Random Unitary Representations of Surface Groups II: The large n limit
RRandom Unitary Representations of Surface Groups II:The large n limit Michael MageeJanuary 12, 2021
Abstract
Let Σ g be a closed surface of genus g ≥ g denote the fundamental group of Σ g .We establish a generalization of Voiculescu’s theorem on the asymptotic ∗ -freeness of Haarunitary matrices from free groups to Γ g . We prove that for a random representation ofΓ g into SU ( n ), with law given by the volume form arising from the Atiyah-Bott-Goldmansymplectic form on moduli space, the expected value of the trace of a fixed non-identityelement of Γ g is bounded as n → ∞ . The proof involves an interplay between Dehn’swork on the word problem in Γ g and classical invariant theory. Contents U ( n ) and SU ( n ) . . . . . . . . . . . . . . . . . . . . . 82.3 The Weingarten calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Free groups and surface groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Witten zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Results of the prequel paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 k = (cid:96) = 0 . . . . . . . . . . . . . . . . . . . . . . . 153.3 A projection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 A combinatorial integration formula . . . . . . . . . . . . . . . . . . . . . . . . 18 a r X i v : . [ m a t h . R T ] J a n .5 Proof of Proposition 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 In a foundational series of papers [Voi85, Voi86, Voi87, Voi90, Voi91], Voiculescu developeda robust theory of non-commuting random variables that became known as non-commutativeprobability theory . One of the initial landmarks of this theory is the following result. Let F r denote the non-commutative free group of rank r . Let U ( n ) denote the group of n × n complexunitary matrices. For any w ∈ F r we obtain a word map w : U ( n ) r → U ( n ) by substitutingmatrices for generators of F r . Let µ Haar U ( n ) r denote the probability Haar measure on U ( n ) r andtr : U ( n ) → C the standard trace.A simplified version of Voiculescu’s result [Voi91, Thm. 3.8] can be formulated as follows : Theorem 1.1 (Voiculescu) . For any non-identity w ∈ F r , as n → ∞ lim n →∞ (cid:90) tr( w ( x )) dµ Haar U ( n ) r ( x ) = o w ( n ) . (1.1)We describe the interpretation of Theorem 1.1 as convergence of non-commutative randomvariables momentarily. Before this, we explain the main result of the current paper.Another way to think about the integral (1.1), that invites generalization, is to identify U ( n ) r with Hom( F r , U ( n )) and µ Haar U ( n ) r as a natural probability measure on this representationvariety . Now it is natural to ask whether there are other infinite discrete groups rather than F r such that Hom( F r , U ( n )) has a natural measure, and whether similar phenomena as inTheorem 1.1 may hold. The main point of this paper is to establish the analog of Theorem1.1 when F r is replaced by the fundamental group of a compact surface of genus at least 2. We now explain this generalization of Theorem 1.1; for technical reasons it superficiallylooks slightly different. For g ≥ g denote a closed topological surface of genus g . Welet Γ g denote the fundamental group of Σ g with explicit presentationΓ g = (cid:104) a , b , . . . , a g , b g | [ a , b ] · · · [ a g , b g ] (cid:105) .
1. The integral (1.1) is equal to 0 if w / ∈ [ F r , F r ], the commutator subgroup of F r [MP15,Claim 3.1], and if w ∈ [ F r , F r ], the value of (1.1) is for n ≥ n ( w ) the same as thecorresponding integral over SU ( n ) r ≤ U ( n ) r , where SU ( n ) is the subgroup of determinantone matrices [Mag21, Prop. 3.1]. So in all cases of interest we can replace U ( n ) by SU ( n )in (1.1).2. Since tr ◦ w is invariant under the diagonal conjugation action of SU ( n ) onHom( F r , SU ( n )) ∼ = SU ( n ) r , the integral (cid:82) tr( w ( x )) dµ Haar SU ( n ) r ( x ) can be written as anintegral over Hom( F r , SU ( n )) / PSU ( n ). Voiculescu’s result in [Voi91, Thm. 3.8] is more general than what we state here, also involving a deter-ministic sequence of unitary matrices.
2. Now turning to Γ g , the most natural measure on Hom(Γ g , SU ( n )) / PSU ( n ) to replacethe measure induced by Haar measure on Hom( F r , SU ( n )) / PSU ( n ) is called the Atiyah-Bott-Goldman measure. The definition of this measure involves removing singular partsof Hom(Γ g , SU ( n )) / PSU ( n ). Indeed, let Hom(Γ g , SU ( n )) irr denote the collection of ho-momorphisms that are irreducible as linear representations. Then M g,n def = Hom(Γ g , SU ( n )) irr / PSU ( n )is a smooth manifold [Gol84]. Moreover there is a symplectic form ω g,n on M g,n calledthe Atiyah-Bott-Goldman form after [AB83, Gol84]. This symplectic form gives, in theusual way, a volume form on M g,n denoted by Vol M g,n . For many more details seeGoldman [Gol84] or the prequel paper [Mag21, §§ γ ∈ Γ, we obtain a function tr γ : Hom(Γ g , SU ( n )) → C defined bytr γ ( φ ) def = tr( φ ( γ )) . This function descends to a function tr γ : M g,n → C . We are interested in the expected value E g,n [tr γ ] def = (cid:82) M g,n tr γ d Vol M g,n (cid:82) M g,n d Vol M g,n . The main theorem of this paper is the following.
Theorem 1.2.
Let g ≥ . If γ ∈ Γ g is not the identity, then E g,n [tr γ ] = O γ (1) as n → ∞ . The non-commutative probabilistic consequences of Theorem 1.2 will be discussed in thenext section.
We follow the book [VDN92]. A non-commutative probability space is a pair ( B , τ ) where B is a unital algebra and τ is a linear functional on B such that τ (1) = 1. A random variable in ( B , τ ) is an element of B . If { ( B , τ n ) } ∞ n =1 are a family of non-commutative probabilityspaces and ( X ( n )1 , . . . , X ( n ) r ) ∈ B r is a family of random variables in ( B , τ n ) for each n , wedefine the notion of a joint limiting distribution as follows. Let C (cid:104) x , . . . , x r (cid:105) denote thefree non-commutative unital algebra in indeterminates x , . . . , x r . For a linear functional τ ∞ : C (cid:104) x , . . . , x r (cid:105) → C with τ (1) = 1, we say that ( X ( n )1 , . . . , X ( n ) r ) converge as n → ∞ tothe limit distribution τ ∞ if for all p ∈ N , and i , . . . , i p ∈ [ r ],lim n →∞ τ n ( X ( n ) i X ( n ) i . . . X ( n ) i p ) = τ ∞ ( x i x i . . . x i p )A very concrete example of this phenomenon is as follows. The function τ n : F r → C , τ n ( w ) def = 1 n (cid:90) tr( w ( x )) dµ Haar U ( n ) r ( x )extends to a linear functional τ n on the algebra C [ F r ] with τ n (id) = 1. From this point ofview, Theorem 1.1 implies the following statement:3 heorem 1.3 (Voiculescu) . Let r ≥ and X , . . . , X r denote the generators of F r , and ¯ X , . . . . ¯ X r denote their inverses, i.e. ¯ X i = X − i . The random variables X , . . . , X r , ¯ X , . . . . ¯ X r in the non-commutative probability spaces ( C [ F r ] , τ n ) converge as n → ∞ to a limiting dis-tribution τ ∞ : C (cid:104) x , . . . , x r , ¯ x , . . . , ¯ x r (cid:105) → C that is completely determined by (1.1). Indeed, if w is any monomial in x , . . . , x r , ¯ x , . . . , ¯ x r ,then τ ∞ ( w ) = 1 if and only if after identifying ¯ x i with x − i , w reduces to the identity in F r = (cid:104) x , . . . , x r (cid:105) , and τ ∞ ( w ) = 0 otherwise. In the language of [Voi91], this implies that in the limiting non-commutative probabilityspace ( C (cid:104) x , . . . , x r , ¯ x , . . . , ¯ x r (cid:105) , τ ∞ ), the subalgebras A = C (cid:104) x , ¯ x (cid:105) , . . . , A r def = C (cid:104) x r , ¯ x r (cid:105) are a free family of subalgebras : if a j ∈ A i j for j ∈ [ q ], i (cid:54) = i (cid:54) = · · · (cid:54) = i q , and τ ∞ ( a j ) = 0 for j ∈ [ q ] then τ ∞ ( a a · · · a q ) = 0 . Accordingly, it is said [Voi91, Thm. 3.8] that if { u j ( n ) : j ∈ [ r ] } are independent Haar-random elements of U ( n ), the family {{ u j ( n ) , u ∗ j ( n ) } : j ∈ [ r ] } of sets of random variables are asymptotically free .Asymptotic freeness does not correctly capture the asymptotic behavior of the expectedvalues E g,n [tr γ ], however, an analog of Theorem 1.3 is implied by Theorem 1.2. For γ ∈ Γ g let ˜ τ n ( γ ) def = 1 n E g,n [tr γ ] . Corollary 1.4.
