CCONJUGATOR LENGTH IN THOMPSON’S GROUPS
JAMES BELK AND FRANCESCO MATUCCI
Abstract.
We prove Thompson’s group F has quadratic conjugator lengthfunction. That is, for any two conjugate elements of F of length n or less,there exists an element of F of length O ( n ) that conjugates one to the other.Moreover, there exist conjugate pairs of elements of F of length at most n such that the shortest conjugator between them has length Ω( n ). This latterstatement holds for T and V as well. Let G be a group with finite generating set S , and let (cid:96) : G → N be the wordlength function on G with respect to S . If g and h are conjugate elements of G ,the conjugator distance from g to h iscd( g, h ) = min { (cid:96) ( k ) | k ∈ G and h = k − gk } . The conjugator length function for G (with respect to S ) is the functionCLF : N → N defined byCLF( n ) = max { cd( g, h ) | g, h ∈ G are conjugate and (cid:96) ( g ) + (cid:96) ( h ) ≤ n } . This definition first appeared in A. Sale’s doctoral dissertation [17], where it iscredited to T. Riley. Note that CLF depends on the finite generating set S , but ifCLF has polynomial growth then the degree of the polynomial is independent of S .The function CLF can be viewed as measuring the difficulty of the conjugacyproblem in G . If G has solvable word problem, then the conjugacy problem issolvable in G if and only if CLF is a computable function, or equivalently if and onlyif CLF has a computable upper bound. The conjugator length function has beenestimated for many classes of groups. It has been shown to be linear in hyperbolicgroups [15], mapping class groups [1, 16, 21] and some metabelian groups [19, 20],including lamplighter groups Z q (cid:111) Z and solvable Baumslag-Solitar groups. It isat most quadratic in fundamental groups of prime 3-manifolds [19], at most cubicin free solvable groups [18], and at most exponential in CAT(0)-groups [7] andin certain semidirect products Z d (cid:111) Z k [19]. Sale also gives examples in [18] ofwreath products whose conjugator length functions have quadratic lower bounds.In upcoming work [8], Bridson, Riley, and Sale give examples of finitely presentedgroups whose conjugator length function is polynomial of arbitrary degree, as wellan example of a finitely presented group whose conjugator length function growslike 2 n . The first author has been partially supported by EPSRC grant EP/R032866/1 during thecreation of this paper.The second author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geomet-riche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM)and gratefully acknowledges the support of the Funda¸c˜ao para a Ciˆencia e a Tecnologia (CEMAT-Ciˆencias FCT projects UIDB/04621/2020 and UIDP/04621/2020) and of the Universit`a degliStudi di Milano–Bicocca (FA project ATE-2016-0045 “Strutture Algebriche”). a r X i v : . [ m a t h . G R ] J a n JAMES BELK AND FRANCESCO MATUCCI
Thompson’s group F is the group defined by the presentation (cid:104) x , x , x . . . . | x n x k = x k x n +1 for n > k (cid:105) . This is one of three groups introduced by Richard J. Thompson in the 1960’s,which have since become important examples in geometric group theory. See [11]for a general introduction to Thompson’s groups. Since x n = x − n x x n − for all n ≥
2, Thompson’s group F is generated by the elements { x , x } . In fact there isa presentation for F with these generators and two relations (see [11]).Our main theorem is the following. Main Theorem.
