Hindmans Theorem, Elliss Lemma, and Thompsons group $F$

HINDMAN’S THEORE M, ELLI S’S LEMMA, AND THOMPSON’S GROU P F JUSTIN T A TCH MOORE Abstract. The purpose of this article is to formulat e conjectural general- izations of Hindman’s Theorem and Ellis’ s Lemma for nonassociative binary systems and relate them to the amena bility problem for Thompson’s group F . Pa rtial results are obta ined for b oth conjectures. The pap er w i ll also cont ain some general analysis of the conjectures. 1. Introduction In [17] a connection was established b e t ween the a menability o f discrete gr o ups and structural Ramse y theory . The purp ose of this a rticle is to examine, from this per sp ective, the problem o f whether Richard Tho mpson’s group F is a mena ble. Spec ific a lly , it will be demonstrated that the question of whether F is a mena ble is closely related to the gener alization of Hindman’s Theorem and Ellis ’s Lemma to nonasso cia tive binary systems. Recall tha t a gr oup G is amenable if there is a finitely additive (left) translation inv ariant probability measur e µ which measures all subsets of G . P robably the most famous example of a group whose a menability is unkno wn is Richard Thompson’s group F . The question of its amenability was co nsidered by R. Thompson himself [24] but w as rediscovered a nd p opularize d by Geo ghegan in 197 9; it first appea red in the published litera ture in [6, p. 549]. The motiv a tion for this que s tion stems from the fact that F do es not contain a copy of F 2 , the free group on t w o g enerator s [2]. It was a longsta nding open problem of von Neumann to deter mine whether every no na menable group contains a copy of F 2 (it is easily demonstr ated that any discrete g roup whic h contains F 2 is nona- menable). A finitely g enerated counterexample w as constructed by O l ′ shanskii [19] and a finitely pr e sented example w as co nstructed only more recently b y Ol ′ shanskii and Sa pir [20]. V ery recently , Mono d constructed a new example o f nona menable group not containing F 2 which is closely related to Thompso n’s group F [15]; see 2010 Mathema tics Subje ct Classific ation. 03E02, 03E50, 05D10, 05C55, 20F38, 43A07. Key wor ds and phr ases. Amenable, Ell is’s Lemma, finitely additiv e measure, idemp otent mea- sure, Hindman’s Theorem, Thompson’s group. The researc h presen ted in th is paper w as partially supp orted by NSF gran ts DM S–0757507 and DMS–1262019. I would like to thank Stev o T o dorcevic and the math departmen t at the Unive rsity of Paris VI I for their hospitality during my visit in July 2011. The presen t article underw en t considerable revisions during that peri od. I would also lik e to thank Miodr ag Soki ˇ c for reading an early draft of this paper and offering a n umber of suggestions. 1 2 JUSTIN T A TCH MOORE also [1 4] for a finitely presented nona menable subgroup of Mono d’s g roup. Th us the or iginal mo tiv ation for co nsidering whether F is a menable is no long er v alid. In the meantime, howev er, the pr oblem of F ’s amenability to ok on a life of its own, owing to the fact that it is simple to define and serves as an impor tant example in group theory . In this pap er we will see that it is related to a natural problem in Ramsey theory , giving the problem renewed motiv ation. Building on work of I. Schur and R. Rado and co nfir ming a conjecture of Graham and Rothschild [8], Hindman prov ed the following result. Hindman’s Theorem. [1 1] If c : N → k is a co lo ring of N with k color s , then there is an infinite X ⊆ N such that c is mono chromatic on the sums of finite subse ts of X . Hindman’s original pro o f of this theorem w as element ary and com binatorial but quite complex. Galvin and Glazer later g av e a simple pro o f using top o logical dy- namics, which I will now describ e (see [12, p. 102-1 03], [25, p. 3 0-33]). The op eration o f addition o n N can b e extended to its ˇ Cech-Stone compactification β N to yield a compact le ft top olog ical semigro up. Galvin rea lized that the exis- tence of an ide mp o ten t U in ( β N , +) allow ed for a simple recursive co nstruction of infinite mono chromatic se ts a s in the conclus io n o f Hindman’s Theore m. Glazer then obser ved that the existence of suc h ide mp o tents follows immediately from the following lemma of Ellis. Ellis ’s Lemma. [5] If ( S, ⋆ ) is a compact left top olo gical semigro up, then S con- tains an idempo tent . W e will now exa mine to what exten t b oth Hindman’s Theo rem and Ellis’s Lemma ca n b e generalized to a nonasso ciative setting. Let ( T , b , 1 ) denote the free binary s ystem on one generator. The algebra ( T , b , 1 ) can be represented in the following manner which will be useful later when defining our mo del for Thompson’s group F . If a and b ar e subsets of (0 , 1], define a b b = 1 2 a ∪ 1 2 ( b + 1 ) . Observe that, as a function, b is injectiv e. Conse q uently , the binar y system gener- ated by 1 = { 1 } is free; we will tak e this as our mo del of T . Just a s in the ca se of addition o n N , a binary op eration ⋆ on a set S ca n b e extended to β S as fo llows: W is in U ⋆ V if and only if { u ∈ S : { v ∈ S : u ⋆ v ∈ W } ∈ V } ∈ U Let us first o bserve that ( β T , b ) doe s not contain an idempotent and that Hind- man’s theo rem is false if we re place N by T . T o see this, define l on T recurs ively by: l ( 1 ) = 0 and l ( a b b ) = l ( a ) + 1 . If U is an ultra filter on T , then { t ∈ T : l ( t ) is even } ∈ U ⇔ { t ∈ T : l ( t ) is o dd } ∈ U b U 3 and in particular, U b U 6 = U for any U ∈ β T . Similarly , if a , b ∈ T , then l ( a ) and l ( a b b ) hav e differ ent parity and hence the naive generaliza tion of Hindman’s theorem to ( T , b ) fails. Similarly , if a, b, c ∈ T , l (( a b b ) b c ) and l ( a b ( b b c )) have a different par ity . It is infor mative to co mpare this situation to a reformulation of Hindman’s The- orem. Theorem 1.1 . If c : FIN → k is a c olo ring of FIN with k c olors, then ther e is an infinite se quen c e x 0 << x 1 << . . . of element s of FIN s u ch that c is m ono chr omatic on al l finite unions of memb ers of this se qu enc e. Here FIN deno tes the nonempt y finite subsets o f N and x < < y abbreviates max( x ) < min( y ). This ca n b e reg arded as the corre c ted form of the following false statement: If c : FIN → k , then ther e is an infinite set X such that c is mono chr omatic on al l nonempty fin ite subsets of X . The r eason this statement is false is that every infinite subset o f N co ntains finite nonempty subsets of bo th even and o dd ca rdinalities. Obs erve that this sta tement is equiv alen t to the mo difica- tion of Theo rem 1.1 where we require x i to b e a sing leton for a ll i . Th us we ca n av oid this trivial coun terexample b y allowing sing letons to b e “ glued” together in to blo cks. F o r the nonasso c iative analog of Hindman’s Theorem, I pr op ose a different form of “ gluing.” Define T n to b e all elements of T of cardinality n . These cor resp ond to the wa ys to a sso ciate a s um of n ones. In pa rticular, e a ch T n is finite and in fact the car dinalities of thes e sets are given by the Catalan num b er s. Let A n denote the colle c tion of all pr o bability measures on T n . Notice that A n can b e viewed as a conv ex s ubset of the vector spa ce gener ated b y T n and T n can b e regar ded as the set of extre me p oints o f A n . In par ticula r, if c : T n → R is an y function, then c extends linea rly to a function whic h maps A n int o R ; such extensions will be taken without further mention. Define A to b e the (disjoint) union of the sets A n and define # : A → N by #( ν ) = n if ν ∈ A n (note that while e a ch A n is co nv ex, A is not). The ope r ation of b on T extends bilinear ly to a function defined o n A : µ b ν ( E ) = X a b b ∈ E µ ( { a } ) ν ( { b } ) Observe tha t # is a homomor phism from ( A , b ) to ( N , +). A sequence µ i ( i < ∞ ) of elemen ts of A is incr e asing if i < j implies #( µ i ) < #( µ j ). In this paper, I will prov e the following par tial extens ion of Hindman’s theorem to ( A , b ). Theorem 1.2 . If c : T → [0 , 1] and ǫ > 0 , then ther e is an r ∈ [0 , 1 ] and an incr e asing se quenc e µ i ( i < ∞ ) of elements of A such that for al l i , | c ( µ i ) − r | < ǫ and al l i < j , | c ( µ i b µ j ) − r | < ǫ . In order to state the nonass o ciative form of Hindman’s theorem we will need the nonasso cia tive analo g of a finite sum. If t is in T m , then t defines a function from T m → T by s ubstitution: t ( u 0 , . . . , u m − 1 ) is obtained b y sim ultaneously substitut- ing u i for the i th o ccurrence o f 1 in the term cor r esp onding to t . This op era tion 4 JUSTIN T A TCH MOORE extends to an m -multilinear function whic h maps A m int o A . W e are now rea dy to state the conjectured genera lization of Hindman’s Theorem. Conjecture 1 .3. If c : T → [0 , 1] a nd ǫ > 0, then there is a n r ∈ [0 , 1] a nd an increasing sequence µ i ( i < ∞ ) of elements of A such that whenev er t is in T m and i 0 < . . . < i m − 1 is admissible for t | c ( t ( µ i 0 , . . . , µ i m − 1 )) − r | < ǫ Admissibilit y is a technical condition which will b e defined later. F or now it is sufficient to mention t w o of its prop erties. First, if m − 1 6 i 0 < . . . < i m − 1 , then i 0 < . . . < i m − 1 is admissible for any element o f T m . Additionally any increas ing sequence of m integers is admissible for so me elemen t o f T m . In particular, if c ( t ) is required to dep end only on #( t ), then the a b ov e conjecture reduces to Hindman’s Theorem. It w as s hown in [17 ] tha t even a weak fo r m o f Hindman’s theorem for ( A , b ) is sufficient to prov e that Tho mps o n’s gro up F is amenable. The rela tio nship b etw een F ’s a menability and the Ramsey theory o f ( A , b ) b ecomes even mor e a pparent when one a ttempts to genera lize Ellis’s L e mma. The extension of a binary op era tion ⋆ on a se t S mentioned ab ove can b e gener alized so as to extend ⋆ to the space ℓ ∞ ( S ) ∗ , which contains the set Pr( S ) of all finitely a dditive pr obability measures on S : µ ⋆ ν ( f ) = Z Z f ( x ⋆ y ) dν ( y ) dµ ( x ) This leads to the following conjectural extension of Ellis’s Lemma. Conjecture 1. 4. If ( S, ⋆ ) is a binary sys tem and C ⊆ Pr( S ) is a compact co nv ex subsystem, then there is a µ in C such that µ ⋆ µ = µ . The following result pr ovides an in triguing stra teg y for proving F ’s amena bility . Prop ositi on 1.5. If µ ∈ Pr( T ) is idemp otent, t hen µ is F -invariant. The pa pe r is or g anized as follows. Section 2 contains a r eview of the no tation and background ma terial which will b e needed fo r the rest of the pap er. A pr o of of Theorem 1.2 will b e given in Section 3. This will serve as a w arm-up for the more inv olv ed pr o of that C o njecture 1 .4 implies Conjecture 1.3 in Section 4. Section 3 will also contain a proo f that idemp otent meas ures in Pr( T ) a re F -inv ariant. The remaining sections contain an a nalysis of idemp otent measures a nd compa ct conv ex subsystems of Pr( T ). The s e results fit roughly into tw o ca tegories: those which offer s ome evidence which ma kes Conjectur e 1.4 plausible and those which reveal wha t sort of difficulties need to b e addressed in proving Co njecture 1.4. 