Let g ≥ , a , b , . . . , a g , b g denote the generators of Γ g , and ¯ a , ¯ b , . . . , ¯ a g , ¯ b g denote their inverses. The random variables a , b , . . . , a g , b g , ¯ a , ¯ b , . . . , ¯ a g , ¯ b g in the non-commutative probability spaces ( C [Γ g ] , ˜ τ n ) converge as n → ∞ to a limiting distribution ˜ τ ∞ : C (cid:104) x , . . . , x g , y , . . . , y g , ¯ x , . . . , ¯ x g , ¯ y , . . . , ¯ y g (cid:105) → C , where x i (resp. y i , ¯ x i , ¯ y i ) corresponds to a i (resp. b i , ¯ a i , ¯ b i ) . This can be described explicitly asfollows. If w is any monomial in x , . . . , x g , y , . . . , y g , ¯ x , . . . , ¯ x g , ¯ y , . . . , ¯ y g , then ˜ τ ∞ ( w ) = 1 if and only if w maps to the identity under the map C (cid:104) x , . . . , x g , y , . . . , y g , ¯ x , . . . , ¯ x g , ¯ y , . . . , ¯ y g (cid:105) → C [Γ g ] obtained by identifying x i , y i , ¯ x i , ¯ y i with the corresponding elements of Γ g . If w does not mapto the identity under this map, then ˜ τ ∞ ( w ) = 0 . Notice that the estimate given in Theorem 1.2 is stronger than needed to establish Corol-lary 1.4.
The most closely related existing result to Theorem 1.2 is a theorem of the author and Puder[MP20, Thm. 1.2] that establishes Theorem 1.2 when the family of groups SU ( n ) is replaced by4he family of symmetric groups S n , and tr is replaced by the character fix given by the numberof fixed points of a permutation. In this case, the result is phrased in terms of integratingover Hom(Γ g , S n ) with respect to the uniform probability measure. The corresponding resultfor Hom( F r , S n ) was proved much longer ago by Nica in [Nic94].The problem of integrating geometric functions like tr γ over M g,n is also connected tothe work of Mirzakhani since, as Goldman explains in [Gol84, § g Riemann surfaces. In [Mir07], Mirzakhani developed a method for integratinggeometric functions on moduli spaces of Riemann surfaces with respect to the Weil-Peterssonvolume form. Although there is certainly a similarity between (ibid.) and the current work,here the emphasis is on n → ∞ , whereas (ibid.) caters to the regime g → ∞ ; the target groupplaying the role of SU ( n ) is always PSL (2 , R ).We now take the opportunity to mention some questions that Theorem 1.2 leads to. Inthe paper [Voi91], Voiculescu is able to boost Theorem 1.1 from a convergence in distributionresult to a result on convergence in probability, that is, for any (cid:15) >
0, and fixed w ∈ F r , theHaar measure of the set { φ ∈ Hom( F r , U ( n )) : | tr( φ ( w )) | ≤ (cid:15)n } tends to one as n → ∞ [Voi91, Thm. 3.9]. To do this, Voiculescu uses that the familyof measure spaces (Hom( F r , U ( n )) , µ Haar U ( n ) r ) form a Levy family in the sense of Gromov andMilman [GM83]. This latter fact relies on an estimate for the first non-zero eigenvalue ofthe Laplacian on Hom( F r , U ( n )). It is interesting to ask whether a similar phenomenonholds for the family of measure spaces ( M g,n , µ ABG g,n ) where µ ABG g,n is the probability measurecorresponding to Vol M g,n . The fact that M g,n is non-compact seems to be a significantcomplication in answering this question using isoperimetric inequalities.In the prequel to this paper [Mag21] it was proved that for any fixed γ ∈ Γ g , there is aninfinite sequence of rational numbers a − ( γ ) , a ( γ ) , a ( γ ) , . . . ∈ Q such that for any M ∈ N , E g,n [tr γ ] = a − ( γ ) n + a ( γ ) + a ( γ ) n + · · · + a M − ( γ ) n M − + O γ,M (cid:18) n M (cid:19) (1.2)as n → ∞ . Theorem 1.2 implies that a − ( γ ) = 0 if γ (cid:54) = id. It is also interesting to understandthe other coefficients of this series. This has been accomplished when Γ g is replaced by F r bythe author and Puder in [MP19] where in fact it is proved that E F r ,n [tr w ] def = (cid:90) tr( w ( x )) dµ Haar U ( n ) r ( x )is given by a rational function of n and in particular can be expanded as in (1.2). Thecorresponding coefficients of the Laurent series of E F r ,n [tr w ] are explained in terms of Eulercharacteristics of subgroups of mapping class groups. One corollary is that as n → ∞ E F r ,n [tr w ] = O (cid:18) n w ) − (cid:19) (1.3)where cl( w ) is the the commutator length of w : the minimal number of commutators that w can be written as a product of, or ∞ if w / ∈ [ F r , F r ]. We guess that an estimate like (1.3)5hould hold for E g,n [tr γ ] where commutator length in F r is replaced by commutator length inΓ g . Another strengthening of Theorem 1.1 is the strong asymptotic freeness of Haar unitaries.This states that for any complex linear combination (cid:88) w a w w ∈ C [ F r ] , almost surely w.r.t. Haar random φ ∈ Hom( F r , U ( n )) as n → ∞ , we have (cid:107) (cid:88) w a w φ ( w ) (cid:107) → (cid:107) (cid:88) w a w w (cid:107) Op( (cid:96) ( F r )) where the left hand side is the operator norm on C n with standard Hermitian inner productand the norm on the right hand side is the operator norm in the regular representation of F r . This result was proved by Collins and Male in [CM14]. It is probably very hard toextend this result to Γ g ; the proof of Collins and Male relies on seminal work of Haagerupand Thorbjørnsen [HT05] in a way that does not obviously extend to Γ g .We finally mention that the expected values E g,n [tr γ ] arise as a limiting form of expectedvalues of Wilson loops in 2D Yang-Mills theory, when the coupling constant is set to zero.This will not be discussed in detail here, we refer the reader instead to the introduction of[Mag21]. We write N for the natural numbers and N = N ∪ { } . We write [ n ] def = { , . . . , n } for n ∈ N and [ k, (cid:96) ] def = { k, k + 1 , . . . , (cid:96) } for k, (cid:96) ∈ N . If A and B are two sets we write A \ B for theelements of A not in B . If H is a group and h , h ∈ H we write [ h , h ] def = h h h − h − . Welet id denote the identity element of a group. We let [ H, H ] be the subgroup of H generated byelements of the form [ h , h ]; this is called the commutator subgroup of H . If V is a complexvector space, for q ∈ N we let V ⊗ q def = V ⊗ V ⊗ · · · ⊗ V (cid:124) (cid:123)(cid:122) (cid:125) q . We use Vinogradov notation as follows. If f and h are functions of n ∈ N , we write f (cid:28) h to mean that there are constants n ≥ C ≥ n ≥ n , f ( n ) ≤ C h ( n ).We write f = O ( h ) to mean f (cid:28) | h | . We write f (cid:16) h to mean both f (cid:28) h and h (cid:28) f . If inany of these statements the implied constants depend on additional parameters we add theseparameters as subscript to (cid:28) , O, or (cid:16) . Throughout the paper we view the genus g as fixedand so any implied constant may depend on g . Acknowledgments
We thank Benoˆıt Collins, Doron Puder, Sanjaye Ramgoolam, and Calum Shearer for discus-sions about this work. This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (grantagreement No 949143). 6
Background
Let S k denote the symmetric group of permutations of [ k ] def = { , . . . , k } , and C [ S k ] denotethe group algebra. The group S is by definition the group with one element.If we refer to S (cid:96) ≤ S k with (cid:96) ≤ k we always view S (cid:96) as the subgroup of permutationsthat fix every element of [ (cid:96) + 1 , k ] def = { (cid:96) + 1 , . . . , k } . We write S (cid:48) r ≤ S k for the subgroup ofpermutations that fix every element of [ k − r ]. As a consequence we obtain fixed inclusions C [ S (cid:96) ] ⊂ C [ S k ] for (cid:96) and k as above. When we write S (cid:96) × S k − (cid:96) ≤ S k , the first factor is S (cid:96) andthe second factor is S (cid:48) k − (cid:96) .A Young diagram λ is a left-aligned contiguous collection of identical square boxes in theplane, such that the number of boxes in each row is non-increasing from top to bottom. Wewrite λ i for the number of boxes in the i th row of λ and say λ (cid:96) k if λ has k boxes. We write (cid:96) ( λ ) for the number of rows of λ . For each λ (cid:96) k there is a Young subgroup S λ def = S λ × S λ × · · · × S λ (cid:96) ( λ ) ≤ S k where the factors are subgroups in the obvious way, according to the increasing order of [ k ].The equivalence classes of irreducible representations of S k are in one-to-one correspon-dence with Young diagrams λ (cid:96) k . Given λ , the construction of the corresponding irreduciblerepresentation V λ can be done for example using Young symmetrizers as in [FH91, Lec. 4].We write χ λ for the character of S k associated to V λ and d λ def = χ λ (id) = dim V λ . Given λ (cid:96) k , the element p λ def = d λ k ! (cid:88) σ ∈ S (cid:96) χ λ ( σ ) σ ∈ C [ S k ]is a central idempotent in C [ S k ].If G is a compact group, ( ρ, W ) is an irreducible representation of G , and ( π, V ) is anyfinite-dimensional representation of G , the ( ρ, W )-isotypic subspace of ( π, V ) is the invariantsubspace of V spanned by all irreducible direct summands of ( π, V ) that are isomorphic to( ρ, W ). When ρ and π can be inferred from W and V we call this simply the W -isotypicsubspace of V . If H ≤ G is a subgroup, and ( ρ, W ) is an irreducible representation of H ,then the W -isotypic subspace of V for H is the W -isotypic subspace of the restriction of ( π, V )to H .If ( π, V ) is any finite-dimensional unitary representation of S k , and λ (cid:96) k , then V is alsoa module for C [ S k ] by linear extension of π and π ( p λ ) is the orthogonal projection onto the V λ -isotypic subspace of V .For any compact group G we write (triv G , C ) for the trivial representation of G . Thefollowing lemma can be deduced for example by combining Young’s rule [FH91, Cor. 4.39]with Frobenius reciprocity. Lemma 2.1.
