CONJUGACY IN THOMPSON S GROUP F. 1. Introduction

CONJUGACY IN THOMPSON S GROUP F NICK GILL AND IAN SHORT Abstract. We complete the program begun by Brn and Squer of charactersng conjugacy n Thompson s group F usng the standard acton of F as a group of pecewse lnear homeomorphsms of the unt nterval. 1. Introducton The object of ths paper s to extend the methods of Brn and Squer descrbed n [3] to solve the conjugacy problem n Thompson s group F. Let f : (a, b) (a, b) be a pecewse lnear order-preservng homeomorphsm of the open nterval (a, b); the ponts at whch f s not locally affne are called the nodes of f. We wrte PLF + (a, b) to denote the group of all pecewse lnear order-preservng homeomorphsms of the open nterval (a, b) whch have fntely many nodes. Thompson s group F s the subgroup of PLF + (0, 1) defned as follows: an element f of PLF + (0, 1) les n F f and only f the nodes of f le n the rng of dyadc ratonal numbers, Z [ 1 2], and f (x) s a power of 2 whenever x s not a node. In [3] Brn and Squer analysed conjugacy n PLF + (a, b) for (a, b) any open nterval. For (a, b) equal to (0, 1) we can restate ther prmary result [3, Theorem 5.3] as follows: we have a smple quantty Σ on PLF + (0, 1) such that two elements f and g of PLF + (0, 1) are conjugate f and only f Σ f = Σ g. If f and g are elements of F then Σ f and Σ g can be computed and compared usng a smple algorthm. Brn and Squer comment on ther constructon of Σ that, Our goal at the tme was to analyze the conjugacy problem n Thompson s group F. In ths paper we acheve Brn and Squer s goal by defnng a quantty on F such that the followng theorem holds. Theorem 1.1. Let f, g F. Then f and g are conjugate n F f and only f (Σ f, f ) = (Σ g, g ). Ths s not the frst soluton of the conjugacy problem n F. In partcular the conjugacy problem n F was frst solved by Guba and Sapr n [5] usng dagram groups. More recently, Belk and Matucc [1, 2] have another soluton usng strand dagrams. Kassabov and Matucc [6] also solved the conjugacy problem, and the smultaneous conjugacy problem. Our analyss s dfferent to all of these as we buld on the geometrc nvarants ntroduced by Brn and Squer. Note that Thm. 1.1 reduces the study of roots and centralzers n F (frst completed n [5]) to a set of easy computatons; we leave ths to the nterested reader. Date: December 10, 2011. 2010 Mathematcs Subject Classfcaton. Prmary: 20E45. Secondary: 20F10, 37E05. Key words and phrases. Conjugacy, pecewse lnear, Thompson. 1

We wll not prove Theorem 1.1 drectly. Rather we prove the followng proposton whch, gven [3, Theorem 5.3], mples Theorem 1.1: Proposton 1.2. Two elements f and g of F that are conjugate n PLF + (0, 1) are conjugate n F f and only f f = g. Our paper s structured as follows. In 2 we ntroduce some mportant background concepts, ncludng the defnton of Σ. In 3 we defne. In 4 we prove Proposton 1.2. In 5 and 6 we outlne formulae whch can be used to calculate. 2. The defnton of Σ Let f be a member of Thompson s group F, embedded n PLF + (0, 1). Followng Brn and Squer [3] we defne the nvarant Σ f to be a tuple of three quanttes, Σ 1, Σ 2 and Σ 3, whch depend on f. The frst quantty, Σ 1, s a lst of ntegers relatng to values of the sgnature of f, ϵ f. We defne ϵ f as follows: 1, f(x) > x; ϵ f : (0, 1) { 1, 0, 1}, x 0, f(x) = x; 1, f(x) < x. If f s an element of PLF + (0, 1) then there s a sequence of open ntervals I 1, I 2,..., I m, I j = (p j 1, p j ), p 0 = 0, p m = 1, (2.1) such that ϵ f s constant on each nterval, and the values of ϵ f on two consecutve