2. Preliminaries Before b eg inning, let us fix some notational co nven tions. In this pap er, N will be taken to b e the p os itive natural num bers a nd ω will denote N ∪ { 0 } . Elements of ω are iden tified with the set of their predece ssors: 0 = ∅ and n = { 0 , . . . , n − 1 } . If S is a set, then the p ow erset of S will b e deno ted by P ( S ). 5 Next w e will recall the definition of a pro duct op era tion on finitely additiv e probability mea sures which is a n extensio n o f the F ubini pro duct of filters. W e will need so me standa rd definitions from functional ana ly sis; further reading and background ca n b e found in [21] [22]. If X is a Bana ch spa ce, let X ∗ denote the collection of contin uous line a r functionals on X . If S is a set, ℓ ∞ ( S ) deno tes the space of b ounded functions from S in to R with the supremum no rm. T he space ℓ ∞ ( S ) ∗ will primarily b e given the we ak* top olo gy : the weakest top olo gy which makes the ev aluation maps f 7→ f ( g ) contin uous for each g in ℓ ∞ ( S ). W e will ident ify the co llection Pr( S ) of all finitely additiv e pro bability measur es o n S with the subspa ce o f ℓ ∞ ( S ) ∗ consisting o f those f suc h that f ( g ) > 0 for all g > 0 a nd such that f ( ¯ 1) = 1, where ¯ 1 is the function which is constantly 1 ; if µ is in P r( S ), then µ will b e identified with the b ounded linea r functiona l f 7→ R f dµ . Dep ending on the con text, w e will sometimes write f ( µ ) for µ ( f ). The elemen ts of Pr( S ) with finite suppo r t are dense in Pr( S ) in the weak* topolog y and this will be used frequently witho ut further mention. Suppo se that S 0 and S 1 are nonempty sets. Define ⊗ : Pr( S 0 ) × Pr( S 1 ) → Pr( S 0 × S 1 ) by µ ⊗ ν ( f ) = Z Z f ( x, y ) dν ( y ) dµ ( x ) , where f is in ℓ ∞ ( S 0 × S 1 ). It should b e noted that the order of in tegration is significant when measures are required to measur e all subsets of S 0 × S 1 . This will be discussed further in Section 5. Prop ositi on 2.1. If S 0 and S 1 ar e nonempty sets, then for every ν ∈ Pr( S 1 ) , µ 7→ µ ⊗ ν is c ontinuous. Mor e over if µ ∈ P r( S 0 ) is finitely supp orte d, then the map ν 7→ µ ⊗ ν is c ontinuous. Prop ositi on 2.2. If S 0 , S 1 , and S 2 ar e nonempty sets and µ i ∈ P r( S i ) for i < 3 , then ( µ 0 ⊗ µ 1 ) ⊗ µ 2 = µ 0 ⊗ ( µ 1 ⊗ µ 2 ) up to the identific ation of ( S 0 × S 1 ) × S 2 with S 0 × ( S 1 × S 2 ) . Now supp os e that ( S, ⋆ ) is a bina ry system. Extend ⋆ to Pr( S ) a s follows: µ ⋆ ν ( f ) = µ ⊗ ν ( f ◦ ⋆ ) = µ ⊗ ν (( x, y ) 7→ f ( x ⋆ y )) . It follows from P rop ositio n 2.1 that if ν ∈ P r( S ), then µ 7→ µ ⋆ ν is c o ntin uo us. If µ is finitely supp orted, then mor eov er ν 7→ µ ⋆ ν is contin uo us. Thompson’s group F can b e descr ib ed as follows. If s , t ∈ T have e qual cardinal- it y , then the incre asing function from s to t extends linear ly to an auto morphism of ([0 , 1] , 6 ). W e will write ( s → t ) to denote this map. The collection of all such functions with the op eratio n of comp osition is F . The gro up F acts partially on T by set-wise application with the stipulation tha t f · t is o nly defined when f is linear on each interv al contained in the c o mplement of t . The sta ndard generators for F a re given b y: x 0 =  ( 1 b 1 ) b 1 → 1 b ( 1 b 1 )  x 1 =  1 b (( 1 b 1 ) b 1 ) → 1 b ( 1 b ( 1 b 1 ))  6 JUSTIN T A TCH MOORE If we view elemen ts o f T a s terms, then the partial a c tion of F on T is by re- asso ciatio n: x 0 ·  ( a b b ) b c  = a b ( b b c ) x 1 ·  s b (( a b b ) b c )  = s b ( a b ( b b c )) The pa rtial ac tion o f F on T es sentially corresp o nds to the action o f F o n its p ositive elements with r esp ect to the gener ating set x k ( k ∈ ω ), where x k +1 = x k 0 x 1 x − k 0 . It is w ell known that F is a menable if and only if there is a µ in Pr( T ) such that µ ( { t ∈ T : x 0 · t and x 1 · t ar e defined } ) = 1 , µ ( x 0 · E ) = µ ( x 1 · E ) = µ ( E ) whenever E ⊆ T (details can be found in, e.g., [16]). A gener al intro ductio n to F and Thompson’s other gro ups can be found in [3]. 