Let k ∈ N , and λ (cid:96) k . The (triv S λ , C ) -isotypic subspace of V λ for the group S λ is one-dimensional, i.e. (triv S λ , C ) occurs with multiplicity one in the restriction of V λ to S λ . .2 Representation theory of U ( n ) and SU ( n ) Every irreducible representation of U ( n ) restricts to an irreducible representation of SU ( n ),and all equivalence classes of irreducible representations of SU ( n ) arise in this way. The equiv-alence classes of irreducible representations of U ( n ) are parameterized by dominant weights,that can be thought of as non-increasing sequencesΛ = (Λ , . . . , Λ n ) ∈ Z n , also known as the signature. We write W Λ for the irreducible representation of U ( n ) corre-sponding to the signature Λ. Let T ( n ) denote the maximal torus of U ( n ) consisting of diagonalmatrices diag(exp( iθ ) , . . . , exp( iθ n )) where all θ j ∈ R . Associated to the signature Λ is thecharacter ξ Λ of T ( n ) given by ξ Λ (diag(exp( iθ ) , . . . , exp( iθ n ))) def = exp i n (cid:88) j =1 Λ j θ j . The highest weight theory says that, among other things, the ξ Λ -isotypic subspace of W Λ for T ( n ) is one-dimensional. Any vector in this subspace is called a highest weight vector of W Λ .Given k, (cid:96) ∈ N and fixed Young diagrams µ (cid:96) k , ν (cid:96) (cid:96) , we define a family of representa-tions of U ( n ) as follows. For n ≥ (cid:96) ( µ ) + (cid:96) ( ν ) defineΛ [ µ,ν ] ( n ) def = ( µ , µ , . . . , µ (cid:96) ( µ ) , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n − (cid:96) ( µ ) − (cid:96) ( ν ) , − ν (cid:96) ( ν ) , − ν (cid:96) ( ν ) − , . . . , − ν ) . We let ( ρ [ µ,ν ] n , W [ µ,ν ] n ) denote the irreducible representation of U ( n ) corresponding to Λ [ µ,ν ] ( n )when n ≥ (cid:96) ( µ )+ (cid:96) ( ν ). We let D [ µ,ν ] ( n ) def = dim W [ µ,ν ] n and s [ µ,ν ] ( g ) def = tr( ρ [ µ,ν ] n ( g )) for g ∈ U ( n ).If µ (cid:96) k and ν (cid:96) (cid:96) then as n → ∞ D [ µ,ν ] ( n ) (cid:16) n k + (cid:96) (2.1)by [Mag21, Cor. 2.3] (alternatively [EI16, Lem. 3.5]).We now present a version of Schur-Weyl duality for mixed tensors due to Koike [Koi89].The very definition of U ( n ) makes C n into a unitary representation of U ( n ) for the standardHermitian inner product. We let { e , . . . , e n } denote the standard basis of C n . If ( ρ, W )is any finite dimensional representation of U ( n ) we write ( ˇ ρ, ˇ W ) for the dual representationwhere ˇ W is the space of complex linear functionals on W . The vector space ˇ C n has a dualbasis { ˇ e , . . . , ˇ e n } given by ˇ e j ( v ) def = (cid:104) v, e j (cid:105) . Throughout the paper we frequently use certaincanonical isomorphisms e.g. ˇ( C n ) ⊗ p ∼ = ( ˇ C n ) ⊗ p , End( W ) ∼ = W ⊗ ˇ W to change points of view on representations; if we use non-canonical isomorphisms we pointthem out.Let T k,(cid:96) n def = ( C n ) ⊗ k ⊗ ( ˇ C n ) ⊗ (cid:96) . With the natural inner product induced by that on C n , thisis a unitary representation of U ( n ) under the diagonal action and also a unitary representationof S k × S (cid:96) where S k acts by permuting the indices of ( C n ) ⊗ k and S (cid:48) (cid:96) acts by permuting the8ndices of ( ˇ C n ) ⊗ (cid:96) . We write π k,(cid:96)n : U ( n ) → End[ T k,(cid:96)n ] and ρ k,(cid:96)n : C [ S k × S (cid:96) ] → End[ T k,(cid:96)n ] forthese representations. By convention ( C n ) ⊗ = C . Recall that S (cid:48) (cid:96) is our notation for thesecond factor of S k × S (cid:96) . The actions of U ( n ) and S k × S (cid:96) on T k,(cid:96)n commute. We use thenotation, for I = ( i , . . . , i k ) and J = ( j , . . . , j (cid:96) ) e I def = e i ⊗ · · · ⊗ e i k ∈ ( C n ) ⊗ k , ˇ e J def = ˇ e j ⊗ · · · ⊗ ˇ e j (cid:96) ∈ ( ˇ C n ) ⊗ (cid:96) ,e JI def = e I ⊗ ˇ e J ∈ T k,(cid:96)n where ˇ e j is the dual element to e j . We write I (cid:116) J for the concatenation ( i , . . . , i k , j , . . . , j (cid:96) ).For k, (cid:96) ≥ T k,(cid:96)n denote the intersection of the kernels of the mixed contractions c pq : T k,(cid:96)n → T k − ,(cid:96) − , p ∈ [ k ], q ∈ [ (cid:96) ] given by c pq ( e i ⊗ · · · ⊗ e i k ⊗ ˇ e j ⊗ · · · ⊗ ˇ e j (cid:96) ) def = δ i p j q e i ⊗ · · · ⊗ e i p − ⊗ e i p +1 ⊗ · · · ⊗ e i k ⊗ ˇ e j ⊗ · · · ⊗ ˇ e j q − ⊗ ˇ e j q +1 ⊗ · · · ⊗ ˇ e j (cid:96) , (2.2)where δ i p j q is the Kronecker delta. If k = 1 or (cid:96) = 1 then the definition is extended in thenatural way, interpreting an empty tensor of e i or ˇ e i as 1. If either k = 0 or (cid:96) = 0 then˙ T k,(cid:96)n = T k,(cid:96)n by convention. The space ˙ T k,(cid:96)n is an invariant subspace under U ( n ) × S k × S (cid:96) andhence a unitary subrepresentation of T k,(cid:96)n . On ˙ T k,(cid:96)n there is an analog of Schur-Weyl dualitydue to Koike. Theorem 2.2. [Koi89, Thm. 1.1] There is an isomorphism of unitary representations of U ( n ) × S k × S (cid:96) ˙ T k,(cid:96)n ∼ = (cid:77) µ (cid:96) k,ν (cid:96) (cid:96)(cid:96) ( µ )+ (cid:96) ( ν ) ≤ n W [ µ,ν ] n ⊗ V µ ⊗ V ν . (2.3)Next we explain how to construct U ( n )-subrepresentations of ˙ T k,(cid:96)n isomorphic to W [ µ,ν ] n .Suppose that ξ ∈ ˙ T k,(cid:96)n is a non-zero vector such that under the isomorphism (2.3), ξ ∼ = w ⊗ v (2.4)for w ∈ W [ µ,ν ] n and v ∈ V µ ⊗ V ν . Then U ( n ) .