ntervals dffer. We defne Σ 1 = (ϵ f (x 1 ),..., ϵ f (x m )) where x I for = 1,..., m. Let fx(f) be the set of fxed ponts of f and observe that the ponts p 0,..., p m from (2.1) all le n fx(f). We say that the nterval I j s a bump doman of f f ϵ f s non-zero on that nterval. Our next two nvarants consst of lsts wth entres for each bump doman of f. If k s a pecewse lnear map from one nterval (a, b) to another, then the ntal slope of k s the dervatve of k at any pont between a and the frst node of k, and the fnal slope of k s the dervatve of k at any pont between the fnal node of k and b. The nvarant Σ 2 s a lst of postve real numbers. The entry for a bump doman I j = (p j 1, p j ) s the value of the ntal slope of f n I j. To defne Σ 3 we need the noton of a fnte functon; ths s a functon [0, 1) R + whch takes the value 1 at all but fntely many values. The nvarant Σ 3 s a lst of equvalence classes of fnte functons. We calculate the entry for a bump doman I j = (p j 1, p j ) as follows. Suppose frst of all that Σ 1 = 1 n I j. Defne, for x I j, the slope rato f (x) = f + (x) f (x). Thus f (x) = 1 except when x s a node of f. Now defne ϕ f,j : I j R, x n= f (f n (x)). Snce f has only fntely many nodes, only fntely many terms of ths nfnte product are dstnct from 1. Let p be the smallest node of f n I j and let p be the smallest 2

node of f n I j such that ϕ f,j (p ) 1 (such a node must exst). Defne, for s [0, 1), ψ f,j (s) = ϕ f (λ s (r p j 1 ) + p j 1 ). Here λ s the entry n Σ 2 correspondng to I j and r s any pont n the nterval (0, p) whch satsfes the formula r = f n (p ) for n some negatve nteger. Note that ψ f,j s a fnte functon; furthermore, n our defnton of ψ f,j, we have chosen a value for r whch guarantees that ψ f,j (0) 1; we can do ths by vrtue of [3, Lemma 4.4]. The entry for Σ 3 correspondng to I j s the equvalence class [ψ f,j ], where two fnte functons c 1 and c 2 are consdered equvalent f c 1 = c 2 ρ where ρ s a translaton of [0, 1) modulo 1. If f(x) < x for each x I j then the entry for Σ 3 correspondng to I j s the equvalence class [ψ f 1,j]. 3. The defnton of The quantty wll also be a lst, ths tme a lst of equvalence classes of tuples of real numbers. To begn wth we need the concept of a mnmum cornered functon. 3.1. The mnmum cornered functon. Take f PLF + (0, 1); n ths subsecton we focus on the restrcton of f to one of ts bump domans D = (a, b). We adjust one of the defntons of Brn and Squer [3]: for us, a cornered functon n PLF + (0, 1) s an element l whch has a sngle bump doman (a, b) and whch satsfes the followng property: Σ 1 takes value 1 (resp. 1) n relaton to (a, b) and there exsts a pont x (a, b) such that all nodes of l whch le n (a, b) le n (x, l(x)) (resp. (l(x), x)). We wll sometmes abuse notaton and consder such a cornered functon as an element of PLF + (a, b). Fgure 1. A cornered functon. We say that a cornered functon l corresponds to a fnte functon c f ψ l = c (we drop the subscrpt here, snce there s only one bump doman). Roughly speakng ths means that the frst node of l corresponds to c(0). For a gven ntal slope λ there s a unque cornered functon n PLF + (a, b) such that ψ l = c (ths follows from [3, Prop. 4.9]; t can also be deduced from the proof of Lemma 5.2). 3

