Homeo + ( R ) is said to be repetitive , if every g ∈ G is repetitive. Proposition 2.2.6. G λ is repetitive.Proof. Consider a nontrivial element g = u v . . . u n v n ∈ G λ , where u i ∈ T and v i ∈ T λ . We will show that g is repetitive. Let ǫ > 0 . Applying Lemma 2.2.3,there is a δ > 0 such that for each δ ∈ (− δ , δ ) , the element g δ = u ( t − v t δ ) ...u n ( t − v n t δ ) satisfies that d g,g δ ([
0, 1 ]) < ǫ .Using Lemma 2.2.4 we find an N ∈ N such that in every interval I of length atleast N there are m, k ∈ Z such that [ m, m + ] ⊂ I and | m − kλ | < δ . For such m, k there is a δ ∈ (− δ , δ ) such that g ↾ [ m, m + ] = f − g δ f m where t · f m = t + m Combining this with the fact that d g,g δ ([
0, 1 ]) < ǫ , we obtain d g ([
0, 1 ] , [ m, m + ]) < ǫ (cid:3) JAMES HYDE, YASH LODHA, AND CRISTÓBAL RIVAS
For the rest of the section, we denote hh g ii G λ as simply hh g ii . We shall now focusour attention on the pointwise stabilizer of Z in G λ . Recall that the pointwisestabilizer of Z in T is F , defined in the preliminaries above. Proposition 2.2.7.
Let g ∈ G λ \ { id } . There is an open interval J ⊂ (
0, 1 ) whoseclosure is also contained in (
0, 1 ) , and a nontrivial element f ∈ hh g ii such that: Supp ( f ) ⊂ [ n ∈ Z ( n + J ) Proof.
First we argue that hh g ii must contain elements that do not lie in h t → t + n | n ∈ Z i . If g is itself not in this subgroup, then we are done. Otherwise,we can find an element f ∈ T λ such that the element [ g, f ] = id and satisfies therequired property.We assume in the rest of the proof that g / ∈ h t → t + n | n ∈ Z i . It follows thatthere is an m ∈ Z and an open interval J such that J ⊂ ( m, m + ) and g ↾ J isnot the restriction of an integer translation.We choose a open subinterval I ⊂ J such that for some ǫ > 0 it holds that | inf ( I ) − inf ( J ) | , | sup ( J ) − sup ( I ) | > ǫ Since g is Lipshitz, for ǫ > 0 , there is a δ > 0 such that given an interval I , if | I | < δ , then | I · g | < ǫ . We then choose an open interval I ⊂ I such that | I | < δ and ( I · g ) ∩ [ n ∈ Z ( n + I ) = ∅ From our assumption, we know that for any n , n ∈ Z , if ( n + I ) · g ∩ ( n + I ) = ∅ ,then ( n + I ) · g ⊂ ( n + J ) .Let f = [ f , f ] ∈ F be such that Supp ( f ) , Supp ( f ) , Supp ( f ) ⊂ [ n ∈ Z ( n + I ) Then the element h = [ f , [ f , g − ]] is the required element that satisfies the con-clusion of the Proposition. That is, Supp ( h ) ⊂ [ n ∈ Z ( n + J ) (cid:3) We define a map ν : F → T as the obvious extension of the natural map ν : F → F ≤ T . For an open interval I ⊂ (
0, 1 ) and N ∈ N , an element f ∈ G λ is said tobe ( I, N ) -stable , if the following holds.(1) Supp ( f ) ⊂ S n ∈ Z ( I + n ) . WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 9 (2) There is an element g ∈ F \ { id } such that for each each interval I, | I | > N ,there is an interval [ m, m + ] ⊂ I for m ∈ Z , such that ν ( g ) ↾ [ m, m + ] = f ↾ [ m, m + ] .Note that the element f that emerges in the conclusion of Proposition 2.2.7 sat-isfies the first of the two conditions above. We show the following. Proposition 2.2.8.