ξ linearly spans a U ( n )-subrepresentation of ˙ T k,(cid:96)n isomorphic to W [ µ,ν ] n . The following argument to construct such a vector ξ , given µ (cid:96) k, ν (cid:96) (cid:96) ,appears implicitly in [Koi89] and is elaborated in [BCH + n ≥ (cid:96) ( µ ) + (cid:96) ( ν ) let˜ θ n [ µ,ν ] def = e ⊗ · · · ⊗ e (cid:124) (cid:123)(cid:122) (cid:125) µ ⊗ e ⊗ · · · ⊗ e (cid:124) (cid:123)(cid:122) (cid:125) µ ⊗ · · · ⊗ e (cid:96) ( µ ) ⊗ · · · ⊗ e (cid:96) ( µ ) (cid:124) (cid:123)(cid:122) (cid:125) µ (cid:96) ( µ ) ⊗ ˇ e n ⊗ · · · ⊗ ˇ e n (cid:124) (cid:123)(cid:122) (cid:125) ν ⊗ ˇ e n − ⊗ · · · ⊗ ˇ e n − (cid:124) (cid:123)(cid:122) (cid:125) ν ⊗ · · · ⊗ ˇ e n − (cid:96) ( ν )+1 ⊗ · · · ⊗ ˇ e n − (cid:96) ( ν )+1 (cid:124) (cid:123)(cid:122) (cid:125) ν (cid:96) ( ν ) (2.5)This vector is in the ξ [ µ,ν ] -isotypic subspace of ˙ T k,(cid:96)n for the maximal torus T ( n ) of U ( n ) where ξ [ µ,ν ] is the character of T ( n ) corresponding to the highest weight in W [ µ,ν ] n .Let p µ ∈ C [ S k ], p ν ∈ C [ S (cid:48) (cid:96) ] be the projections defined in §§ ρ kn : S k → End( T k,(cid:96)n )denote the representation of S k described above and ˆ ρ kn : S (cid:48) (cid:96) → End( T k,(cid:96)n ) that of S (cid:48) (cid:96) . Clearly9hese two representations commute. Now let θ n [ µ,ν ] def = ρ kn ( p µ ) ˆ ρ (cid:96)n ( p ν )˜ θ n [ µ,ν ] ∈ ˙ T k,(cid:96)n . (2.6)Now this is in the same isotypic subspace for T ( n ) as before since S k × S (cid:96) commutes with U ( n ). Moreover it is in the subspace of ˙ T k,(cid:96)n corresponding to W [ µ,ν ] n ⊗ V µ ⊗ V ν under theisomorphism (2.3). The intersection of the two subspaces of ˙ T k,(cid:96)n just discussed correspondsvia (2.3) to C w ⊗ V µ ⊗ V ν where w is a highest-weight vector in W [ µ,ν ] n and hence θ n [ µ,ν ] takesthe form of (2.4) as we desired.Of course we also want to know θ n [ µ,ν ] (cid:54) = 0. Lemma 2.3.
Suppose that k, (cid:96) ∈ N , µ (cid:96) k, ν (cid:96) (cid:96) , and θ n [ µ,ν ] is as in (2.6) for n ≥ (cid:96) ( µ ) + (cid:96) ( ν ) .We have (cid:107) θ n [ µ,ν ] (cid:107) > and (cid:107) θ n [ µ,ν ] (cid:107) does not depend on n , only on µ, ν .Proof. Recall the definition of Young subgroups S µ , S ν from §§ θ = ˜ θ n [ µ,ν ] (as in(2.5)) and θ = θ n [ µ,ν ] we have θ = ρ kn ( p µ ) ˆ ρ (cid:96)n ( p ν )˜ θ = d µ d ν k ! (cid:96) ! (cid:88) σ =( σ ,σ ) ∈ S k × S (cid:96) χ µ ( σ ) χ ν ( σ ) ρ kn ( σ ) ˆ ρ (cid:96)n ( σ )˜ θ = d µ d ν k ! (cid:96) ! (cid:88) [ σ ] ∈ S k /S µ [ σ ] ∈ S (cid:96) /S ν (cid:88) τ ∈ S µ τ ∈ S ν χ µ ( σ τ ) χ ν ( σ τ ) ρ kn ( σ ) ˆ ρ (cid:96)n ( σ )˜ θ. = d µ d ν k ! (cid:96) ! (cid:88) [ σ ] ∈ S k /S µ [ σ ] ∈ S (cid:96) /S ν (cid:88) τ ∈ S µ χ µ ( σ τ ) (cid:88) τ ∈ S ν χ ν ( σ τ ) ρ kn ( σ ) ˆ ρ (cid:96)n ( σ )˜ θ. The second equality used that ˜ θ is invariant under S µ × S ν .By Lemma 2.1, there is a one dimensional subspace of invariant vectors for S µ in V µ . If v µ ∈ V µ is a unit vector in this space then (cid:88) τ ∈ S µ χ µ ( σ τ ) = | S µ |(cid:104) σ v µ , v µ (cid:105) . (2.7)Since the vectors ρ kn ( σ ) ˆ ρ (cid:96)n ( σ )˜ θ for [ σ ] ∈ S k /S µ and [ σ ] ∈ S (cid:96) /S ν are orthogonal unit vectors,this gives (cid:107) θ (cid:107) = (cid:18) d µ d ν k ! (cid:96) ! (cid:19) (cid:88) [ σ ] ∈ S k /S µ [ σ ] ∈ S (cid:96) /S ν (cid:88) τ ∈ S µ χ µ ( σ τ ) (cid:88) τ ∈ S ν χ ν ( σ τ ) . This clearly does not depend on n . To see that (cid:107) θ (cid:107) (cid:54) = 0, the contribution from any [ σ ] , [ σ ] isnon-negative and the contribution from σ = id , σ = id is by (2.7) equal to (cid:16) d µ d ν k ! (cid:96) ! (cid:17) | S µ | | S ν | (cid:54) =0. Recall that we write π k,(cid:96)n : U ( n ) → End( T k,(cid:96)n ) for the diagonal representation of U ( n ) on T k,(cid:96)n . By the remarks surrounding (2.4), Lemma 2.3 implies the following corollary.10 orollary 2.4. Suppose n ≥ (cid:96) ( µ ) + (cid:96) ( ν ) . The subspace W n ( θ n [ µ,ν ] ) def = span { π k,(cid:96)n ( u ) θ n [ µ,ν ] : u ∈ U ( n ) } ⊂ ˙ T k,(cid:96)n is, under π k,(cid:96)n , a U ( n ) -subrepresentation of ˙ T k,(cid:96)n isomorphic to W [ µ,ν ] n . The Weingarten calculus is a method based on Schur-Weyl duality that allows one to calculateintegrals of products of matrix coefficients in the defining representation of U ( n ) in terms ofsums over permutations. It was developed initially by physicists in [Wei78, Xu97] and thenmore precisely by Collins and Collins-´Sniady in [Col03, C´S06].We present two formulations of the Weingarten calculus. Given k ∈ N , n ∈ N , the Weingarten function with parameters n, k is the following element of C [ S k ] [C´S06, eq. (9)]Wg n,k def = 1( k !) (cid:88) λ (cid:96) k(cid:96) ( λ ) ≤ n d λ D λ ( n ) (cid:88) σ ∈ S k χ λ ( σ ) σ. (2.8)We write Wg n,k ( σ ) for the coefficient of σ in (2.8). The following theorem was proved byCollins and ´Sniady [C´S06, Cor. 2.4]. Theorem 2.5.
For k ∈ N and for i , i (cid:48) , j k , j (cid:48) k , . . . , i k , i (cid:48) k , j k , j (cid:48) k ∈ [ n ] (cid:90) u ∈ U ( n ) u i j · · · u i k j k ¯ u i (cid:48) j (cid:48) · · · ¯ u i (cid:48) k j (cid:48) k dµ Haar U ( n ) ( u ) (2.9)= (cid:88) σ,τ ∈ S k δ i i (cid:48) σ (1) · · · δ i k i (cid:48) σ ( k ) δ j j (cid:48) τ (1) · · · δ j k j (cid:48) τ ( k ) Wg n,k ( τ σ − ) , where δ pq is the Kronecker delta function. It is sometimes more flexible to reformulate Theorem 2.5 in terms of projections. Here u ∈ U ( n ) acts on A ∈ End(( C n ) ⊗ k ) by A (cid:55)→ π kn ( u ) Aπ kn ( u − ), π kn : U ( n ) → End(( C n ) ⊗ k )the diagonal action. Let ρ kn : C [ S k ] → End(( C n ) ⊗ k ) be the map induced by S k permutingcoordinates of ( C n ) ⊗ k . Write P n,k for the orthogonal projection in End(( C n ) ⊗ k ) onto the U ( n )-invariant vectors. The following proposition is due to Collins and ´Sniady [C´S06, Prop.2.3]. Proposition 2.6 (Collins-´Sniady) . Let n, k ∈ N . Suppose A ∈ End(( C n ) ⊗ k ) . Then P n,k [ A ] = ρ kn (cid:0) Φ[ A ] · Wg n,k (cid:1) where Φ[ A ] def = (cid:88) σ ∈ S k tr( Aρ kn ( σ − )) σ. Later we will need the following bound for the Weingarten function due to Collins and´Sniady [C´S06, Prop. 2.6]
Proposition 2.7.