Now let c : [0, 1) R be a fnte functon such that [c] s the entry n Σ 3 assocated wth D. Wthn ths equvalence class [c] we can defne a mnmum fnte functon c m as follows. Frst defne C = {c 1 [c] c 1 (0) 1} and defne an orderng on C as follows. Let c 1, c 2 C and let x be the smallest value such that c 1 (x) c 2 (x). Wrte c 1 < c 2 provded c 1 (x) < c 2 (x). We defne c m to be the mnmum functon n C under ths orderng. Fgure 2. Two equvalent fnte functons. The mnmum fnte functon s on the left. Suppose that λ s the entry n Σ 2 assocated wth D. Suppose that l s the cornered functon n PLF + (a, b), wth ntal slope λ, whch corresponds to c m. We say that l s the mnmum cornered functon assocated wth f over D. 3.2. The quantty. Now let us fx f to be n F and defne accordngly. A bump chan s a subsequence I t, I t+1,..., I u of (2.1) such that each nterval s a bump doman, and of the ponts p t 1, p t,..., p u only p t 1 and p u are dyadc. Thus I 1, I 2,..., I m can be parttoned nto bump chans and open ntervals of fxed ponts of f (whch have dyadc numbers as end-ponts). In [3], conjugatng functons n PLF + (0, 1) are constructed by dealng wth one bump doman at a tme. We wll construct conjugatng functons n F by dealng wth one bump chan at a tme. Consder a partcular bump chan D 1,..., D s and let f j be the restrcton of f to D j = (a j, b j ). Accordng to [3, Theorem 4.18], the centralzer of f j wthn PLF + (a j, b j ) s an nfnte cyclc group generated by a root f j of f j. We defne λ j to be the ntal slope of f j and µ j to be the fnal slope of f j. (Let m j be the nteger such that f j m j = f j ; then λ j and µ j are the postve m j th roots of the ntal and fnal slopes of f j.) Next, let k j be a member of PLF + (a j, b j ) that conjugates f j to the assocated mnmum cornered functon, l j, n PLF + (a j, b j ). Thus k j s some functon satsfyng the equalty k j f j k 1 j = l j. Let α j be the ntal slope of k j and let β j be the fnal slope. 4

Consder the equvalence relaton on R s where (x 1,..., x s ) s equvalent to (y 1,..., y s ) f and only f there are ntegers m, n 1,..., n s such that 2 m x 1 = λ n 1 1 y 1 µ n 1 1 x 2 = λ n 2 2 y 2 µ n 2 2 x 3 = λ n 3 3 y 3. µ n s 1 s 1 x s = λ n s s y s. It s possble to check whether two s-tuples of real numbers are equvalent accordng to the above relaton n a fnte amount of tme because the quanttes λ and µ j are ratonal powers of 2. We assgn to the chan D 1,..., D s the equvalence class of the s-tuple ( α1, α 2 w 1,..., α ) s w s 1 w 1 w 2 β 1 w s β s 1 where w j = b j a j. We defne f to consst of an ordered lst of such equvalence classes; one per bump chan. 4. Proof of Proposton 1.2 We prove Proposton 1.2 after the followng elementary lemma. Lemma 4.1. Let f and g be maps n F, and let h be an element of PLF + (0, 1) such that hfh 1 = g. Let D = (a, b) be a bump doman of f and suppose that the ntal slope of h n D s an nteger power of 2. Then all slopes of h n D are nteger powers of 2 and all nodes of h n D occur n Z[ 1 2 ]. Proof. Let (a, a + δ) be a small nterval over whch h has constant slope; suppose that ths slope s greater than 1. We may assume that f has ntal slope greater than 1 otherwse replace f wth f 1 and g wth g 1. Now observe that hf n h 1 = g n for all ntegers n and so h = g n hf n. Now, for any x (a, b) there s an nterval (x, x + ϵ) and an nteger n so that f n (x, x + ϵ) (a, a + δ). Then the equaton h = g n hf n mples that, where defned, the dervatve of h over (x, x + ϵ) s an ntegral power of 2. Furthermore any node of h occurng n (x, x + ϵ) must le n Z[ 1 ] as requred. 2 If h does not have slope greater than 1 then apply the same argument to h 1 usng the equaton h 1 gh = f. We have two elements f and g of F and a thrd element h of PLF + (0, 1) such that hfh 1 = g. We use the notaton for f descrbed n the prevous secton, such as the quanttes I j, p j, f j, f j, k j, l j, α j, β j, w j, λ j, and µ j. We need exactly the same quanttes for g, and we dstngush the quanttes for g from those for f by addng a after each one. In partcular, we choose a bump chan D 1,..., D s of f and defne D = h(d ) for = 1,..., s. Note that D 1,..., D s are bump domans but need not form a bump chan for g accordng to our assumptons, because h s not necessarly a member of F. 5