The conjugator length function for Thompson’s group F hasquadratic growth. That is, there exist constants < a < b so that an ≤ CLF( n ) ≤ bn . for all sufficiently large n . The conjugacy problem for F was first solved by V. Guba and M. Sapir as a spe-cial case of the solution for diagram groups [14]. In [5], the authors gave a differentdescription of this solution using the language of strand diagrams, and in [3, 4]it was shown that the solution to the conjugacy problem could be implementedin linear time. All of our work here is phrased using strand diagrams, but ourproof of the upper bound in Section 2 essentially follows Guba and Sapir’s proof[14, Theorem 15.23] while keeping track of the lengths of the conjugators.We prove the lower bound by exhibiting a sequence of pairs of conjugate elements( f n , g n ) whose lengths grow linearly with n but whose conjugator distance growsquadratically. The main idea is that the “area” of a conjugating strand diagram canbe forced to be much larger than areas of the strand diagrams for the two conjugateelements, as shown in Figure 8. The elements we choose have cyclic centralizers,which makes it easy to compute an explicit lower bound for the conjugator distance.All of the arguments in this paper can be modified to work for the generalizedThompson groups F n (see [9]), and more generally for diagram groups over finitepresentations of finite semigroups (see [14]).We prove that a quadratic lower bound holds for T and V as well (see Theo-rem 3.5), and we would conjecture that a quadratic upper bound holds for T as wellusing a modified version of the arguments in Section 2. In Thompson’s group V the word length is not comparable to the complexity of a strand diagram (see Re-mark 1.3), so the methods in Section 2 cannot be modified to obtain an upperbound better than CLF( n ) ≤ C ( n log n ) for some constant C .This paper is organized as follows. In Section 1 we establish a linear relationshipbetween the word length of an element and the number of nodes in the corre-sponding strand diagram. In Section 2 we prove a quadratic upper bound for theconjugator length function using the known solution to the conjugacy problem in F .Finally, in Section 3 we prove a quadratic lower bound for the conjugator lengthfunction by exhibiting the aforementioned sequence of pairs ( f n , g n ) and analyzingtheir length and conjugator distance. Acknowledgements.
The authors would like to thank Timothy Riley and AndrewSale for drawing our attention to the question addressed in this paper, and forsuggesting the inclusion of Theorem 3.5. We would also like to thank Collin Bleakfor several helpful suggestions and comments.
ONJUGATOR LENGTH IN THOMPSON’S GROUPS 3 x x Figure 1.
Strand diagrams for the generators { x , x } of F .1. Strand Diagrams, Norm, and Length
We will use the description of F as the group of reduced (1 , F using reduced annular strand diagrams. We will also make use of the groupoidof ( m, n )-strand diagrams described in [5]. We write f · g for the concatenation oftwo diagrams with f on the top and g on the bottom (the opposite of the conventionin [5]), with the sinks of f attached to the sources of g . The product fg of two reducedstrand diagrams is the strand diagram obtained by reducing the concatenation f · g .Furthermore, if f is an ( m, n )-strand diagram and g is an ( m (cid:48) , n (cid:48) )-strand diagram,we will let f ⊕ g denote the ( m + m (cid:48) , n + n (cid:48) )-strand diagram obtained by placing g to the right of f .The reduced strand diagrams for the generators x and x of F are shown inFigure 1. For f ∈ F , let (cid:96) ( f ) denote the word length of f with respect to the { x , x } generating set. An explicit formula for (cid:96) ( f ) was first given by Fordham [12], andvariants of Fordham’s formula were subsequently published by Belk and Brown [2]and Guba [13].Another strand diagram that we will use repeatedly is the right vine with k leaves. This is the (1 , k )-strand diagram t k shown in Figure 2.The norm (cid:107) f (cid:107) of a strand diagram f is its number of interior nodes (i.e. mergesand splits). A strand diagram f is trivial if (cid:107) f (cid:107) = 0. Note that (cid:107) f − (cid:107) = (cid:107) f (cid:107) , (cid:107) f · g (cid:107) = (cid:107) f (cid:107) + (cid:107) g (cid:107) , and (cid:107) fg (cid:107) ≤ (cid:107) f (cid:107) + (cid:107) g (cid:107) for any strand diagrams f and g . Note alsothat if f ∈ F , then the strand diagram for f always has the same number of mergesas splits, and therefore the norm (cid:107) f (cid:107) must be even. The following propositionrelates the norm of such an element to its length. Figure 2.
The right vine with k leaves, denoted t k . JAMES BELK AND FRANCESCO MATUCCI
Proposition 1.1.
For any f ∈ F , we have (cid:107) f (cid:107) − ≤ (cid:96) ( f ) ≤ (cid:107) f (cid:107) . Proof.