For any fixed σ ∈ S k , Wg n,k ( σ ) = O k (cid:0) n − k −| σ | (cid:1) as n → ∞ , where | σ | isthe minimum number of transpositions that σ can be written as a product of. .4 Free groups and surface groups Let F g def = (cid:104) a , b , . . . , a g , b g (cid:105) be the free group on 2 g generators a , b , . . . , a g , b g and R g def =[ a , b ] · · · [ a g , b g ] ∈ F g . Therefore we have a quotient map F g → Γ g given by reductionmodulo R g . We say that w ∈ F g represents the conjugacy class of γ ∈ Γ g if the projectionof w to Γ g is in the conjugacy class of γ in Γ g .Given w ∈ F g we view w as a combinatorial word in a , a − , b , b − , . . . , a g , a − g , b g , b − g by writing it in reduced (shortest) form; i.e., a does not follow a − etc. We say that w is cyclically reduced if the first letter of its reduced word is not the inverse of the last letter. Thelength | w | of w ∈ F g is the length of its reduced form word. We say w ∈ F g is a shortestelement representing the conjugacy class of γ ∈ Γ g if it has minimal length among all elementsrepresenting the conjugacy class of γ . If w is a shortest element representing some conjugacyclass in Γ g then w is cyclically reduced.For any group H , the commutator subgroup [ H, H ] ≤ H is the subgroup generated by allelements of the form [ h , h ] def = h h h − h − with h , h ∈ H . It is not hard to see that if γ ∈ [Γ g , Γ g ], and w represents the conjugacy class of γ , then w ∈ [ F g , F g ]. Witten zeta functions appeared first in Witten’s work [Wit91] and were named by Zagier in[Zag94]. The
Witten zeta function of SU ( n ) is defined, for s in a half-plane of convergence,by ζ ( s ; n ) def = (cid:88) ( ρ, W ) ∈ (cid:92) SU ( n ) W ) s (2.10)where (cid:92) SU ( n ) denotes the equivalence classes of irreducible representations of SU ( n ). Indeed,the series (2.10) converges for Re( s ) > n by a result of Larsen and Lubotzky [LL08, Thm.5.1] (see also [HS19, § s > n →∞ ζ ( s ; n ) = 1 . (2.11) By [Mag21, Prop. 1.5], if γ / ∈ [Γ g , Γ g ], E g,n [tr γ ] = 0 for n ≥ n ( γ ). This proves Theorem1.2 in this case. Hence in the rest of the paper we need only consider γ ∈ [Γ g , Γ g ] and hence w ∈ [ F g , F g ] if w ∈ F g represents the conjugacy class of γ . For each w ∈ F g , we have a word map w : U ( n ) g → U ( n ) obtained by substituting matri-ces for the generators of F g . For example, if u , v , . . . , u g , v g ∈ U ( n ) then R g ( u , v , . . . , u g , v g ) =[ u , v ] · · · [ u g , v g ]. We begin with the following result from the prequel paper [Mag21, Cor.1.8]. Proposition 2.8.
Suppose that g ≥ , γ ∈ Γ g , and w ∈ F g represents the conjugacy classof γ . For any B ∈ N we have as n → ∞ E g,n [tr γ ] = ζ (2 g − n ) − (cid:88) µ,ν Young diagrams (cid:96) ( µ ) , (cid:96) ( ν ) ≤ B, µ , ν ≤ B D [ µ,ν ] ( n ) I n ( w, [ µ, ν ]) + O B,w,g (cid:16) n | w | n − B (cid:17) . (2.12)12 here I n ( w, [ µ, ν ]) def = (cid:90) tr( w ( x )) s [ µ,ν ] ( R g ( x ))) dµ Haar SU ( n ) g ( x ) (2.13)Notice that for n ≥ B the right hand side of (2.12) makes sense, i.e. D [ µ,ν ] , s [ µ,ν ] arewell-defined. We also have by [Mag21, Prop. 3.1] and s [ µ,ν ] = s ν,µ the following proposition. Proposition 2.9.
Let w ∈ [ F g , F g ] . Then for any fixed µ, ν and n ≥ (cid:96) ( µ ) + (cid:96) ( ν ) I n ( w, [ µ, ν ]) = J n ( w, [ ν, µ ]) def = (cid:90) tr( w ( x )) s [ ν,µ ] ( R g ( x )) dµ Haar U ( n ) g ( x ) . This is convenient as it will allow us to use the Weingarten calculus directly as it is pre-sented in §§ U ( n ) rather than SU ( n ). By using Proposition 2.9, taking a representative w ∈ F g of the conjugacy class of γ and taking B such that | w | − B ≤ − Corollary 2.10.
Let γ ∈ [Γ g , Γ g ] and w ∈ [ F g , F g ] be a representative of the conjugacyclass of γ ∈ Γ . Then there exists a finite set ˜Ω of pairs ( µ, ν ) of Young diagrams that E g,n [tr γ ] = ζ (2 g − n ) − (cid:88) ( µ, ν ) ∈ ˜Ω D [ µ,ν ] ( n ) J n ( w, [ µ, ν ]) + O w,g (cid:18) n (cid:19) . As we know lim n →∞ ζ (2 g − , n ) = 1 by (2.11), we have now reduced the proof of Theorem1.2 to establishing suitable bounds for the integrals J n ( w, [ µ, ν ]) where we can view µ, ν asfixed Young diagrams since ˜Ω is finite. The main result of this § Theorem 3.1.
For γ ∈ Γ g , γ (cid:54) = id , if w ∈ F g is a cyclically shortest word representing theconjugacy class of γ , we have for µ (cid:96) k , ν (cid:96) (cid:96) fixed, D [ µ,ν ] ( n ) J n ( w, [ µ, ν ]) = O w,k,(cid:96) (1) as n → ∞ . Accordingly, since we know the large n behaviour of D [ µ,ν ] ( n ) from (2.1), in this sectionwe wish to estimate J n ( w, [ µ, ν ]) = (cid:90) tr( w ( x )) s [ µ,ν ] ( R g ( x )) dµ Haar U ( n ) g ( x )for fixed µ (cid:96) k, ν (cid:96) (cid:96) . We begin by discussing why the most straightforward approach to thisproblem leads to serious complications. It is possible to approach the problem by writing s [ µ,ν ] ( h ) as a fixed finite linear combinations of functions p µ (cid:48) ( h ) p ν (cid:48) ( h − )13here p µ (cid:48) ( h ) (resp. p ν (cid:48) ( h − )) is a power sum symmetric polynomial of the eigenvalues of h (resp. h − or ¯ h ). See for example [Mag21, §§ J n ( w, [ µ, ν ]) as a finite linear combinationof integrals of the form (cid:90) tr( w ( x ))tr (cid:16) R g ( x ) k (cid:17) · · · tr (cid:16) R g ( x ) k p (cid:17) tr (cid:16) R g ( x ) − (cid:96) (cid:17) · · · tr (cid:16) R g ( x ) − (cid:96) q (cid:17) dµ Haar U ( n ) g ( x ) (3.1)where (cid:80) k j = | µ | and (cid:80) (cid:96) j = | ν | .The work of the author and Puder in [MP19] gives a full asymptotic expansion for (3.1) as n → ∞ . However these estimates are not sufficient for the current paper and to motivate therest of this § w ∈ F g . Roughly speaking, every term in this expansion comesfrom a homotopy class of map f from an orientable surface Σ f to (cid:87) ri =1 S ; to contribute to(3.1) the surface Σ f has one boundary component that maps to w at the level of the fun-damental groups, p boundary components that map respectively to R k g , . . . , R k p g at the levelof fundamental groups, and q boundary components that map respectively to R − (cid:96) g , . . . , R − (cid:96) q g at the level of fundamental groups. The contribution of the pair ( f, Σ f ) to (3.1) is of theform c ( f, Σ f ) n χ (Σ f ) ; the coefficient c ( f, Σ f ) is an Euler characteristic of a symmetry group of( f, Σ f ) and is not easy to calculate in general. However, one could still hope to get decay of(3.1) by controlling the possible χ (Σ f ) that could appear.There are two issues with this. The first one is that if w is not the shortest elementrepresenting the conjugacy class of γ then we get bounds that are not helpful. For a verysimple example, let w = R (cid:96)g , γ = id Γ g , and consider the potential contribution from p = 0 , q =1 , (cid:96) = (cid:96) . Then for any ν with | ν | = (cid:96) there is contribution to J n ( w, [ ∅ , ν ]) that is a multipleof (cid:90) tr (cid:16) R g ( x ) (cid:96) (cid:17) tr (cid:16) R g ( x ) − (cid:96) (cid:17) dµ Haar U ( n ) g ( x ) . Here, in the theory of [MP19] there is a (Σ f , f ) that is an annulus, one boundary componentcorresponding to w = R (cid:96)g and one corresponding to R − (cid:96)g , so we can only bound the corre-sponding contribution to D [ ∅ ,ν ] ( n ) J n ( w, [ ∅ , ν ]) by using [MP19] on the order of D [ ∅ ,ν ] ( n ) (cid:16) n (cid:96) .On the other hand, any approach that works to establish Theorem 3.1 (for γ (cid:54) = id) shouldextend to show when γ = id D [ ∅ ,ν ] ( n ) J n ( w, [ ∅ , ν ]) (cid:28) n as E g,n [tr id ] = n .Indeed, this phenomenon extends to words of the form w R (cid:96)g and more generally to wordsthat are not shortest representatives of some conjugacy class in Γ g . It means that even ifwe use something similar in spirit to [MP19], to prove Theorem 3.1 we must incorporate thetheory of shortest representative words. This indeed takes place in §§ §§ .