Let I ⊂ (
0, 1 ) be an open interval whose closure is also containedin (
0, 1 ) , and let f ∈ G λ \ { id } such that: Supp ( f ) ⊂ [ n ∈ Z ( n + I ) Then there is a nontrivial element h ∈ hh f ii and an N ∈ N such that h is ( I, N ) -regular.Proof. Assume without loss of generality that f ↾ [
0, 1 ] is nontrivial. (Otherwise,we may replace f with a conjugate of f by an integer translation and proceed.)Let J ⊂ I ⊂ (
0, 1 ) be an open interval such that either sup ( J · f ) < inf ( J ) or sup ( J ) < inf ( J · f ) . It follows that there is an ǫ > 0 such that for any g ∈ Homeo + [
0, 1 ] such that d g,f ([
0, 1 ]) < ǫ , we have that J · g ∩ J = ∅ .Since f is repetitive, there is an N ∈ N such that for each interval I, | I | > N thereis an interval [ k, k + ] ⊂ I for k ∈ Z , such that d f ([
0, 1 ] , [ k, k + ]) < ǫ Let g = [ g , g ] , g , g ∈ F be non trivial elements such that Supp ( g ) , Supp ( g ) , Supp ( g ) ⊂ J Then the element h = [ ν ( g ) , [ ν ( g ) , f − ]] ∈ hh f ii has the property that for each interval I, | I | > N there is an interval [ k, k + ] ⊂ I for k ∈ Z such that ν ( g ) ↾ [ k, k + ] = h ↾ [ k, k + ] Moreover,
Supp ( h ) ⊂ [ k ∈ Z ( I + k ) This finishes the proof. (cid:3)
An element f ∈ Homeo + ( R ) is said to be ǫ -advancing for some ǫ > 0 , if foreach x ∈ R we have that x · f ≥ x − ǫ . (The analagous notion is defined for ahomeomorphism of a compact interval). Lemma 2.2.9.
Let α ∈ Homeo + [
0, 1 ] \ { id } such that Supp ( α ) ⊆ I for an openinterval I ⊂ (
0, 1 ) whose closure is also contained in (
0, 1 ) . For any ǫ > 0 and x, y ∈ (
0, 1 ) , x < y , there exist elements g , ..., g n ∈ F, n ∈ N such that: (1) The element h = ( α g ) l ... ( α g n ) l n satisfies that x · h > y for some l , ..., l n ∈ Z . (2) For any homeomorphism β ∈ Homeo + [
0, 1 ] , Supp ( β ) ⊆ I (where I is thesame interval as above), we have that ( β g ) l ... ( β g n ) l n is ǫ -advancing for any l , ..., l n ∈ Z .Proof. Recall that the action of F on (
0, 1 ) has the following two features, whichwe shall use. The first is that for any pair of closed intervals I , I ⊂ (
0, 1 ) withdyadic rational endpoints, there is an element f ∈ F such that I · f = I . Thesecond is that for any triple of closed intervals I ⊂ I ⊂ I ⊂ (
0, 1 ) with dyadicrationals as endpoints, we can find an f ∈ F such that Supp ( f ) ⊂ I and that I ⊂ I · f . We use these features to construct elements g , ..., g n ∈ F such that J = I · g ... J n = I · g n are intervals satisfying:(1) For each ≤ i ≤ n − , we have inf ( J i ) < inf ( J i + ) < sup ( J i ) < sup ( J i + ) sup ( J i ) < inf ( J i + ) (for i < n − ) (2) For each ≤ i ≤ n , | J i | < ǫ .(3) x ∈ J , y ∈ J n .Moreover, we choose g , ..., g n ∈ F that also satisfy that for some l , ..., l n ∈ Z we have x · ( α g ) l ... ( α g n ) l n > y Note that condition ( ) above guarantees that for any β ∈ Homeo + [
0, 1 ] suchthat Supp ( β ) ⊆ I , we have that ( β g ) l ... ( β g n ) l n is ǫ -advancing for any l , ..., l n ∈ Z . (cid:3) Proof of Proposition 2.2.1.