Observe that each tree in the reduced tree pair diagram for f has (cid:107) f (cid:107) / (cid:96) ( f ) ≤ (cid:107) f (cid:107) .For the lower bound, observe that (cid:107) t − x t (cid:107) = (cid:107) t − x t (cid:107) = 2, where t is theright vine with 3 leaves. Given a word f = s (cid:15) · · · s (cid:15) n n where each s i ∈ { x , x } andeach (cid:15) i = ±
1, we can write f = t (cid:0) t − s t (cid:1) (cid:15) · · · (cid:0) t − s n t (cid:1) (cid:15) n t − and hence (cid:107) f (cid:107) ≤ (cid:107) t (cid:107) + n (cid:88) i =1 (cid:107) t − s i t (cid:107) + (cid:107) t (cid:107) = 2 n + 4 . Thus (cid:107) f (cid:107) ≤ (cid:96) ( f ) + 4, so (cid:96) ( f ) ≥ (cid:107) f (cid:107) / − (cid:3) Remark . In fact we have (cid:107) f (cid:107) − ≤ (cid:96) ( f ) ≤ (cid:107) f (cid:107) − (cid:107) f (cid:107) ≥
6, since the leftmost ∧ -node in a tree pair always has weight 0 andthe two rightmost ∧ -nodes each have weight at most 2. Both bounds are sharp,with the lower bound realized by the elements x n and the upper bound realized bythe elements x (cid:0) x x − (cid:1) n x − (cid:0) x x − (cid:1) n x − . Remark . Burillo, Clear, Stein, and Taback have proven analog of Proposi-tion 1.1 for Thompson’s group T [10, Theorem 5.1], but no analogous result holdsfor Thompson’s group V . The trouble is that V allows arbitrary permutations ofthe leaves of a tree diagram, so there are at least ( n + 1)! different elements f ∈ V with (cid:107) f (cid:107) ≤ n , and therefore (cid:96) ( f ) is not bounded above by any linear functionof (cid:107) f (cid:107) . However, Birget has proven that there exists a constant C > (cid:96) ( f ) ≤ C (cid:107) f (cid:107) log (cid:107) f (cid:107) for all f ∈ V [6, Theorem 3.8].2. An Upper Bound
Recall that the closure of an ( n, n )-strand diagram f is the annular stranddiagram obtained by attaching the sources and sinks of f together and eliminatingthe resulting bivalent vertices. Note that the closure of a reduced strand diagramneed not be reduced. We say that a strand diagram f is strongly cyclicallyreduced if its closure is reduced and has the same number of connected componentsas f . (These correspond to the “absolutely reduced normal diagrams” defined byGuba and Sapir in [14].) Lemma 2.1.
Let f be a nontrivial reduced ( i, i ) -strand diagram with (cid:107) f (cid:107) = n . Thenthere exists a reduced ( i, j ) -strand diagram h so that f (cid:48) = h − fh is strongly cyclicallyreduced, (cid:107) f (cid:48) (cid:107) ≤ n , and (cid:107) h (cid:107) ≤ n ( n + 4 i − . ONJUGATOR LENGTH IN THOMPSON’S GROUPS 5 = Figure 3.
The equality f ⊕ f = ( e j ⊕ f )( f ⊕ e i − k ), where e j and e i − k are trivial strand diagrams. Proof.
We proceed by induction on i + n . The base case is i + n = 1, for which f isthe trivial (1 , f has fewer componentsthan f . This occurs when f can be written as f = f ⊕ f , where f is a ( j, k )-stranddiagram with j (cid:54) = k and f is an ( i − j, i − k )-strand diagram. Without loss ofgenerality, suppose that j < k . Then we can rewrite f as a product f = ( e j ⊕ f )( f ⊕ e i − k )as shown in Figure 3, where e j and e i − k denote trivial strand diagrams with j strands and i − k strands, respectively. Let k be whichever of ( e j ⊕ f ) and ( f ⊕ e i − k ) − has fewer interior nodes, and let f (cid:48) = k − fk = ( f ⊕ e i − k )( e j ⊕ f ). Then f (cid:48) is an ( i + j − k, i + j − k )-strand diagram with (cid:107) f (cid:48) (cid:107) ≤ n (there may be fewer than n interior nodes if f (cid:48) is not initially reduced), and (cid:107) k (cid:107) ≤ n/
2. Since i + j − k ≤ i − i + j − k ) + (cid:107) f (cid:48) (cid:107) < i + n . Therefore, it follows from our inductionhypothesis that there exists an h with (cid:107) h (cid:107) ≤ n ( n + 4( i − − n ( n + 4 i − . such that h − f (cid:48) h is strongly cyclically reduced. Then ( kh ) − f ( kh ) is strongly cycli-cally reduced and (cid:107) kh (cid:107) ≤ (cid:107) k (cid:107) + (cid:107) h (cid:107) ≤ n n ( n + 4 i − n ( n + 4 i − . Now consider the case where the closure of f has the same number of componentsas f . Note then that any trivial strands of f (i.e. edges whose endpoints are a sourceand a sink) must correspond to free loops in the closure. If the closure of f isreduced then we are done, so it must be possible to apply a reduction of type I, II,or III to the closure of f as described in [5] (see Figure 4).Suppose first that the closure of f is subject to a type I reduction. Since all thetrivial strands of f correspond to free loops, there must be a 1 ≤ j ≤ i − j and j + 1 are connected to a merge and sinks j and j + 1 are connected toa split. Let k be the ( i, i − j and j + 1 and sink j . Then f (cid:48) = k − fk is a nontrivial ( i − , i − (cid:107) f (cid:48) (cid:107) = n −
2. Since ( i −
1) + ( n − < i + n , our induction hypothesis JAMES BELK AND FRANCESCO MATUCCI I −−→ II −−→ III −−→
Figure 4.