3. A p ar tition theorem concerning sums of a t most two elements of A In this section we will prov e Theorem 1 .2. The first step is to define an appr o- priate limit of the seq uence A m ( m ∈ N ). Let U b e a fixed idempotent in ( β N , +). Define A U to b e the set of a ll µ ∈ Pr( T ) suc h that if W is open a b o ut µ , then { m ∈ N : W ∩ A m 6 = ∅} ∈ U . Define T U = A U ∩ β T . Notice that A U is the U -limit of the sequence A m ( m ∈ N ) in the space K (Pr( T )) consis ting of all nonempt y compact subsets of Pr( T ) equipped with the Vietoris top olo gy . In particular , A U is compact, convex, and nonempty . Lemma 3.1 . If µ and ν ar e in A U , so is µ b ν . Pr o of. Let W b e op en a b out µ b ν for µ and ν in A U . Let Z = { p ∈ N : W ∩ A p 6 = ∅ } . Since U is an idemp otent, we need to prov e that there is a set X in U such that for every m in X , { n ∈ N : m + n ∈ Z } ∈ U Applying Prop ositio n 2.1, there is an op en U ab out µ such that if µ ′ is in U , then µ ′ b ν is in W . By a ssumption, there is an X in U such that if m is in X , then U ∩ A m 6 = ∅ . F or eac h m in X , let µ m be an ele ment of U ∩ A m . Again by Prop os ition 2 .1, there is, for each m in X , an op en V m ab out ν such that if ν ′ is in V m , then µ m b ν ′ is in W . B y our assumption tha t ν is in A U , w e have that for each m in X Y m = { n ∈ N : V m ∩ A n 6 = ∅ } is in U . If n is in Y m , fix an element ν ′ of A n ∩ V m . It follows that µ m b ν ′ is in W ∩ A m + n . Th us Y m ⊆ { n ∈ N : m + n ∈ Z } and hence { n ∈ N : m + n ∈ Z } is also in U .  Theorem 1.2 will b e derived from the following prop o sition whic h is of indep e n- dent interest. Prop ositi on 3.2. If B is a finite c ol le ctio n of su bsets of T and U ∈ β N is idem- p otent , than ther e is a µ in A U such that µ b µ ↾ B = µ ↾ B . 7 Pr o of. By enla rging B if necessa ry , we ma y assume that it is a finite Bo o lean subalgebra of P ( T ). Let A cons ist of the atoms A of B such that { m ∈ N : A ∩ T m 6 = ∅} is in U . Notice that if A is an ato m of B whic h is not in A , then ξ ( A ) = 0 whenever ξ is in A U . On the other hand, if A is in A , then there is a ξ in T U such that ξ ( A ) = 1. Let X ⊆ T U be a set of car dinality |A| such that for each A in A , there is a unique ξ in X such that ξ ( A ) = 1. Define a binary o p eration ⋆ : X × X → X by ξ ⋆ η = ζ if ζ ( A ) = 1 where A is the a tom of B such that ξ b η ( A ) = 1. Notice that w e hav e ξ ⋆ η ↾ B = ξ b η ↾ B . Extend ⋆ to a bilinear op eration on the vector space ge ne r ated by X and le t C denote the con vex hull of X , noting that ⋆ maps C × C into C . Notice that µ ⋆ ν ↾ B = µ b ν ↾ B . Since X is finite, µ 7→ µ ⋆ µ is a contin uous map defined on C . Thus there is a µ in C suc h that µ ⋆ µ = µ . It follows that µ b µ ↾ B = µ ⋆ µ ↾ B = µ ↾ B .  W e ar e now r eady to complete the pro of of Theorem 1 .2. Let c : T → [0 , 1 ] and ǫ > 0 b e g iven. Fix a δ > 0 a nd a finite B ⊆ P ( T ) such that if ξ , η ∈ P r( T ) and | ξ ( E ) − η ( E ) | < δ for a ll E in B , then | ξ ( c ) − η ( c ) | < ǫ . If ξ is in ℓ ∞ ( T ) ∗ , let || ξ || B denote ma x E ∈B | ξ ( E ) | . Fix an element µ of A U such that µ ↾ B = µ b µ ↾ B a nd set r = c ( µ ), recalling the conv en tion that c ( µ ) = µ ( c ). Constr uct an increasing seq ue nce µ i ( i ∈ N ) of elements of A by induction so that if i < j , then || µ ( E ) − µ i ( E ) || B < δ and || µ b µ − µ i b µ || B < δ / 2 || µ i b µ − µ i b µ j || B < δ / 2 . This is p ossible by Pro p o sition 2.1 and the definition of A U . It follows that if i ∈ N , then || µ i − µ || B < δ and hence | c ( µ i ) − r | < ǫ . Simiarly if i < j , then || µ i b µ j − µ || B 6 || µ − µ i b µ || B + || µ i b µ − µ i b µ j || B < δ and consequently | c ( µ i b µ j ) − r | < ǫ . This