Let the functon h = h D have ntal slope γ and fnal slope δ. Let u be the member of PLF + (0, 1) whch, for = 1,..., m, s affne when restrcted to I, and maps ths nterval onto I. Notce that, restrcted to D, ul u 1 s a cornered functon whch s conjugate to l (by the map k h k 1 u 1 ), and whch satsfes ψ l = ψ ul u 1. Therefore = l, k g k 1 = l, and ul u 1 = l. Combne ths equaton wth the equatons k f k 1 h f h 1 = g to yeld (k 1 u 1 k h )f (k 1 u 1 k h ) 1 = f. Therefore k 1 u 1 k h s n the centralzer of f, so there s an nteger N such that h = (k ) 1 uk f N for each = 1,..., s. Then by comparng ntal and fnal slopes n ths equaton we see that γ = λ N α w w α, δ = µ N β w w for = 2,..., s. We are now n a poston to prove Proposton 1.2. β (4.1) Proof of Proposton 1.2. Suppose that h F. Then there are ntegers M 1,..., M s such that γ 1 = 2 M 1 and γ = δ 1 = 2 M for = 2,..., s. Substtutng these values nto (4.1) we see that 2 M 1 α 1 w 1 for = 2,..., s, as requred. = λ N 1 1 α 1 w 1, µ N 1 1 α w w 1 β 1 = λ N α w 1, w β 1 Conversely, suppose that f = g. We modfy h so that t s a member of F. If I j s an nterval of fxed ponts of f then modfy h j so that t s any pecewse lnear map from I j to I j whose slopes are nteger powers of 2, and whose nodes occur n Z[ 1 ]. (It 2 s straghtforward to construct such maps, see [4, Lemma 4.2].) Now we modfy h on a bump chan D 1,..., D s. Snce f = g we know that there are ntegers m and n 1,..., n s such that, for = 2,..., s, 2 m α 1 = λ n 1 1 w 1 α 1 w 1, µ n 1 1 α w 1 = λ n w β 1 α w w 1. (4.2) β 1 Consder the pecewse lnear map h : D h (D ) gven by h n N = h f. The ntal slope γ of h s γ λ n N and the fnal slope δ = δ µ n N. From (4.1) and (4.2) we see that γ 1 = 2 m, γ = δ 1. for = 2,..., s. We modfy h by replacng h wth h on D. Then h does not have a node at any of the end-ponts of D 1,..., D s other than the frst and last end-pont. By Lemma 4.1, the nodes of h 1 occur n Z[ 1] and the slopes of h 2 1 are all powers of 2. Snce the ntal slope of h 1 concdes wth the fnal slope of h 1, the same can be sad of h 2. Smlarly, for = 2,..., s, the ntal slope of h concdes wth the fnal slope of h 1. We repeat these modfcatons for each bump chan of f; the resultng conjugatng map s a member of F. 6

5. Calculatng α and β It may appear that, n order to calculate, t s necessary to construct varous conjugatng functons. In partcular to calculate α one mght have to construct the functon n PLF + (a, b ) whch conjugates f to the conjugate mnmum cornered functon n PLF + (a, b ). It turns out that ths s not the case. The values for α and β can be calculated smply by lookng at the entres n Σ 1, Σ 2 and Σ 3 whch correspond to D. In ths secton we gve a formula for α ; we then observe how to use the formula for α to calculate β. In what follows we take f to be a functon n PLF + (a, b) such that f(x) x for x (a, b). Let l be the mnmum cornered functon whch s conjugate to f n PLF + (a, b). 