Let g ∈ G λ \ { id } . By combining Propositions 2.2.7 and2.2.8, we obtain an element h ∈ hh g ii , an open interval I ⊂ (
0, 1 ) whose closureis also contained in (
0, 1 ) , and an N ∈ N such that h is ( I, N ) -regular. Replacing h by a conjugate of an integer translation if necessary, we assume that for [
0, N ] ,the interval [
0, 1 ] realizes condition ( ) of the definition of ( I, N ) -regular.We apply Lemma 2.2.9 to α = h ↾ [
0, 1 ] for ǫ = , x = , y = to obtainelements g , ..., g n ∈ F, n ∈ N such that:(1) The element γ = ( α g ) l ... ( α g n ) l n satisfies that · γ > for some l , ..., l n ∈ Z .(2) For any homeomorphism β ∈ Homeo + [
0, 1 ] , Supp ( β ) ⊆ I (where I isthe same interval as above), we have that ( β g ) l ... ( β g n ) l n is ǫ -advancingfor any l , ..., l n ∈ Z . WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 11
Let ζ = ( h ν ( g ) ) l ... ( h ν ( g n ) ) l n and ζ = Y ≤ k ≤ N,k ∈ Z f − ζ f n where t · f k = k + Then ζ is the required element that satisfies the conditions of Proposition 2.2.1for c = , that is:(1) x · ζ = x for all x ∈ Z .(2) And ( + n ) · ζ > 12 + n ∀ n ∈ Z (cid:3) Proof of Corollary 1.0.2.
We shall define an action of G λ on R that commuteswith the natural action of Z on R , and preserves a lamination on R . We definehomomorphisms η , η : T → Homeo + ( R ) as follows. ( x, y ) · η ( g ) = ( x + ( y · g − y ) , y · g )( x, y ) · η ( g ) = ( x · g, y + λ ( x · g − x )) Note that here x · g, y · g refers to the action of T on R . The group generated by η ( T ) , η ( T ) provides a homomorphism η : T ∗ T → Homeo + ( R ) whose imageis G λ . It is an easy exercise to check that this preserves the foliation obtained bylines that have an angle θ with the x -axis, where tan ( θ ) = λ , and the action onthe leaves is conjugate to the original action on R . (cid:3) The second family
Our goal in this section will be to prove Theorem 1.0.4.3.1.
Preliminaries.
We recall from the introduction the n -ring configuration ofintervals { J , ..., J n } and homeomorphisms { f , ..., f n } that satisfy the dynamicalcondition ( ∗ ) . Note that if the above is satisfied for some ≤ i ≤ n , then it issatisfied for all ≤ i ≤ n . As before, we denote the resulting group, called thefast n -ring group, as G n = h f , ..., f n i . The following was proved in [2]. Theorem 3.1.1.
Given an n -ring configuration of intervals and homeomorphisms thatsatisfies condition ( ∗ ) , the (marked) isomorphism type of the group G n (with generatingset { f , ..., f n } ) does not depend on the choice of homeomorphisms f , ..., f n . First, recall the following well known result (Theorem in [15]).
Theorem 3.1.2. If G <
Homeo + ( S ) then precisely one of the following holds: (1) There is a finite orbit. (2)
All the orbits are dense. (3)
There exists a copy of the cantor set C ⊂ S , which is G -invariant, and suchthat G ↾ C is minimal. In this case, the given action is semiconjugate to aminimal action, i.e. there is a degree one continuous map Φ : S → S and agroup homomorphism ψ : G → H onto some group H such that ∀ f ∈ G Φ ◦ f = ψ ( f ) ◦ Φ The resulting minimal action of H on S is called the minimalisation of theaction of G . Remark 3.1.3.
Note that in case , the minimalisation is an action of H , but since H is a quotient of G (possibly with trivial kernel), it may be viewed as an actionof G . In some cases (for instance if G is the fast n -ring group), one can showthat φ : G → H must be an isomorphism.Given a group of homeomorphisms of the circle, we say that the action is prox-imal , if for every interval I ⊂ S such that S \ I has nonempty interior, and anyopen set J ⊂ S , there is an element f in the group such that I · f ⊂ J .3.2. The proof of Theorem 1.0.4.