Reductions of type I, II or III for annular strand dia-grams.tells us that there exists a reduced strand diagram h with (cid:107) h (cid:107) ≤ n − (cid:0) ( n −
2) + 4( i − − (cid:1) n − n + 4 i − h − f (cid:48) h is strongly cyclically reduced. Then ( kh ) − f ( kh ) is strongly cycli-cally reduced and (cid:107) kh (cid:107) ≤ (cid:107) k (cid:107) + (cid:107) h (cid:107) ≤ n − n + 4 i − ≤ n ( n + 4 i − , where the last inequality follows from the fact that n ≥ i ≥ f is subject to a type II reduction. Again, sinceall the trivial strands of f correspond to free loops, there must exist a 1 ≤ j ≤ i sothat source j of f is connected to a split and sink j of f is connected to a merge.Let k be the ( i, i + 1)-strand diagram with exactly one split connected to source j and sinks j and j + 1. Then f (cid:48) = k − fk is a nontrivial ( i + 1 , i + 1)-strand diagramwith (cid:107) f (cid:48) (cid:107) = n −
2. Since ( i + 1) + ( n − < i + n , our induction hypothesis tells usthat there exists a reduced strand diagram h with (cid:107) h (cid:107) ≤ n − (cid:0) ( n −
2) + 4( i + 1) − (cid:1) n − n + 4 i − h − f (cid:48) h is strongly cyclically reduced. Then ( kh ) − f ( kh ) is strongly cyclicallyreduced and (cid:107) kh (cid:107) ≤ (cid:107) k (cid:107) + (cid:107) h (cid:107) ≤ n − n + 4 i − ≤ n ( n + 4 i − , where the last inequality follows from the fact that i ≥ f is subject to a type III reduction. Thenthere exists a 1 ≤ j ≤ i − j is connected directly to sink j andsource j + 1 is connected directly to sink j + 1 in f . Let k be the ( i, i − j and j + 1 and sink j . Then f (cid:48) = k − fk is an ( i − , i − (cid:107) f (cid:48) (cid:107) = n . Since ( i −
1) + n < i + n ,our induction hypothesis tells us that there exists a reduced strand diagram h with (cid:107) h (cid:107) ≤ n (cid:0) n + 4( i − − (cid:1) n ( n + 4 i − h − f (cid:48) h is strongly cyclically reduced. Then ( kh ) − f ( kh ) is strongly cyclicallyreduced and (cid:107) kh (cid:107) ≤ (cid:107) k (cid:107) + (cid:107) h (cid:107) ≤ n ( n + 4 i − ≤ n ( n + 4 i − . The last inequality follows from the fact that f is nontrivial, and hence n ≥ (cid:3) ONJUGATOR LENGTH IN THOMPSON’S GROUPS 7
Figure 5.
The deck transformation δ takes f ∪ h to h ∪ g , andhence fh = hg . Corollary 2.2.
Let f ∈ F with (cid:107) f (cid:107) = n . Then there exists a reduced (1 , j ) -stranddiagram h so that f (cid:48) = h − fh is strongly cyclically reduced, (cid:107) f (cid:48) (cid:107) ≤ n , and (cid:107) h (cid:107) ≤ ( n − + 78 . Lemma 2.3.
