The growth of torus link groups
aa r X i v : . [ m a t h . G R ] J a n THE GROWTH OF TORUS LINK GROUPS
YOSHIYUKI NAKAGAWA, MAKOTO TAMURA, AND YASUSHI YAMASHITA
Abstract.
Let G be a finitely generated group with a finite generating set S . For g ∈ G , let l S ( g ) be the length of the shortest word over S representing g . The growth series of G with respect to S is the series A ( t ) = P ∞ n =0 a n t n ,where a n is the number of elements of G with l S ( g ) = n . If A ( t ) can beexpressed as a rational function of t , then G is said to have a rational growthfunction.We calculate explicitly the rational growth functions of ( p, q )-torus linkgroups for any p, q > . As an application, we show that their growth rates arePerron numbers. Introduction and the main result
Let G be a finitely generated group with a finite generating set S . For g ∈ G , wedenote by l S ( g ) the length of the shortest word over S representing g . The growthseries of G with respect to S is the series A ( t ) = P ∞ n =0 a n t n , where a n is the numberof elements of G with l S ( g ) = n . In many important and interesting cases, A ( t ) canbe expressed as a rational function of t . However, it is often very difficult to compute A ( t ) explicitly. Note that the radius of convergence R of A ( t ) = P ( t ) /Q ( t ) is themodulus of a zero of the denominator Q ( t ) closest to the origin 0 ∈ C of all zerosof Q ( t ). Recall that the growth rate ω = lim sup n →∞ n √ a n is equal to 1 /R . AfterCannon-Wagreich [1], nice algebraic properties of R or ω (e.g., Salem numbers,Pisot numbers, Perron numbers) were investigated.In this paper, we consider torus link group: the fundamental group of the com-plement of a torus link in 3-sphere. They are important objects in knot theory. The( p, q )-torus link group has a presentation T p,q = h x, y | x p = y q i . T p,q has a nontrivialcenter generated by x p = y q and we add this element, say z , to the generating set.We denote this presentation by T ′ p,q = h x, y, z | x p = y q = z i . The case for p = 2and q = 3 (the trefoil group) has some history. In [8], T , and T ′ , were studiedas a three stranded braid group and the rational growth functions for both caseswere given. Also, the rational growth function for T ′ , was calculated in [4] andin chapter 14 of [7], which is the starting point of this paper. We note that T , and T , were studied in [6] as examples of groups for which the growth rates arerealized. (See [6] for the definition of “realized.”)The aim of this paper is to give the explicit expression of the rational growthfunction of T ′ p,q for any p, q >
1, and study the algebraic property of the growth rate.We denote by C n ( t ) the growth series (polynomial) of the cyclic group of order n , Date : June 14, 2018.2010
Mathematics Subject Classification.
Primary 20E06; Secondary 20F05, 57M25.
Key words and phrases. growth function, torus link group, free product with amalgamation,Perron number.This work was supported by JSPS KAKENHI Grant Number 23540088. i.e., C n ( t ) = 1+2 t + · · · +2 t ( n − / if n is odd, and C n ( t ) = 1+2 t + · · · +2 t n/ − + t n/ if n is even, and C ∞ ( t ) = (1 + t ) / (1 − t ) for infinite cyclic case. Here is our maintheorem: Theorem 1.1.
The rational growth function of T ′ p,q = h x, y, z | x p = y q = z i forany p, q > is A ( t ) = C ∞ ( t ) C p ( t ) C q ( t ) C p ( t ) + C q ( t ) − C p ( t ) C q ( t ) + m q ( t ) C p ( t ) + m p ( t ) C q ( t ) ( C p ( t ) + C q ( t ) − C p ( t ) C q ( t )) , where m r ( t ) = t r/ if r is even, if r is odd. The proof will be given in the next section.
Remark . The rational growth function of h x, y | x p = y q i was calculated by Gill.See Theorem 2.3.2 in [2]. (It was pointed out that it has a misprint in the formula.See [6] [7].) Remark . Our result should have a generalization to the groups h x , x , . . . , x r , z | x p = x p = · · · = x p r r = z i , where 2 ≤ p ≤ p ≤ · · · ≤ p r . See [2]. For example, if p i is odd for all i , we canapply Theorem 2.2 in the next section one by one.Here is a direct consequence of the main result. A real number ω is called aPerron number if it is greater than 1 and an algebraic integer whose conjugateshave moduli less than the modulus of ω . Corollary 1.4.