finishes the pro of of Theorem 1.2. Extending this result even to right (o r left) a sso ciated sums o f at mo st three elements of A see ms to require new idea s. In particular , it is not clear ho w to find, for a given c ∈ ℓ ∞ ( T ), a µ in A U such that c ( µ ) = c ( µ b µ ) = c ( µ b ( µ b µ )). I will conclude this section with the following prop osition which relates the nonas- so ciative for m of Ellis’s Lemma to the amenability pro blem for F . Prop ositi on 3.3. If µ ∈ Pr( T ) is an idemp otent me asur e, then µ is F -invariant. Pr o of. Supp ose that µ ∈ Pr( T ) satisfies µ b µ = µ ; w e need to sho w that µ is F -inv ariant. First obse r ve that µ ( { 1 } ) = µ b µ ( { 1 } ) = µ ⊗ µ ( ∅ ) = 0 . Also µ b µ ( { t ∈ T : ∃ a ( t = a b 1 ) } ) = µ ( T ) · µ ( { 1 } ) = 1 · 0 = 0 . Since µ is an idemp otent, followin g ident ities hold: µ = µ b ( µ b µ ) = ( µ b µ ) b µ µ = µ b ( µ b ( µ b µ )) = µ b (( µ b µ ) b µ ) 8 JUSTIN T A TCH MOORE Now supp os e that E ⊆ T . µ ( E ) = µ ( { t ∈ E : ∃ a ∃ b ∃ c ( t = ( a b b ) b c ) } ) = ( µ b µ ) b µ ( { t ∈ E : ∃ a ∃ b ∃ c ( t = ( a b b ) b c ) } ) = ( µ ⊗ µ ) ⊗ µ ( { ( a, b, c ) ∈ T 3 : ( a b b ) b c ∈ E } ) = µ ⊗ ( µ ⊗ µ )( { ( a, b, c ) ∈ T 3 : a b ( b b c ) ∈ x 0 · E } ) = µ b ( µ b µ )( { t ∈ x 0 · E : ∃ a ∃ b ∃ c ( t = a b ( b b c )) } ) = µ ( x 0 · E ) . A similar computation shows that µ ( x 1 · E ) = µ ( E ).  4. Conjecture 1.4 implies Conjecture 1 .3 In this section I will prove that the nona s so ciative form of Ellis ’s Lemma implies the nonasso cia tive form of Hindman’s Theorem. This will b e an elabo ration on the pro of of Theorem 1.2 from Pr op osition 3.2. Before pr o ceeding, it is necessar y to define the no tion of admissibil ity fro m the statement of Conjectur e 1 .3. In order to motiv ate the definition o f admissibility , consider the natural adaptation of the pro o f in the previous sec tion: at stage n , we hav e constructed measures µ i ( i < n ) and wish to pick a measure µ n in A such that if t is an y element of T and i k ( k < m ) is an increa s ing s equence of length at most #( t ), then c ( t ( µ i 0 , . . . , µ i m − 1 , µ, µ, . . . , µ )) and c ( t ( µ i 0 , . . . , µ i m − 1 , µ n , µ, . . . , µ ) differ by less than ǫ 2 − n − 1 . The problem is that there are infinitely man y t ’s to consider. The fix to this problem is to consider o nly those t o f car dinality at mos t n at sta ge n . This rea dily allows us to prov e the for m of Conjecture 1.3 in whic h one obtains the conclusion for those seq uences i k ( k < m ) satisfying m − 1 6 i 0 . It is p o ssible to do b etter, how ever, by no ticing tha t if µ is idemp otent, then expressions such as t ( µ i 0 , . . . , µ i m − 1 , µ, µ, . . . , µ ) may allo w for some algebraic sim- plification. This is made pr ecise as follows. F or eac h k < #( t ), define t 7→ t k by t k =      1 if t = 1 and k = 0 a k b 1 if t = a b b and k < #( a ) a b b k − #( a ) if t = a b b and #( a ) 6 k < #( t ) Notice that if we regard t as a ro oted order d binary tree, then t k is the res ult of iteratively remo ving all carets in t which inv olve only leav es of index greater than k (wher e leav es are indexed in increasing or de r from left to right). Set l k ( t ) = #( t k ) − 2. If t is in T m , then an increasing sequence i k ( k < m ) is admissible for t if for all k < m , l k ( t ) 6 i k . Notice that l k ( t ) < #( t ) for ea ch k and in particula r a sequence i k ( k < m ) is admissible for a ny element of T m provided that m 6 i 0 . Also, the v alue o f l k at any right assoc ia ted p ow er of 1 is a t mos t k and hence any increas ing sequence is admissible for some element of T m . It will b e helpful to ado pt the following notation: if t is in T m , k 6 m , and ν i ( i < k ) and µ are in Pr( T ), let t ( ν 0 , . . . , ν k − 1 ; µ ) denote t ( ν 0 , . . . , ν k − 1 , µ, . . . , µ ) (i.e. the sequence ν i ( i < k ) is extended to a s equence of length m b y adding on a sequence of m − k man y µ ’s and then substituting in to t ). The k ey prop er ty of the 9 definition of t k is that whenever t in T , ν i ( i < k ) are in A , and µ is an idemp otent in Pr( T ), then t ( ν 0 , . . . , ν k − 1 ; µ ) = t k ( ν 0 , . . . , ν k − 1 ; µ ); this is easily established by induction on #( t ). W e are now r eady to prov e the main result of the section. Theorem 4.1 . Conje ctur e 1.4 implies Conje cture 1.3. Pr o of. In Section 3, w e pr ov ed that if U is an idempotent in ( β N , + ), then A U is a nonempty compact co nv ex subsystem of (Pr( T ) , b ). Thus if Conjecture 1.4 is true, then A U contains a µ such that µ b µ = µ . Fix a c : T → [0 , 1 ] and set r = c ( µ ). Construct an increa sing sequence µ i ( i ∈ ω ) in A by recur sion such that, if µ i ( i < n ) hav e b ee n constructed, then for all k < m 6 n + 2 a nd i 0 < . . . < i k − 1 < n and t in T m , | c ( t ( µ i 0 , . . . , µ i k − 1 ; µ )) − c ( t ( µ i 0 , . . . , µ i k − 1 , µ n ; µ )) | < ǫ 2 − n − 1 . This is p ossible by applying the definition of A U and the following claim. Claim 4.2. If t is in T m and ν i ( i < m ) ar e s uch that ν i is in Pr( T ) and has finite suppo rt if i < k , then the function F defined by F ( ζ ) = t ( ν 0 , . . . , ν k − 2 , ζ , ν k , . . . , ν m − 1 ) is contin uo us. Pr o of. The pro of is by induction on m . If m = 1 , then ther e is no thing to show since then F is just the identit y . If m > 1 and t is in T m , then there are a and b such that t = a b b . If #( a ) = l 6 k , then t ( ν 0 , . . . , ν k − 2 , ζ , ν k , . . . ) = a ( ν 0 , . . . , ν l − 1 ) b b ( ν l , . . . , ν k − 1 , ζ , ν k , . . . ) Letting ν = a ( ν 0 , . . . , ν l − 1 ), we have that F ( ζ ) = ν b b ( ν l , . . . , ν k − 1 , ζ , ν k , . . . , ν m − 1 ) . Contin uit y of F no w follows fro m Prop os ition 2 .1 and the induction hypothesis applied to b . A similar arg ument handles the ca se #( a ) > k .  Now w e will verify that µ k ( k ∈ ω ) satisfies the conclusio n of Co njecture 1 .3. T o this end, let t b e an element of T m and let i 0 < . . . < i m − 1 be admis s ible for t . By construction we ha v e | c ( t ( µ i 0 , . . . , µ i m − 1 )) − c ( t ( µ, . . . , µ )) | 6 X k 1 / 2 } , then { U r : r ∈ 2 N } is an uncountable family of nonempty pa irwise disjo int op en subsets of K . Hence K is no t separable.  7. Concluding remarks A t the time this article w as written, it is still unknown if F is amenable. Never- the-less, I feel Conjectures 1.3 and 1.4 are based on sound heuristics fro m Ramsey theory . It is rare in the Ramsey theory o f countably infinite sets that there are difficult counterexamples to Ra msey-theoretic statements (there are exceptions, per haps most no tably [18]; see also [23, § 9]). O n the other hand, ther e are ma ny deep a nd often difficult po sitive results in Ramsey theor y at this level: the Dual Ramsey Theor em [8], Hindman’s Theor em [11], Gow e rs’s FIN k Theorem [7 ], the Hales-Jewett Theorem [9], a nd the Halp er n-L¨ auchli Theo rem [10]. See [25] for further reading on these theorems as well as many others. Also, while w e do not know whether (Pr( S ) , ⋆ ) contains an idemp otent if ( S, ⋆ ) is a n arbitra ry binar y system, we do k now that there are quite different examples of binary systems which admit idemp otent me asur es : semigr oups, finite binar y systems, and binary systems dep ending o n only one v ariable. The results of this pa p er also sug gest several test ques tio ns which allow for an incremental approach to proving Conjectures 1.3 and 1.4: Question 7.1. Is Conjecture 1.3 true for sums of d elements, for a fixed d > 3? What ab out the case d = 3? Question 7. 