Observe that while a given action of G n maynot be minimal, by part ( ) of Theorem 3.1.2, it is semiconjugate to a minimalaction on S . (Clearly the group action has no finite orbit.) It is clear that thedynamical condition ( ∗ ) holds for the new minimal action as well. Since thisdynamical condition guarantees a stable isomorphism type (by Theorem 3.1.1),it follows that this new minimal action of G n is also faithful. Actually, the mainTheorem of [2] in fact guarantees that we can choose the homeomorphisms f , ..., f n satisfying the dynamical condition ( ∗ ) such that the action of G n on S is minimal. Therefore, for the rest of this section we shall assume that the actionof G n on S is minimal. We denote as before H n = G ′ n and assume that n ≥ . Lemma 3.2.1.
The action of H n on S is proximal.Proof. First we shall prove that the action of G n on S is proximal. Let I ⊂ S such that S \ I has nonempty interior, and consider an open set J ⊂ S .By minimality, we find an element g ∈ G n such that inf ( J ) · g ⊂ J . By con-tinuity, there is an open interval I containing inf ( J ) such that I · g ⊂ J . It isan elementary exercise using the definition of f , ..., f n , and minimality, to con-struct an element g ∈ G n such that I · g ⊂ Supp ( f ) . There is an n ∈ Z suchthat I · g · f n1 ⊂ I . It follows that I · g f n1 g ⊂ J , finishing the proof. WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 13
First we show that the action of H n is minimal. Using proximality of G n as above,let g i ∈ G n be an element such that J i · g i ⊂ J ci . We define the elements l i = g − f − g i f i Clearly, l i ∈ H n and l i ↾ Supp ( f i ) = f i ↾ Supp ( f i ) . It follows that the orbits of theactions of H n , G n are the same, hence the action of H n is minimal. We show thatthe action of H n is proximal in a similar way as done above for G n , replacing theelements f i by l i . (Note that one may construct l i above so that Supp ( l i ) ∩ J ci liesin any given open interval J ⊂ J ci .) (cid:3) Proposition 3.2.2. H n is simple.Proof. To prove simplicity, we must show that hh g ii H n = H n for an arbitrary g ∈ H n \ { id } . First we show that hh g ii H n contains a nontrivial element f suchthat Supp ( f ) c has nonempty interior. Let J be an open interval such that J ∩ ( J · g ) = ∅ and ( J ∪ ( J · g )) c has nonempty interior. It is easy to find an element γ ∈ H n such that Supp ( γ ) c has nonempty interior. (Take [ f , f ] , for instance.)Using proximality, we find an element h ∈ H n such that Supp ( γ ) · h ⊂ J . Theelement f = [ g − , γ h ] has the feature that f ∈ hh g ii H n \ { id } and that Supp ( f ) c has nonempty interior.Since hh f ii H n ≤ hh g ii H n , showing that hh f ii H n = H n finishes the proof. It sufficesto show that [ f i , f j ] h = [ f hi , f hj ] ∈ hh f ii H n for each ≤ i, j ≤ n and h ∈ G n .We denote β = f hi , β = f hj and K = Supp ( β ) , K = Supp ( β ) . It is easy to seethat ( K ∪ K ) c has nonempty interior since Supp ([ f i , f j ]) c has the same feature.Let I , I , I be disjoint open intervals such that I · f ∩ I = ∅ I ∪ I ⊂ Supp ( f ) c Using proximality, we find g , g , g ∈ H n such that ( K ∪ K ) · g ⊂ I K · g ⊂ I K · g ⊂ I Let α = [ g − , β ][ β , g − ] = β g ( β − ) g α = [ g − , β ][ β , g − ] = β g ( β − ) g Note that α , α ∈ H n .We obtain [ β , β ] = [ α , [ α , f − ]] g − ∈ hh f ii H n This proves our claim. (cid:3)
Definition 3.2.3.