5; the topologicalresult proved there hinges on this theory.The second issue is a little more subtle and only appears for ‘mixed’ representations, i.e.,both µ, ν (cid:54) = ∅ . In this case, suppose w is a shortest element representing some conjugacyclass in Γ g and w ∈ [ F g , F g ]. This means that there is a pair ( f , Σ f ) where Σ f has oneboundary component that maps to w at the level of the fundamental groups. Let us take[ µ, ν ] = [( k ) , ( k )], i.e each Young diagram has one row of k boxes. This means we get a14otential contribution to D [ µ,ν ] ( n ) J n ( w, [ µ, ν ]) that is a constant multiple of D [( k ) , ( k )] ( n ) (cid:90) x ∈ U ( n ) g tr( w ( x ))tr (cid:16) R g ( x ) k (cid:17) tr (cid:16) R g ( x ) − k (cid:17) dµ Haar U ( n ) g ( x ) (3.2)Now, for every k ∈ N , there is ( f, Σ f ) contributing to (3.2) with one component that is( f , Σ f ) and the other is an annulus with boundary components corresponding to R kg , R − kg .Since the annulus has Euler characteristic 0, and D [( k ) , ( k )] (cid:16) n k , the order of this contributionto D [( k ) , ( k )] ( n ) J n ( w, [( k ) , ( k )]) is potentially (cid:29) n k n χ (Σ f ) . For large enough k the exponenthere is arbitrarily large, which is clearly catastrophic. In reality, this contribution must cancelwith some other contribution but we do not know how to see these cancellations.To bypass this we produce a refined version of the Weingarten calculus that leads to arestricted set of surfaces, for instance, not including the ones causing the problem aboveas well as all generalizations of this issue. The restriction we obtain is summarized in the forbidden matching property below ( §§ P4 ( §§ k = (cid:96) = 0 Here we give a proof of Theorem 3.1 when k = (cid:96) = 0. This will allow us to bypass the slightlyconfusing issue of using the Weingarten function Wg n,k + (cid:96) when k + (cid:96) = 0 in §§ k = (cid:96) = 0 then the only possible µ (cid:96) k , ν (cid:96) (cid:96) are empty Young diagrams µ = ν = ∅ ,and W [ ∅ , ∅ ] n is the trivial representation of U ( n ), so D [ ∅ , ∅ ] ( n ) = 1 for all n ≥ s [ ∅ , ∅ ] ( h ) = 1for all h ∈ U ( n ). We then have D [ ∅ , ∅ ] ( n ) J n ( w, [ ∅ , ∅ ]) = J n ( w, [ ∅ , ∅ ]) = (cid:90) tr( w ( x )) dµ Haar U ( n ) g ( x ) . (3.3)If w ∈ F g is a cyclically shortest word representing the conjugacy class of γ ∈ Γ g with γ (cid:54) = id,then w (cid:54) = id. It then follows from (1.1) that D [ ∅ , ∅ ] ( n ) J n ( w, [ ∅ , ∅ ]) = o w ( n ) as n → ∞ , but infact, (3.3) is given by a rational function of n for n ≥ n ( w ) by a straightforward applicationof the Weingarten calculus [MP19]. This implies D [ ∅ , ∅ ] ( n ) J n ( w, [ ∅ , ∅ ]) = O w (1) as n → ∞ , asrequired. This proves Theorem 3.1 when k = (cid:96) = 0 . Hence in the rest of this § k + (cid:96) > . Here we develop an integral calculus that is more powerful than the usual Weingarten calculusand allows us to directly tackle J n ( w, [ µ, ν ]) without writing it in terms of integrals as in (3.1).The key point is that our method leads to the forbidden matchings property of §§ P4 of §§ k, (cid:96) , µ (cid:96) k, ν (cid:96) (cid:96) as fixed, assume k + (cid:96) > n ≥ (cid:96) ( µ ) + (cid:96) ( ν ), and write θ = θ n [ µ,ν ] as in (2.6), suppressing the dependence on n . Let W n ( θ ) be defined as in Corollary2.4. Thus W n ( θ ) is an irreducible summand of ˙ T k,(cid:96)n isomorphic to W [ µ,ν ] n for the group U ( n ).Our first task is to compute the orthogonal projection q θn onto W n ( θ ). Let P θ denotethe orthogonal projection in T k,(cid:96)n onto θ . We also view P θ as an element of End( ˙ T k,(cid:96)n ) byrestriction. 15nder the canonical isomorphism End( ˙ T k,(cid:96)n ) ∼ = ˙ T k,(cid:96)n ⊗ ˇ˙ T k,(cid:96)n we have P θ ∼ = θ ⊗ ˇ θ (cid:107) θ (cid:107) and alsofrom (2.6) P θ = 1 (cid:107) θ (cid:107) ρ kn ( p µ ) ˆ ρ (cid:96)n ( p ν )[˜ θ n [ µ,ν ] ⊗ ˇ˜ θ n [ µ,ν ] ] ρ kn ( p µ ) ˆ ρ (cid:96)n ( p ν ); (3.4)here the inner square bracket is interpreted as an element of End( ˙ T k,(cid:96)n ). By Schur’s lemmawe have q θn = D [ µ,ν ] ( n ) (cid:90) h ∈ U ( n ) π n ( h ) P θ π n ( h − ) dµ Haar U ( n ) ( h ) (3.5)since the right hand side is an element of End( W n ( θ )) ⊂ End( T k,(cid:96)n ) that commutes with π k,(cid:96)n ( U ( n )), so it is a multiple of q θn , and it has the correct trace.On the other hand, we can view T k,(cid:96)n ⊗ ˇ T k,(cid:96)n ∼ = T k + (cid:96),k + (cid:96)n by the canonical isomorphism T k,(cid:96)n ⊗ ˇ T k,(cid:96)n ∼ = ( C n ) ⊗ k ⊗ ( ˇ C n ) ⊗ (cid:96) ⊗ ( ˇ C n ) ⊗ k ⊗ ( C n ) ⊗ (cid:96) followed by the following fixed isomorphism ϕ : e JI ⊗ ˇ e J (cid:48) I (cid:48) (cid:55)→ e I (cid:116) J (cid:48) ⊗ ˇ e I (cid:48) (cid:116) J . (3.6)Finally, there is a canonical isomorphism T k + (cid:96),k + (cid:96)n ∼ = End(( C n ) ⊗ k + (cid:96) ). So combining these wefix isomorphisms End( T k,(cid:96)n ) ∼ = T k,(cid:96)n ⊗ ˇ T k,(cid:96)n ∼ −→ ϕ T k + (cid:96),k + (cid:96)n ∼ = End(( C n ) ⊗ k + (cid:96) ) . (3.7)We view the outer two isomorphisms as fixed identifications. These isomorphisms are ofunitary representations of U ( n ) when everything is given its natural inner product. Moreoverfor σ = ( σ , σ ) ∈ S k × S (cid:96) and τ = ( τ , τ ) ∈ S k × S l we have for A ∈ End( T k,(cid:96)n ) ϕ [ ρ kn ( σ ) ˆ ρ (cid:96)n ( σ ) Aρ kn ( τ ) ˆ ρ (cid:96)n ( τ )] = ρ k + (cid:96)n ( σ , τ − ) ϕ [ A ] ρ k + (cid:96)n ( τ , σ − ) , (3.8)recalling that ρ k + (cid:96)n : C [ S k + (cid:96) ] → End(( C n ) ⊗ k + (cid:96) ) is the representation by permuting coordi-nates.We now return to the calculation of q θ in (3.5). We have q θ = D [ µ,ν ] ( n ) ϕ − [ P n,k + (cid:96) [ ϕ ( P θ )]] (3.9)where P n,k + (cid:96) is projection onto the U ( n )-invariant vectors (by conjugation) in End(( C n ) ⊗ k + (cid:96) ).This can now be done using the classical Weingarten calculus. By Proposition 2.6 we have P n,k + (cid:96) [ ϕ ( P θ )] = ρ k + (cid:96)n (cid:0) Φ[ ϕ ( P θ )] · Wg n,k + (cid:96) (cid:1) (3.10)where Φ[ ϕ ( P θ )] = (cid:88) σ ∈ S k + (cid:96) tr( ϕ ( P θ ) ρ k + (cid:96)n ( σ − )) σ.