2. F or which classes of binary systems is Co njecture 1.4 true? Question 7. 3. If t w o binary systems satisfy Conjecture 1.4, do es their pro duct? Question 7 .4. F o r whic h sp ecific v a lues o f m can one prov e that there is an n such that if c : T n → { 0 , 1 } then ther e is a copy of T m in A n on which c is constant? What b ounds (upper or low er) can be prov ed on n for a g iven v alue of m ? References [1] J. E. Baumgartner. A short proof of Hindman’s theorem. J. Combinatorial The ory Ser. A , 17:384–386, 1974. [2] M. G. Brin and C. C. Squier. Groups of piecewise li near homeomorphisms of the real line. Invent. Math. , 79(3):485–498, 1985. [3] J. W. Cannon, W. J. Floyd , and W. R. Parry . Introductory notes on Richard Thompson ’s groups. Enseign. Math. (2) , 42(3-4):215 –256, 1996. [4] P . Dehornoy . Braid groups and left distributive operations. T r ans. Amer. Math. So c. , 345(1):115 –150, 1994. [5] R. Ellis . Distal transform ation groups. Pacific J. Math. , 8:401 –405, 1958. 16 JUSTIN T A TCH MOORE [6] S. M. Gersten and J. R. Stallings, editors. Combinatorial gr oup the ory and t op olo gy , v olume 111 of Anna ls of Mathematics Studies . Princeton Uni versity Pr ess, Princeton, NJ, 1987. Pa pers fr om the conference held in Alta, Utah, July 15–18, 198 4. [7] W. T. Gow ers. Lipschitz functions on classical spaces. Eur op e an J. Combin. , 13(3) :141–151, 1992. [8] R. L. Graham and B. L. Rothsc hild. Ramsey’s theo rem for n -parameter sets. T r ans. Amer. Math. So c. , 159:257–292 , 1971 . [9] A. W. Hales and R. I. Jew ett. Regularity an d p ositional games. T r ans. Amer. Math. So c. , 106:222–229 , 1963. [10] J. D. H alper n and H. L¨ auchli. A partition theorem. T r ans. Am er. Math. So c. , 124:360–367, 1966. [11] N. Hi ndman. Finite sums fr om sequences wi thin ce lls of a partition of N . J. Combinatorial The ory Ser. A , 17:1–11, 197 4. [12] N. Hi ndman and D. Strauss. Algebr a in the St one- ˇ Ce ch c omp actific ation , volume 27 of de Gruyter Exp ositions in Mathematics . W alter de Gruyter & Co., Berlin, 1998. Theory and applications. [13] R. Lav er. On the algebra of elementary embeddings of a rank in to itself. A dv. Math. , 110:334– 346, 1995. [14] Y. Lo dha, J. T atch Mo ore. A geometric sol ution to the von Neumann-Da y problem for finitely presen ted groups. ArXiv preprint 1308.4250, 2013. [15] N. Mono d. Groups of piecewise pro jective homeomorphisms. Pr o c. Natl. A c ad. Sci . USA , 110(12) :4524–4527, 2013. [16] J. T atc h M o or e. F ast growth in Følner function for Thompson’s group F . Gr oups, Geo m. Dyn. , 7(3):633 –651, 2013. [17] J. T atc h Mo ore. Amenability and Ramsey theory . F und. Math. , 220(3) :263–280, 201 3. [18] E. Odell and Th. Sc hlumprec h t. The distortion problem. A cta Math. , 173(2):259– 281, 1994. [19] A. Ju. Ol ′ ˇ sanski ˘ ı. O n the question of the existence of an inv ariant mean on a group. Usp ek hi Mat. Nauk , 35(4(214)):199–2 00, 1980. [20] A. Y u. Ol ′ shanskii and M. V. Sapir. Non-amenable finitely presen te d torsion-by-cyclic groups. Publ. M ath. Inst. Hautes ´ Etudes Sc i . , (96):43–1 69 (2003), 2002. [21] A. L. T. P aterson. Am enability , volume 29 of M athematica l Survey s and Mono gr aphs . A mer- ican Mathematical So ciety , Pro vidence, RI, 1988. [22] W. Rudin. F unctional analysis . In ternational Series in Pure and Applied M athematics. McGraw-Hill Inc., New Y ork, second edition, 1991. [23] S. Solecki. Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem. A dv. Math. , 248:1156– 1198, 2013. [24] Letter from Richa rd Thompson to George F rancis, dated Septem ber 26 , 1973. [25] S. T o dorcevic. Intr o duction to Ramsey sp ac es , vo lume 174 of Annals of Mathematics St udies . Princeton Univ ersity Press, Princeton, NJ, 2010. Justin M oore, 555 Malott Hall, Dep ar tment of Ma thema tics, Cornell Univ ersity, Ithaca, NY 1485 3-4201