We say that an open interval I ⊂ S is small if there exists ≤ k ≤ n such that I ⊂ J k . We find a collection of pairwise disjoint smallintervals I = { L i,j ⊂ S | ≤ i, j ≤ n } satisfying that L i,j ⊂ int ( S \ J j ) ≤ i, j ≤ n We say that a small open interval I ⊂ S is I -small if for each ≤ i, j ≤ n thefollowing holds:(1) If I ∩ L i,j = ∅ then I ∩ J j = ∅ .(2) If I ∩ J j = ∅ , then I ∩ L i,j = ∅ .It is an easy exercise to show that for any such I , there is an ǫ > 0 such that anyinterval I with | I | < ǫ is I -small. More generally, we say that a subset of S is I -small, if it is contained in a I -small interval.Using proximality from Lemma 3.2.1, we construct a set of n elements { λ i,j | ≤ i, j ≤ n } ⊂ G n satisfying that J i · λ i,j ⊂ L i,j ∀ ≤ i, j ≤ n We define ν i,j = λ − f i λ i,j . Note that Supp ( ν i,j ) ⊆ L i,j . Since the intervals { L i,j | ≤ i, j ≤ n } are pairwise disjoint, the elements of the set { ν i,j | ≤ i, j ≤ n } generate a free abelian group of rank n .We define the set X = { ν − f i | ≤ i, j ≤ n } ⊂ H n Observe that { ν i,j ν − | ≤ i, j , j ≤ n } ⊂ h X i since ν i,j ν − = ( ν − f i )( ν − f i ) − . Also, observe that ( f i ν − ) ∈ h X i since ( ν − ν i,j )( ν − f i )( ν i,i ν − ) = ( ν − f i )( ν i,i ν − ) = ( f i ν − )( ν i,i ν − ) = ( f i ν − ) An element of the form fν − ∈ h X i is called a special element if Supp ( f ) ∩ Supp ( ν − ) = ∅ and Supp ( f ) is I -small. Lemma 3.2.4.
Let fν − ∈ h X i be a special element and let g ∈ { f ± , ..., f ± } be suchthat Supp ( f g ) is I -small. Then there is a ≤ k ≤ n such that f g ν − is also a specialelement of h X i . WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 15
Proof.
Assume that g = f l . The proof where g = f − is similar. If Supp ( f ) ∩ Supp ( f l ) = ∅ , then f g ν − = fν − and we are done. If Supp ( f ) ∩ Supp ( f l ) = ∅ ,then we consider the special element fν − = ( fν − )( ν i,j ν − ) ∈ h X i Note that this is a special element since
Supp ( f ) is I -small, and Supp ( f ) ∩ Supp ( f l ) = ∅ , hence Supp ( f ) ∩ Supp ( ν − ) = ∅ . It follows that ( fν − ) f l ν − = f f l ν − ν − = ( f ν − ) f l ν − = f f l ν − ∈ h X i is a special element. (cid:3) Lemma 3.2.5.
Let J ⊂ S be an open interval such that J c has nonempty interior. Foreach ≤ i ≤ n, s ∈ { ± } , there exists an element γ ∈ h X i such that γ ↾ J = f si ↾ J .Proof. We fix i ∈ {
1, ..., n } and s = − (the proof for s = + is similar). Considerthe element ν i,j for some ≤ j ≤ n . We know that Supp ( ν i,j ) ⊂ J k for some ≤ k ≤ n . Consider the element ν i,j ν − ∈ h X i .By minimality of the action of G n on S , we can find an element g = g ...g m for g i ∈ { f , ± , ..., f ± } such that inf ( J k ) · g ⊂ J c . By continuity, we find a I -smallopen interval I containing inf ( J k ) such that:(1) I · g ⊂ J c .(2) For each ≤ s ≤ m , the interval I · g ...g s is I -small.We let h = f k ν − ∈ h X i . It follows that for some large l ∈ N the element ( ν i,j ν − ) h l = ν f lk i,j ν − ∈ h X i has the property that if γ = ν f lk i,j , then Supp ( γ ) ⊂ I and γν − is a special element.Note that this means that for each ≤ s ≤ m , Supp ( γ ) · g ...g s is I -small, hence Supp ( γ g ...g s ) is I -small.Applying Lemma 3.2.4 to γν − , we conclude that there is a ≤ k ≤ n such that γ g ν − ∈ h X i is a special element. Proceeding inductively, applying Lemma3.2.4 each time, we find ≤ k , ..., k m ≤ n such that γ g ...g s ν − s ∈ h X i is a special element for ≤ s ≤ m In particular, γ g ν − m ∈ h X i and ( γ g ν − m )( ν i,k m f − ) = γ g f − ∈ h X i is an elementwhich satisfies the conclusion of the Lemma, since Supp ( γ g ) ⊂ J c . (cid:3) Now we prove our main theorem.