Let f and g be strongly cyclically reduced strand diagrams whoseclosures are the same, and let n = (cid:107) f (cid:107) = (cid:107) g (cid:107) . Then there exists a reduced stranddiagram h so that g = h − fh and (cid:107) h (cid:107) ≤ n .Proof. Suppose first that f and g are connected. Let f ∞ be the lift of the clo-sure of f to universal cover of the annulus. Then f ∞ can be viewed as an infiniteconcatenation of f ’s, i.e. f ∞ = (cid:91) k ∈ Z f k where each f k is a copy of f and the sinks of each f k are the same as the sourcesof f k +1 . We can also decompose f ∞ as an infinite concatenation (cid:83) k ∈ Z g k of copiesof g . Indeed, we can choose such a decomposition so that g ⊆ (cid:83) ∞ k =1 f k and g intersects f . Let h be the strand diagram that lies between f and g . Then δ ( f ∪ h ) = h ∪ g , where δ : f ∞ → f ∞ is the deck transformation that maps each f k to f k +1 , so the strand diagrams fh and hg are the same (see Figure 5).If f and g are the identity we are done. Otherwise, we claim that g ⊆ n/ (cid:91) k =1 f k . It follows that h ⊆ (cid:83) n/ k =1 f k , so (cid:107) h (cid:107) ≤ (3 n/ (cid:107) f (cid:107) = 3 n / full edges of f to be those that start and end attrivalent vertices, and the half edges of f to be those that have either a source ora sink at one end. (Since f is connected and nontrivial, there are no edges directly JAMES BELK AND FRANCESCO MATUCCI from a source to a sink.) We place a δ -invariant geodesic metric on f ∞ so that eachfull edge of f has length 1 and each half edge has length 1 /
2. Since f has exactly n interior nodes, the total length of all of the edges of f is 3 n/
2. Then the totallength of all of the edges of g must be exactly 3 n/
2, and in particular the diameterof g is at most 3 n/
2. Since the minimum distance from a source to a sink in f isat least 1, the claim follows easily.For the general case, suppose f = f ⊕ · · · ⊕ f k , where each f i is connected and hasthe same number of sources as sinks. Since f and g are strongly cyclically reducedand have the same closure, it follows that g = g ⊕ · · · ⊕ g k , where each g i isconnected, has the same number of sources as sinks, and has the same closure as f i .By the argument above, there exists for each i a reduced strand diagram h i with (cid:107) h i (cid:107) ≤ (cid:107) f i (cid:107) / g i = h − i f i h i . Then g = h − fh , where h = h ⊕ · · · ⊕ h n and (cid:107) h (cid:107) = k (cid:88) i =1 (cid:107) h i (cid:107) ≤ k (cid:88) i =1 (cid:107) f i (cid:107) ≤ (cid:18) k (cid:88) i =1 (cid:107) f i (cid:107) (cid:19) = 32 (cid:107) f (cid:107) . (cid:3) Theorem 2.4.
Let f, g ∈ F , with (cid:107) f (cid:107) = i and (cid:107) g (cid:107) = j . Suppose f and g areconjugate, with the corresponding reduced annular strand diagram having k nodes.Then there exists an h ∈ F so that g = h − f h and (cid:107) h (cid:107) ≤ ( i − + ( j − + 12 k + 148 . Proof.
By Corollary 2.2, there exist reduced strand diagrams h and h with (cid:107) h (cid:107) ≤ ( i − + 78 and (cid:107) h (cid:107) ≤ ( j − + 78so that f = h − f h and g = h − g h are strongly cyclically reduced. Then f and g have the same closure and (cid:107) f (cid:107) = (cid:107) g (cid:107) = i , so by Lemma 2.3 there exists a stranddiagram h with (cid:107) h (cid:107) ≤ k so that g = h − fh . Let h = h h h − . Then g = h − f h which means that h ∈ F , and (cid:107) h (cid:107) ≤ (cid:107) h (cid:107) + (cid:107) h (cid:107) + (cid:107) h (cid:107) ≤ ( i − + ( j − + 12 k + 148 . (cid:3) Corollary 2.5.