16y (3.8) and (3.4), and since e.g. χ µ ( g ) = χ µ ( g − ) we obtain ϕ ( P θ ) = 1 (cid:107) θ (cid:107) ϕ (cid:16) ρ kn ( p µ ) ˆ ρ (cid:96)n ( p ν )[˜ θ n [ µ,ν ] ⊗ ˇ˜ θ n [ µ,ν ] ] ρ kn ( p µ ) ˆ ρ (cid:96)n ( p ν ) (cid:17) = 1 (cid:107) θ (cid:107) ρ k + (cid:96)n ( p µ ⊗ ν ) ϕ (cid:16) ˜ θ n [ µ,ν ] ⊗ ˇ˜ θ n [ µ,ν ] (cid:17) ρ k + (cid:96)n ( p µ ⊗ ν )where p µ ⊗ ν def = d µ d ν k ! (cid:96) ! (cid:88) σ =( σ ,σ ) ∈ S k × S (cid:96) χ µ ( σ ) χ ν ( σ ) σ ∈ C [ S k + (cid:96) ] . Now using that Φ is a C [ S k + (cid:96) ]-bimodule morphism [C´S06, Prop. 2.3 (1)] we obtainΦ[ ϕ ( P θ )] = 1 (cid:107) θ (cid:107) p µ ⊗ ν Φ (cid:104) ϕ (cid:16) ˜ θ n [ µ,ν ] ⊗ ˇ˜ θ n [ µ,ν ] (cid:17)(cid:105) p µ ⊗ ν = 1 (cid:107) θ (cid:107) p µ ⊗ ν (cid:88) σ ∈ S k + (cid:96) tr (cid:16) ϕ (cid:16) ˜ θ n [ µ,ν ] ⊗ ˇ˜ θ n [ µ,ν ] (cid:17) ρ k + (cid:96)n (cid:0) σ − (cid:1)(cid:17) σ p µ ⊗ ν . Now, tr( ϕ (cid:16) ˜ θ n [ µ,ν ] ⊗ ˇ˜ θ n [ µ,ν ] (cid:17) ρ k + (cid:96)n ( σ − )) is equal to 1 if and only if σ is in S µ × S ν ≤ S k × S (cid:96) , andis 0 otherwise. So we obtainΦ[ ϕ ( P θ )] = 1 (cid:107) θ (cid:107) p µ ⊗ ν (cid:88) σ ∈ S µ × S ν σ p µ ⊗ ν , hence from (3.10) P n,k + (cid:96) [ ϕ ( P θ )] = ρ k + (cid:96)n ( z θ )where z θ def = (cid:88) τ ∈ S k + (cid:96) z θ ( τ ) τ def = 1 (cid:107) θ (cid:107) p µ ⊗ ν (cid:88) σ ∈ S µ × S ν σ p µ ⊗ ν Wg n,k + (cid:96) ∈ C [ S k + (cid:96) ] . (3.11)Therefore we obtain the following proposition. Proposition 3.2.
We have q θ = ϕ − [ ρ k + (cid:96)n ( z θ )] . We can use the bound for the coefficients of Wg n,k + (cid:96) from Proposition 2.7 to infer a boundon the coefficients z θ ( τ ) . For σ ∈ S k + (cid:96) , let (cid:107) σ (cid:107) k,(cid:96) denote the minimum m for which σ = σ t t · · · t m where σ ∈ S k × S (cid:96) and t , . . . , t m are transpositions in S k + (cid:96) . Lemma 3.3.
For all τ ∈ S k + (cid:96) and θ = θ n [ µ,ν ] as above, z θ ( τ ) = O k,(cid:96) ( n − k − (cid:96) −(cid:107) τ (cid:107) k,(cid:96) ) as n → ∞ .Proof. Referring to (3.11), as n → ∞ , (cid:107) θ (cid:107) − = O k,(cid:96) (1) by Lemma 2.3 and the coefficients of17 µ ⊗ ν (cid:16)(cid:80) σ ∈ S µ × S ν σ (cid:17) p µ ⊗ ν are clearly O k,(cid:96) (1), so z θ has the form (cid:88) σ ∈ S k × S (cid:96) A ( σ ) σ Wg n,k + (cid:96) where each A ( σ ) is O k,(cid:96) (1). This means z θ ( τ ) = (cid:88) σ ∈ S k × S (cid:96) , σ (cid:48) ∈ S k + (cid:96) : σσ (cid:48) = τ A ( σ )Wg n,k + (cid:96) ( σ (cid:48) ) . The order of any of the finitely many summands above is n − k − (cid:96) −| σ (cid:48) | by Proposition 2.7, andthe minimum possible value of | σ (cid:48) | is (cid:107) τ (cid:107) k,(cid:96) .Before moving on, it is useful to explain the operator ϕ − [ ρ k + (cid:96)n ( π )] for π ∈ S k + (cid:96) . For I = ( i , . . . , i k + (cid:96) ) let I (cid:48) ( I ; π ) def = i π (1) , . . . , i π ( k ) and J (cid:48) ( I ; π ) def = i π ( k +1) , . . . , i π ( k + (cid:96) ) . As anelement of ( C n ) ⊗ k + (cid:96) ⊗ ( ˇ C n ) ⊗ k + (cid:96) , ρ k + (cid:96)n ( π ) is given by (cid:88) I =( i ,...,i k ) ,J =( j k +1 ,...,j k + (cid:96) ) e I (cid:48) ( I (cid:116) J ; π ) (cid:116) J (cid:48) ( I (cid:116) J ; π ) ⊗ ˇ e I (cid:116) J , so from (3.6) ϕ − [ ρ k + (cid:96)n ( π )] = (cid:88) I =( i ,...,i k ) ,J =( j k +1 ,...,j k + (cid:96) ) e JI (cid:48) ( I (cid:116) J ; π ) ⊗ ˇ e J (cid:48) ( I (cid:116) J ; π ) I . (3.12) In this rest of this § g = 2. All proofs extend to g ≥
3. We write { a, b, c, d } for the generators of F and R def = [ a, b ][ c, d ]. Assume both γ and w are not the identity and w ∈ [ F , F ] according to the remarks at the beginning of §§ w in reduced form: w = f (cid:15) f (cid:15) . . . f (cid:15) | w | | w | , (cid:15) u ∈ {± } , f u ∈ { a, b, c, d } , (3.13)where if f u = f u +1 , then (cid:15) u = (cid:15) u +1 . For f ∈ { a, b, c, d } let p f denote the number of occurrencesof f +1 in (3.13). The expression (3.13) implies that for h def = ( h a , h b , h c , h d ) ∈ U ( n ) ,tr( w ( h )) = (cid:88) i j ∈ [ n ] ( h (cid:15) f ) i i ( h (cid:15) f ) i i · · · ( h (cid:15) | w | f | w | ) i | w | i . (3.14)Working with this expression will be cumbersome so we explain a diagrammatic wayto think about (3.14). This will be the starting point for how we eventually understand J n ( w, [ µ, ν ]) in terms of decorated surfaces. We begin with a collection of intervals as follows. w -intervals and the w -loop Firstly, for every j ∈ [ w ], with f j = f as in (3.13) and (cid:15) j = 1 we take a copy of [0 ,
1] anddirect it from 0 to 1.In our constructions, every interval will have two directions: the intrinsic direction (whichis the direction from 0 to 1) and the assigned direction.