Theorem 3.2.6. H n = h X i , and hence it is finitely generated.Proof. Let f ∈ h X i be any nontrivial element such that Supp ( f ) c has nonemptyinterior. Since H n is simple, hh f ii H n = H n . To prove the claim, it suffices to showthat for any element g ∈ H n , we have that ( f ± ) g ∈ h X i . We proceed by inductionon the word length of g . The base case is trivial. Now assume that g = hf l forsome ≤ l ≤ n , and that by the induction hypothesis f h ∈ h X i . The proof forthe case g = hf − shall be similar.Let J = Supp ( f h ) . Note that J c has nonempty interior. Applying Lemma 3.2.5,there exists an element γ ∈ h X i such that γ ↾ J = f l . We obtain that f hf l = f hγ ∈h X i . (cid:3) We finish the proof of Theorem 1.0.4 by proving the following.
Proposition 3.2.7. G n is left orderable.Proof. Recall that a group is left orderable if and only if it admits a faithful ac-tion by orientation preserving homeomorphisms on the real line. Recall thatHomeo + ( S ) admits a lift to R as follows: → Z → ^ Homeo + ( S ) → Homeo + ( S ) → where Z = h t → t + n | n ∈ Z i and ^ Homeo + ( S ) is the centralizer of h t → t + n | n ∈ Z i in Homeo + ( R ) .We claim that the lift f G n of G n to a subgroup of Homeo + ( R ) is isomorphic to G n . Denote the lifts of the generators f i as h i , for each ≤ i ≤ n . It suffices toshow that f G n ∩ h t → t + n | n ∈ Z i = { id R } If this were not the case, then we can find a word g = g ...g m g i ∈ { h , ..., h n } with the property that x · g = x + n for n ∈ Z \ { } . The corresponding word γ = γ ...γ m , where each h ± is replaced by f ± , must satisfy the following condition.For any point in x ∈ S , the sequence x, x · γ , x · γ γ , ..., x · γ ...γ m = x must "go around the circle" a nontrivial number of times to arrive back to it-self. We will show that this is impossible, i.e. the above can only happen with"backtracking".Let x = inf ( J ) . Consider the sequence x, x · γ , x · γ γ , ..., x · γ ...γ m = x WO NEW FAMILIES OF FINITELY GENERATED SIMPLE GROUPS OF HOMEOMORPHISMS 17 If x · γ ...γ i = x · γ ...γ i + we delete the occurrence of γ i + from the word γ ...γ m , and adjust the indices(replacing j by j − for j > i + ) to obtain a new word γ ...γ m − . Whenever wefind backtracking , i.e. x · γ ...γ i − = x · γ ...γ i + in γ ...γ m , we remove γ i γ i + from the word, and adjust the indices (replacing j by j − for j > i + ) to obtain a new word γ ...γ m − . At the end of this process,we obtain a new word γ ...γ k such that x, x · γ , x · γ γ , ..., x · γ ...γ k = x such that for each ≤ i < k x · γ ...γ i = x · γ ...γ i + and for each ≤ i < k − · γ ...γ i = x · γ ...γ i + In this situation, an elementary inductive argument using condition ( ∗ ) impliesthat x · γ ...γ k / ∈ { inf ( J i ) , sup ( J i ) | ≤ i ≤ n } contradicting our assumption that x = inf ( J ) . (cid:3) References [1] J ames Belk, James Hyde, Francesco Matucci Embeddings of Q into Some FinitelyPresented Groups. arXiv:2005.02036[2] Collin Bleak, Matthew G. Brin, Martin Kassabov, Justin Tatch Moore andMatthew C.B. Zaremsky.
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Malott Hall, Room 590, Cornell University, Ithaca NY
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