Let f and g be conjugate elements of F with (cid:96) ( f ) = i and (cid:96) ( g ) = j ,where i ≤ j . Then cd( f, g ) ≤ i + j + 27 i + 3 j + 20 . In particular, the conjugator length function of Thompson’s group F satisfies CLF( n ) ≤ n + 15 n + 20 for all n ∈ N .Proof. By Proposition 1.1, we know that (cid:107) f (cid:107) ≤ i + 4 and (cid:107) g (cid:107) ≤ j + 4. Moreover,since the closure of a nontrivial (1 , f (and hence g ) has at most (cid:107) f (cid:107) − i + 2 nodes. By Theorem 2.4 and Proposition 1.1, there exists an h ∈ F so that g = h − f h and (cid:96) ( h ) ≤ (cid:107) h (cid:107) ≤ i + 3) + (2 j + 3) + 12(2 i + 2) + 148 = 13 i + j + 27 i + 3 j + 20 . ONJUGATOR LENGTH IN THOMPSON’S GROUPS 9
Figure 6.
The (2 n + 1 , n + 1)-strand diagram f n .If n = i + j , it follows that (cid:96) ( h ) ≤ i + j + 27 i + 3 j + 20= 72 n + 15 n + 20 −
12 (19 i + 5 j + 24)( j − i ) ≤ n + 15 n + 20 . (cid:3) A Lower Bound
In this section we prove a quadratic lower bound on conjugator lengths for ele-ments of F . This will require some known results about centralizers in F .For the following proposition, a proper root of an element f ∈ F is an element g ∈ F such that f = g k for some k ≥ Proposition 3.1.
Let f ∈ F and suppose that f has no proper roots in F and thereduced annular strand diagram for f is connected. Then the centralizer of f in F is the cyclic group (cid:104) f (cid:105) generated by f .Proof. Guba and Sapir compute centralizers for elements of diagram groups in [14,Theorem 15.35], and this follows easily from their proof. (cid:3)
For any n ≥
2, let f n and g n be the strand diagrams shown in Figures 6 and 7,and let f n , g n ∈ F be the elements f n = t n +1 f n t − n +1 and g n = t n g n t − n where t k denotes the right vine with k leaves shown in Figure 2. It is tedious butstraightforward to check that f n = x x x · · · x n − x − n − · · · x − x − = x (cid:0) x − x − x x − (cid:1) n − x − (cid:0) x x x x − (cid:1) n − Figure 7.
The (3 n, n )-strand diagram g n . Figure 8.
The (2 n +1 , n )-strand diagram h n lies in the triangularregion between the bottom red curve and the top blue curve. Notethat f n h n = h n g n .and g n = x x x · · · x n − x − n − · · · x − x − = x (cid:0) x − x − x x − (cid:1) n − x − ( x x x ) n − . It follows that (cid:96) ( f n ) ≤ n − (cid:96) ( g n ) ≤ n − Theorem 3.2.
The elements f n and g n satisfy cd( f n , g n ) ≥ n − n − . Proof.
Since the closure of f n is connected and already reduced, the reduced annularstrand diagram for f n is connected. Moreover, since f (cid:48) n (0) = 2, the element f n hasno proper roots in F . By Proposition 3.1, we deduce that the centralizer of f n isjust (cid:104) f n (cid:105) .Now let h n be the (2 n + 1 , n )-strand diagram shown in Figure 8, and observethat f n h n = h n g n . Then the element h n = t n +1 h n t n conjugates f n to g n , so bythe previous paragraph every conjugator from f n to g n must have the form f kn h n for some k ∈ Z . We must prove that (cid:96) ( f kn h n ) ≥ ( n − n − / k ∈ Z . ONJUGATOR LENGTH IN THOMPSON’S GROUPS 11
Observe that the interior nodes of h n in Figure 8 lie on 2 n − h n as a product r r · · · r n − , where each r i includes two rows of interiornodes. (Note that r i depends on both i and n , though we do not make this explicitin our notation.) Then (cid:107) r i (cid:107) = 4( n − i ) + 1 for each i , so (cid:107) h n (cid:107) = n − (cid:88) i =1 (cid:0) n − i ) + 1 (cid:1) = ( n − n + 1) . Now, for any k ∈ Z the concatenation f kn · h n has k (cid:107) f n (cid:107) + (cid:107) h n (cid:107) interior nodes, butis not necessarily reduced. There are three cases: • For k ≥