5.1. Calculatng α. Suppose frst that f(x) > x for x (a, b). Let y j, for j = 0,..., t be the ponts at whch the fnte functon ψ f does not take value 1; let ψ f take the postve value z j at the pont y j and assume that 0 = y 0 < y 1 < < y t < 1. We wll denote ψ f by c t and defne c j = c t (x + y j+1 ). Then c j s a translaton of c t under whch y j s mapped to the last pont of c j whch does not take value 1. Let u j be the cornered functon correspondng to c j and let x j be the fnal node of u j. Note that u j s conjugate to f and, for j equal to some nteger n, u j equals l, the mnmum cornered functon. Defne the elementary functon h x,r to be the functon whch s affne on (0, x) and (x, 1) and whch has slope rato r at x. We defne ζ j to be the ntal slope of the elementary functon h xj,z j. Let p be the frst node of f and let q be the frst node of u t. Lemma 5.1. There exsts k n PLF + (a, b) such that kfk 1 = l and the ntal slope of k s ( ) q a (ζ t ζ t 1... ζ n+1 ). p a Note that, n the formula just gven, p and q stand for the x-coordnates of the correspondng nodes. Before we prove Lemma 5.1 we observe that we can calculate values for the ζ j and q smply by lookng at Σ 2 and Σ 3 and usng the followng lemma: Lemma 5.2. Let l be a cornered functon n PLF + (a, b) wth ntal slope λ > 1, and suppose that the correspondng fnte functon c takes the value 1 at all ponts n [0, 1) except 0 = s 0 < s 1 < < s k < 1, at whch c(s ) = z. Then the frst node q 0 of l s gven by the formula q 0 = a + (b a)(1 [λz 0 z k ]) [λ(1 z 0 )]+[λ s 1 +1 z 0 (1 z 1 )]+ +[λ s k 1 +1 z 0 z k 2 (1 z k 1 )]+[λ s k +1 z 0 z k 1 (1 z k )], (5.1) and the ntal slope ζ of the elementary functon h qk,z k, where q k s the fnal node of l, s gven by b a ζ =. (5.2) λ s k (q0 a)(1 z k ) + (b a)z k Proof. If q 0,..., q k are the nodes of l we have equatons λ s (q 0 a) + a = q, = 0, 1, 2,..., k. (5.3) 7

Defne q k+1 = b and let λ be the slope of l between the nodes q 1 and q for = 1,..., k + 1. Then z = λ /λ 1 for > 1, and we obtan If we substtute (5.3) and (5.4) nto the equaton λ = λz 0... z 1, = 1,..., k + 1. (5.4) b a = λ(q 0 a) + λ 1 (q 1 q 0 ) + λ 2 (q 2 q 1 ) + + λ k+1 (b q k ), then we obtan (5.1). To obtan (5.2), notce that z k ζ s the fnal slope of h qk,z k, therefore b a = ζ(q k a) + z k ζ(b q k ). Substtute the value of q k from (5.3) nto ths equaton to obtan (5.2). Before we prove Lemma 5.1, we make the followng observaton. Let g be a functon such that g(x) > x for all x (a, b) and suppose that g has nodes p 1 < < p s. Now let h = h ps,g (p s ). Then hgh 1 has nodes h(p 1 ),..., h(p s 1 ), hg 1 (p s ) wth (hgh 1 ) takng on values g (p 1 ),... g (p t ) at the respectve nodes. If hg 1 (p s ) = h(p ) for some, then (hgh 1 ) has value g (p )g (p s ). Proof of Lemma 5.1. The formula gven n Lemma 5.1 arses as follows. We start by fndng the conjugator from f to the cornered functon u t ; then we cycle through the cornered functons u j untl we get to u n = l. Thus the q a part of the formula arses p a from the ntal conjugaton to a cornered functon, and the ζ j s arse from the cyclng. Consder ths cyclng part frst and use our observaton above on the cornered functons, u j : we have h xj,z j u j (h xj,z j ) 1 = u j 1. Thus n order to move from u t to u n we repeatedly conjugate by elementary functons wth ntal gradent ζ t,..., ζ n+1. We must now explan why we can use q a for the frst conjugaton whch moves from p a f to u t. It s suffcent to fnd a functon whch conjugates f to u t and whch s lnear on [a, p]. Consder the effect of applyng an elementary conjugaton to a functon f that s not a cornered functon. Suppose that f has nodes p 1 < < p s. So p = p 1. We consder the effect of conjugaton by an elementary functon h = h ps,f (p s ) as above. To reterate, we obtan a functon wth nodes h(p 1 ),..., h(p s 1 ), hf 1 (p s ) Now observe that, snce f (p s ) < 1, h(x) > x for all x and h s lnear on [a, p s ]. So clearly h s lnear on the requred nterval. There are three possbltes: If hf 1 (p s ) < h(p) then f was already a cornered functon; n fact f = u t. We are done. If hf 1 (p s ) > h(p) then we smply terate. We replace f wth hfh 1, p wth h(p) etc. We conjugate by another elementary functon exactly as before. It s clear that the next elementary conjugaton wll be lnear on [a, h(p)] whch s suffcent to ensure that the composton s lnear on [a, p]. If hf 1 (p s ) = h(p) then we need to check f hfh 1 s a cornered functon. If so then hfh 1 = u t, the corner functon we requre. If hfh 1 s not a cornered functon then we terate as above, replacng f wth hfh 1. It s possble that h(p) wll no longer be the frst node of hfh 1, but n ths case we replace p by 8

h(p 2 ). Snce [a, h(p 2 ] [a, h(p)] ths s suffcent to ensure that the composton s lnear on [a, p]. We can proceed lke ths untl the process termnates at a cornered functon. Snce conjugatng a non-cornered functon by h preserves ψ we can be sure that we wll termnate at u t as requred. What s more the composton of these elementary functons s lnear on [a, p]. Suppose next that f(x) < x for all x (a, b) and kfk 1 = l, a mnmum cornered functon. Observe that kf 1 k 1 = l 1 and f 1 (x) > x for all x (a, b). We can now apply the formula n Lemma 5.1, replacng f wth f 1 and l wth l 1, to get a value for the ntal slope of k. 5.2. Calculatng β. The method we have used to calculate α can also be used to calculate β. Defne τ : [a, b] [a, b], x b + a x. Now τ s an automorphsm of PLF + (a, b); the graph of a functon, when conjugated by τ, s rotated 180 about the pont ( b+a, b+a ). Consder the functon τfτ and let k be 2 2 the conjugatng functon from earler, so that kfk 1 = l. Then (τkτ)(τfτ)(τkτ) 1 = (τlτ). The ntal slope of τkτ equals the fnal slope of k. Thus we can use the method outlned above replacng f wth τfτ and l wth τlτ to calculate the ntal slope of τkτ. Note that, for ths to yeld β, we must make an adjustment to the nteger n n the formula n Lemma 5.1: the functon τ lτ s not necessarly the mnmum cornered functon whch s conjugate to τfτ. Thus we choose n to ensure that l s mnmum rather than τlτ. 6. Calculatng λ and µ Let f be a fxed-pont free element of PLF + (a, b). Let f be a generator of the centralzer of f wthn PLF + (a, b). The formula for requres that we calculate the ntal slope and the fnal slope of f. It turns out that ths s easy thanks to the work of Brn and Squer [3]. Let c, c : [0, 1) R be fnte functons. We say that c s the p-th root of c provded that, for all x [0, 1), we have c(x) = c (px). The property of havng a p-th root s preserved by the equvalence used to defne Σ 3. Thus we may talk about the equvalence class [c] havng a p-th root, provded any representatve of [c] has a p-th root. Now [3, Theorem 4.15] asserts that f has a p-th root n PLF + (a, b), for p a postve nteger, f and only f the sngle equvalence class n Σ 3 s a p-th power (followng Brn and Squer we say that ths class has p-fold symmetry). What s more [3, Theorem 4.18] asserts that f must be a root of f. Thus f p s the largest nteger for whch the sngle class n Σ 3 has p-fold symmetry then f s the p-th root of f. The ntal slope of f s the postve p-th root of the ntal slope of f, and the fnal slope of f s the postve p-th root of the fnal slope of f. 9

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