The growth rate of T ′ , is . If ( p, q ) = (2 , , the growth rate of T ′ p,q is a Perron number. To prove this corollary, let us recall a result by Komori-Umemoto [5]. To showthat the growth rates of three-dimensional non-compact hyperbolic Coxeter groupswith four generators are Perron numbers, they proved the next lemma:
Lemma 1.5 (Lemma 1 in [5]) . Consider the polynomial of degree n ≥ g ( t ) = n X k =1 a k t k − , where a k is a non-negative integer. We also assume that the greatest commondivisor of { k ∈ N | a k = 0 } is . Then there exists a real number r , < r < which is the unique zero of g ( t ) having the smallest absolute value of all zeros of g ( t ) .Proof of Corollary 1.4. The growth rate of T ′ , can be calculated directly.Suppose that ( p, q ) = (2 , T ′ p,q to h x, y, z | x = y q = z = id i ≃ Z p ∗ Z q , we see that the growth rate ω is greaterthan 1. (See Theorem 16.12 in [7].) Hence, the radius of convergence R of A ( t ) issmaller than 1. Put g ( t ) = ( C p ( t ) − C q ( t ) − −
1. Clearly, g ( t ) satisfies thecondition of the lemma. By the main theorem, the set of poles of A ( t ) other than1 is contained in the set of zeros of g ( t ). Thus, R is equal to the unique real zero of g ( t ) = 0 between 0 and 1. Then, ω is a zero of the reciprocal g ∗ ( t ) of − g ( t ) whichis monic since the constant term of − g ( t ) is 1, and zeros of g ∗ ( t ) have moduli lessthan the modulus of ω . This completes the proof. (cid:3) HE GROWTH OF TORUS LINK GROUPS 3 Proof
In this section, we prove the main theorem. What we will do is to follow thearguments in chapter 14 of [7] carefully, apply the same idea to the general caseswith some modification, and carry out the calculations.Let p, q be integers greater than one. Let X p = h x, z | x p = z i and Y q = h y, z | y q = z i . They are infinite cyclic groups with one extra generator added. We say that ashortest word in X p (resp. Y q ) of the form z i x j (resp. z i y j ) a normal form.We separate the proof into three cases: (1) p and q are both odd, (2) p is evenand q is odd, (3) p and q are both even.2.1. p and q are both odd. To our surprise, this case was very easy. We can applya general theorem for free product with amalgamation. Let us recall notation anda theorem. See [7] for detail.
Definition 2.1 (Section 14.3 in [7]) . Consider pairs (
G, S ), where G is generatedby S . A pair ( H, T ) is admissible in (
G, S ), if H is a subgroup of G , T is a subsetof S , and there exists a transversal U for H in G , termed an admissible transversal,such that if g = hu with g ∈ G , h ∈ H , u ∈ U , then l S ( g ) = l T ( h ) + l S ( u ). Wealways assume that the transversal contains the identity as the representative for H . Theorem 2.2 (Prop. 14.2 in [7]) . Let ( L, R ) be admissible in both ( H, S ) and ( K, T ) . Let G = H ∗ L K . Let G , H , K , L have growth functions A ( t ) , B ( t ) , C ( t ) and D ( t ) , respectively, relative to the generating set S ∪ T , S , T , R . Then we have A ( t ) = 1 B ( t ) + 1 C ( t ) − D ( t ) . Now, let us begin the proof for this case. Suppose that p = 2 n + 1 and q =2 m + 1. Note that T ′ p,q = h x, y, z | x p = y q = z i = X p ∗ h z i Y q is a free product withamalgamation.Then, ( h z i , { z } ) is admissible in both ( X p , { x, z } ) and ( Y q , { y, z } ). To see this,set U p = { x − n , x − n +1 , . . . , x n − , x n } and V q = { y − m , y − m +1 , . . . , y p − , y m } . Then, U p (resp. V q ) is an admissible transversal for h z i in X p (resp. Y q ).Hence, we can apply Theorem 2.2. Let B ( t ) , C ( t ) and D ( t ) be the growthfunctions for X p , Y q and h z i , respectively. Then, it is easy to see that B ( t ) = C ∞ ( t ) C p ( t ), C ( t ) = C ∞ ( t ) C q ( t ), and D ( t ) = C ∞ ( t ). By Theorem 2.2, we have A ( t ) = C ∞ ( t ) C p ( t ) C q ( t ) C p ( t ) + C q ( t ) − C p ( t ) C q ( t )which gives the claimed formula.2.2. p is even and q is odd. Suppose that p = 2 n and q = 2 m + 1. Unfortunately,( h z i , { z } ) is not admissible in ( X p , { x, z } ). Set U p = { x − n , x − n +1 , . . . , x n − , x n } and V q = { y − m , y − m +1 , . . . , y p − , y m } as