0, the concatenation f kn · h n is reduced, so (cid:107) f kn h n (cid:107) = k (cid:107) f n (cid:107) + (cid:107) h n (cid:107) ≥ (cid:107) h n (cid:107) = ( n − n + 1) . • For 1 ≤ j ≤ n −
1, the product f − jn h n cancels precisely the first 2 j rows r · · · r j of h n . Then (cid:107) f − jn h n (cid:107) = (cid:107) h n (cid:107) + j (cid:107) f n (cid:107) − j (cid:88) i =1 (cid:107) r i (cid:107) = (cid:107) h n (cid:107) + j (4 n ) − j (cid:88) i =1 (cid:0) n − i ) + 1 (cid:1) = (cid:107) h n (cid:107) − j (2 n − j − j = (cid:98) n/ (cid:99) , with a minimum value of (cid:107) h n (cid:107) − n ( n −
1) = n − • For j ≥ n , the product f − jn h n cancels all of h n , so (cid:107) f − jn h n (cid:107) = j (cid:107) f n (cid:107) − (cid:107) h n (cid:107) = j (4 n ) − ( n − n + 1) ≥ n + n + 1 . We conclude that (cid:107) f kn h n (cid:107) ≥ n − k ∈ Z . Since f kn h n = t n +1 f kn h n t − n , itfollows that (cid:107) f kn h n (cid:107) ≥ (cid:107) f kn h n (cid:107) − (cid:107) t n +1 (cid:107) − (cid:107) t n (cid:107) ≥ n ( n − k ∈ Z , so (cid:96) ( f kn h n ) ≥ n ( n − / − n − n − / (cid:3) Remark . Using any of the known length formulas for F [2, 13, 12] together withthe analysis of centralizers in the proof of Theorem 3.2, it is possible to show thatin fact (cid:96) ( f n ) = (cid:96) ( g n ) = 8 n − f n , g n ) = (cid:6) n − n + 4 (cid:7) for all n ≥
3, with f −(cid:98) n/ (cid:99) n h n being the unique minimum-length conjugator for n ≥
4. It follows thatthe conjugator length function for Thompson’s group F satisfiesCLF( n ) ≥ (cid:22) n + 1016 (cid:23) − (cid:22) n + 1016 (cid:23) + 4 ≥ ( n − + 412128for all n ≥ T and V . This dependson the following lemma. Lemma 3.4. If f ∈ F and the reduced annular strand diagram for f is connected,then the centralizer of f in V is the same as the centralizer of f in F . Proof.
Let k ∈ V so that kf = f k . Since the reduced annular strand diagram for f is connected, we know that f has no dyadic fixed points in the interval (0 ,
1) (see [5,Theorem 5.2]). Let 0 = p < p < · · · < p m = 1 be the fixed points of f , whichmust be permuted by k . However, observe that for each x ∈ [0 , \{ p , . . . , p m } , thefull f -orbit { f n ( x ) } n ∈ Z has accumulation points at p i − and p i for some i . Since k maps full f -orbits to full f -orbits, it follows that k ( p i ) = p i for all i , and indeed k maps each interval [ p i − , p i ] to itself.All that remains is to show that k is order-preserving on each interval ( p i − , p i ),and hence k ∈ F . Let (cid:15) > k is linear on ( p i − (cid:15), p i ], and let p i − < x In Thompson’s group T or V , there exists a constant C > so that CLF( n ) ≥ Cn for all n ∈ N .Proof. Recall that elements of Thompson’s group V can also be represented bystrand diagrams (see [5]). If we fix a finite generating set for V , the length (cid:96) V ( f )and norm (cid:107) f (cid:107) of an element f ∈ V are related by the formula (cid:107) f (cid:107) ≤ m (cid:96) V ( f ),where m is the maximum norm of any generator for V . If f ∈ F , it follows fromProposition 1.1 that (cid:96) F ( f )2 m ≤ (cid:107) f (cid:107) m ≤ (cid:96) V ( f ) ≤ (cid:96) F ( f ) . That is, the embedding of F into V is quasi-isometric. A similar argument showsthat the embedding of F into T is quasi-isometric.Now, by Lemma 3.4 the centralizers of the elements f n are the same in T or V as they are in F . It follows that the conjugators from f n to g n in T or V are thesame as they are in F , and since the word lengths of the conjugators are the sameup to a linear factor we obtain a quadratic lower bound on the conjugator lengthfunction. (cid:3) References [1] J. Behrstock and C. 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