in the previous case. Note that V q is anadmissible transversal for h z i in Y q , but U p is not a transversal for h z i in X p because x n = zx − n .2.2.1. The minimal normal form. (See section 14.1 of [7].)Let w be a shortest word in T ′ p,q . Collecting together generators from the samegroup, we obtain a product in which elements from X p and Y q alternate. Wereplace each factor by its normal form in X p or Y q . Then, we get a product in YOSHIYUKI NAKAGAWA, MAKOTO TAMURA, AND YASUSHI YAMASHITA which elements from h z i alternate with elements from U p and V q . If we have inthis product a segment of the form xz j , say, then xz j ∈ X p , and we can replace itby its normal form z j ′ x ′ , without increasing the length. In this way, we representeach element in G by a minimal word of the form w = z i u where the occurrencesof elements in U p and V q alternate in u ,Suppose that i > x − n occurs in u . Then, we can replace x − n by z − x n and send this z − to the left and replace z i by z i − . The case for i < x n is similar. Therefore, we may assume that, if i = 0, then in all occurrences of x ± n , the exponent has the same sign, which is the sign of i .Next, suppose that i = 0. If u contains both x n and x − n , then we can write x − n = z − x n , then move z − to the position of x n , and replace this z − x n by x − n .This means that we can interchange the occurrences of x n and x − n , therefore wemay assume that all terms x n proceed all terms x − n . We assume that our minimalwords with i = 0 are of this form, and call them minimal normal form.We claim that this form (minimal normal form) is unique.To see this, we replace all occurrences of x − n in w = z i u by zx n and send this z to the left. This (possibly not shortest) form is called “canonical form”. Weclaim that this form of the elements in ( p, q )-torus link group G is unique. Indeed,suppose that z i u and z j v are canonical forms with z i u = z j v in G . Note that, byintroducing the relation x p = y q = z = id , we can map G to H = Z p ∗ Z q . Then,we find the equality u = v in H , and since H is a free product, u and v are thesame words, and the equality u = v holds in G , too. It follows that i = j , and theuniqueness of the canonical form is proved.One can show that the map from the set of minimal normal forms to the set ofcanonical forms is injective by the same arguments given in 14.1 in [7]. It followsthat the minimal normal forms are also unique.2.2.2. Counting the minimal normal words. If z i u is a normal form, then u is ofthe form u v u v · · · u r v r , where u j ∈ U p and v k ∈ V q with u = id for i > v r = id for j < r . Let M be the set of all minimal normal words. Define M := { w ∈ M | w = z i u, i > } , M := { w ∈ M | w = z i u, i < } ,M := { w ∈ M | w = z u, u = id } , M := { w ∈ M | w = z u, u = id } . Then, we have M = M ∪ M ∪ M ∪ M and M i ∩ M j = ∅ if i = j. First, we consider the growth function of M . Put U + p = U p \ { x − n } . Then,note that C p ( t ) = 1 + 2 t + · · · + 2 t n − + t n (resp. C q ( t ) = 1 + 2 t + · · · + 2 t m , ) isthe “growth polynomial” for U + p (resp. V q ). Let r be a positive integer. Then, thegrowth polynomial for the set of words M ( r ) := (cid:26) u v u v · · · u r v r (cid:12)(cid:12)(cid:12)(cid:12) u i ∈ U + p , u i = id (for i > ,v j ∈ V q , u j = id (for j < r ) (cid:27) is C p ( t ) (cid:16)Q r − k =1 ( C q ( t ) − C p ( t ) − (cid:17) C q ( t ) . (See, for example, the proof of Prop.1.5 in [7].) Since x − n does not occur in any word of M , any u in w = z i u of M is contained in M ( r ) for some r , and the growth function A ( t ) for M is A ( t ) = ( t + t + t + . . . ) ∞ X r =1 C p ( t )( C q ( t ) − r − ( C p ( t ) − r − C q ( t )= t − t · C p ( t ) C q ( t )1 − ( C p ( t ) − C q ( t ) − . Since x n does not occur in any word of M , the growth function A ( t ) for M isequal to A ( t ),Next, we consider the growth function A ( t ) for M . Suppose again that r is apositive integer and define M ( r ) = (cid:26) u v u v · · · u r v r ∈ M (cid:12)(cid:12)(cid:12)(cid:12) u i ∈ U p , u i = id (for any i ) ,v j ∈ V q , u j = id (for j < r ) (cid:27) . Then, each word in M ( r ) is determined by the corresponding word in M ( r ) (weconsider that x − n corresponds to x n in U + p .) and the number, say k , of occurrencesof x − n . The number k varies between 0 and the number of occurrences of x n inthe corresponding word in M ( r ). Also, the growth series of the image of M ( r )in M ( r ) is ( C p ( t ) − r ( C q ( t ) − r − C q ( t ). Hence, to calculate the growth seriesfor M ( r ), for each term in ( C p ( t ) − r ( C q ( t ) − r − C q ( t ) which corresponds toa word having k occurrences of x n , we have to multiply this term by k + 1. Todo this, put D p ( t ) := 2 t + 2 t + · · · + 2 t n − , and E p ( t ) := t n . Since we have( C p ( t ) − r = P rk =0 (cid:0) rk (cid:1) D p ( t ) r − k E p ( t ) k , and E p ( t ) corresponds to x n , we define F ( r ) ( t ) := r X k =0 ( k + 1) (cid:18) rk (cid:19) D p ( t ) r − k E p ( t ) k . Then, F ( r ) ( t )( C q ( t ) − r − C q ( t ) is the growth function for M ( r ). By the binomialtheorem, we see that F ( r ) ( t ) = ( C p ( t ) − r + rE p ( t )( C p ( t ) − r − . Then, the growthfunction for M is A ( t ) = ∞ X r =1 F ( r ) ( t )( C q ( t ) − r − C q ( t )= ( C p ( t ) − C q ( t )1 − ( C p ( t ) − C q ( t ) −
1) + E p ( t ) C q ( t )(1 − ( C p ( t ) − C q ( t ) − For the last equality, we used P ∞ r =1 rs r − = 1 / (1 − s ) .Similarly, the growth function for M is A ( t ) = ∞ X r =1 F ( r − ( t )( C q ( t ) − r − C q ( t )= C q ( t )1 − ( C p ( t ) − C q ( t ) −
1) + E p ( t ) C q ( t )( C q ( t ) − − ( C p ( t ) − C q ( t ) − . Hence, the growth function for T ′ p,q is A ( t ) = A ( t ) + A ( t ) + A ( t ) + A ( t )= 1 + t − t C p ( t ) C q ( t )( C p ( t ) + C q ( t ) − C p ( t ) C q ( t )) + E p ( t ) C q ( t ) ( C p ( t ) + C q ( t ) − C p ( t ) C q ( t )) which gives the claimed formula. YOSHIYUKI NAKAGAWA, MAKOTO TAMURA, AND YASUSHI YAMASHITA p and q are both even. Suppose that p = 2 n and q = 2 m . We will use thesame notation as in the previous section.Similarly to the previous case, we can define the minimal normal form z i u .However, the case i = 0 needs some modification. Suppose that i = 0. Let a ∈{ x n , y m } and b ∈ { x − n , y − m } . Then, we can interchange the occurrences of a and b in u . Thus, we assume that our minimal words with i = 0 are of the followingtypes:( α ) All terms x n proceed all terms x − n , and u contains only y − m .( β ) u contains only x n , and all terms y m proceed all terms y − m .We assume that our minimal words with i = 0 are of this form. The uniqueness ofthis normal form can be proved similarly. Note that minimal normal words of type( γ ) u contains only x n and y − m .are exactly the intersection of type ( α ) and ( β ).Let M be the set of all minimal normal words defined above. Define M := { w ∈ M | w = z i u, i > } ,M := { w ∈ M | w = z i u, i < } ,M α := { w ∈ M | w = z u, u is of type ( α ) } ,M β := { w ∈ M | w = z u, u is of type ( β ) } ,M γ := { w ∈ M | w = z u, u is of type ( γ ) } . Since C p (resp. C q ) is the growth polynomial for U + p (resp. V + q ), the growthfunction A ( t ) for M is A ( t ) = t − t · C p ( t ) C q ( t )1 − ( C p ( t ) − C q ( t ) − , as in the previous case. The growth function A ( t ) for M is equal to A ( t ). Since M α corresponds to M and M in the previous case, its growth function A α ( t ) is A α ( t ) = ∞ X r =1 F ( r ) ( t )( C q ( t ) − r − C q ( t ) + ∞ X r =1 F ( r − ( t )( C q ( t ) − r − C q ( t )= C p ( t ) C q ( t )1 − ( C p ( t ) − C q ( t ) −
1) + E p ( t ) C q ( t ) (1 − ( C p ( t ) − C q ( t ) − . Similarly, we have A β ( t ) = C p ( t ) C q ( t )1 − ( C p ( t ) − C q ( t ) −
1) + E q ( t ) C p ( t ) (1 − ( C p ( t ) − C q ( t ) − ,A γ ( t ) = C p ( t ) C q ( t )1 − ( C p ( t ) − C q ( t ) − . Then, the rational growth function A ( t ) is A ( t ) + A ( t ) + A α ( t ) + A β ( t ) − A γ ( t )which gives the claimed formula.This completes the proof of the main theorem. Remark . Besides the proof, we have checked our formula for many examplesusing a software called kbmag by Holt [3].
HE GROWTH OF TORUS LINK GROUPS 7
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