In the case just discussed, these agree,but in general they will not. 18igure 3.1: Illustration of the w -loop for w = a ba − b − . The solid intervals are w -intervalsand the dashed intervals are w -intermediate-intervals. We also label each interval by the sete.g. I + a,w to which they belong.We write [0 , f,j,w for such an interval and I + f,w for the collection of these intervals.For every j ∈ [ w ], with f j = f as in (3.13) and (cid:15) j = − ,
1] and directthis interval from 1 to 0. We write [0 , f − ,j,w for such an interval and I − f,w for the collectionof these intervals.All the intervals described above are called w -intervals . There are | w | of these intervalsin total. w-intermediate-intervals. Between each [0 , f (cid:15)jj ,j,w and [0 , f (cid:15)j +1 j +1 ,j +1 ,w we add a new interval connecting 1 f (cid:15)jj ,j,w to0 f (cid:15)j +1 j +1 ,j +1 ,w , where the indices j run mod | w | . These intervals added are called w -intermediate-intervals. Note that these intervals together with the w -intervals now form a closed cyclethat is paved by 2 | w | intervals alternating between w -intervals and w -intermediate-intervals.Starting at [0 , f (cid:15) , ,w , reading the directions and f -labels of the w -intervals so that every w -interval is traversed from 0 to 1 spells out the word w . The resulting circle is called the w -loop and the previously defined orientation of this loop is now fixed. See Figure 3.1 for anillustration of the w -loop in a particular example.We now view the indices i j as an assignment a : { end-points of w -intervals } → [ n ] , a (0 f,j,w ) def = i j , a (1 f,j,w ) = i j +1 , a (0 f − ,j,w ) = i j , a (1 f − ,j,w ) = i j +1 . The condition that a comes from a single collection of i j is precisely that if two end points of w -intervals are connected by a w -intermediate-interval, they are assigned the same value by a . Let A ( w ) denote the collection of such a . If I is any copy of [0 ,
1] we write 0 I for the copy19f 0 and 1 I for the copy of 1 in I . We can now writetr( w ( h )) = (cid:88) a ∈A ( w ) (cid:89) f ∈{ a,b,c,d } (cid:89) i ∈ I + f,w h a (0 i ) a (1 i ) (cid:89) j ∈ I − f,w h a (1 j ) a (0 j ) . Now let v p be an orthonormal basis for W n ( θ ). We have s [ µ,ν ] ( R g ( h a , h b , h c , h d )) = (cid:88) p i (cid:104) h a v p , v p (cid:105)(cid:104) h b v p , v p (cid:105)(cid:104) h − a v p , v p (cid:105)(cid:104) h − b v p , v p (cid:105)(cid:104) h c v p , v p (cid:105)(cid:104) h d v p , v p (cid:105)(cid:104) h − c v p , v p (cid:105)(cid:104) h − d v p , v p (cid:105) . Here we have written e.g. h a v p for π k,(cid:96)n ( h a ) v p to make things easier to read. Next we writeeach v p = (cid:80) I,J β JpI e JI , where β JpI def = (cid:104) v p , e JI (cid:105) . We then have (cid:104) h a v p , v p (cid:105)(cid:104) h b v p , v p (cid:105)(cid:104) h − a v p , v p (cid:105)(cid:104) h − b v p , v p (cid:105)× (cid:104) h c v p , v p (cid:105)(cid:104) h d v p , v p (cid:105)(cid:104) h − c v p , v p (cid:105)(cid:104) h − d v p , v p (cid:105) = (cid:88) r f , R f , V f , v f , U f , u f , s f , S f β V a p s a ¯ β U a p r a β V b p s b ¯ β U b p r b β u a p R a ¯ β v a p S a β u b p R b ¯ β v b p S b β V c p s c ¯ β U c p r c β V d p s d ¯ β U d p r d β u c p R c ¯ β v c p S c β u d p R d ¯ β v d p S d (cid:104) h a e V a s a , e U a r a (cid:105)(cid:104) h b e V b s b , e U b r b (cid:105)(cid:104) h − a e u a R a , e v a S a (cid:105)(cid:104) h − b e u b R b , e v b S b (cid:105)(cid:104) h c e V c s c , e U c r c (cid:105)(cid:104) h d e V d s d , e U d r d (cid:105)(cid:104) h − c e u c R c , e v c S c (cid:105)(cid:104) h − d e u d R d , e v d S d (cid:105) . (3.15)We calculate (cid:104) h f e V f s f , e U f r f (cid:105)(cid:104) h − f e u f R f , e v f S f (cid:105) = (cid:104) h f e s f , e r f (cid:105)(cid:104) h f e V f , e U f (cid:105)(cid:104) h f e S f , e R f (cid:105)(cid:104) h f e v f , e u f (cid:105) = (cid:104) h f e s f (cid:116) v f , e r f (cid:116) u f (cid:105)(cid:104) h f e S f (cid:116) V f , e R f (cid:116) U f (cid:105) . (3.16)We now want a diagrammatic interpretation of (3.15) similarly to before. We make thefollowing constructions. R -intervals. For each j ∈ [ k ] and f ∈ { a, b, c, d } , we make a copy of [0 , f , and also number it by j . We write I + f,R for the collection of these intervals. Thesecorrespond to occurrences of f in R .For each j ∈ [ k ] and f ∈ { a, b, c, d } , we make a copy of [0 , f , and also number it by j . We write I − f,R for the collection of these intervals. Thesecorrespond to occurrences of f − in R .(These two constructions of k intervals correspond to the presence of f and f − eachexactly once in R .)These intervals are called R -intervals. There are 8 k R -intervals in total (for general g ,there are 4 gk of these intervals). R − -intervals. For each j ∈ [ k + 1 , k + (cid:96) ] and f ∈ { a, b, c, d } , we make a copy of [0 , f , and also number it by j . We write I + f,R − for the collection of these intervals.20igure 3.2: Here is shown the R -intervals (left) and the R − -intervals (right). We haveindicated their assigned direction and label (which f they correspond to). We have also, foreach endpoint of an interval, indicated which index function, e.g. r a , has this endpoint in itsdomain.These correspond to occurrences of f in R − .For each j ∈ [ k + 1 , k + (cid:96) ] and f ∈ { a, b, c, d } , we make a copy of [0 , f , and also number it by j . We write I − f,R − for the collection of these intervals.These correspond to occurrences of f − in R − .These intervals are called R − -intervals. There are 8 (cid:96) R − intervals in total (for general g ,there are 4 g(cid:96) of these intervals). See Figure 3.2 for an illustration of the R and R − intervals.We now view (by identifying endpoints of intervals with the given numbers of intervals in[ k + (cid:96) ]) r f : { i : i ∈ I + f,R } → [ n ] , R f : { i : i ∈ I − f,R } → [ n ] , s f : { i : i ∈ I + f,R } → [ n ] , S f : { i : i ∈ I − f,R } → [ n ] , U f : { i : i ∈ I − f,R − } → [ n ] , u f : { i : i ∈ I + f,R − } → [ n ] , V f : { i : i ∈ I − f,R − } → [ n ] , v f : { i : i ∈ I + f,R − } → [ n ] . We obtain from (3.16) (cid:104) h a e V a s a , e U a r a (cid:105)(cid:104) h b e V b s b , e U b r b (cid:105)(cid:104) h − a e u a R a , e v a S a (cid:105)(cid:104) h − b e u b R b , e v b S b (cid:105)(cid:104) h c e V c s c , e U c r c (cid:105)(cid:104) h d e V d s d , e U d r d (cid:105)(cid:104) h − c e u c R c , e v c S c (cid:105)(cid:104) h − d e u d R d , e v d S d (cid:105) = (cid:89) f (cid:89) i + ∈ I + f,R (cid:89) i − ∈ I − f,R (cid:89) j + ∈ I + f,R − (cid:89) j − ∈ I − f,R − h r f (0 i + ) s f (1 i + ) h u f (0 j + ) v f (1 j + ) ¯ h R f (1 i − ) S f (0 i − ) ¯ h U f (1 j − ) V f (0 j − ) . J n ( w, [ µ, ν ])= (cid:88) p i (cid:88) r f , R f , V f , v f , U f , u f , s f , S f (cid:88) a ∈A ( w ) β V a p s a ¯ β U a p r a β V b p s b ¯ β U b p r b β u a p R a ¯ β v a p S a β u b p R b ¯ β v b p S b β V c p s c ¯ β U c p r c β V d p s d ¯ β U d p r d β u c p R c ¯ β v c p S c β u d p R d ¯ β v d p S d (cid:89) f ∈{ a,b,c,d } (cid:90) h ∈ U ( n ) (cid:89) i ∈ I + f,w (cid:89) j ∈ I − f,w (cid:89) i + ∈ I + f,R (cid:89) i − ∈ I − f,R (cid:89) j + ∈ I + f,R − (cid:89) j − ∈ I − f,R − (3.17) h a (0 i ) a (1 i ) ¯ h a (1 j ) a (0 j ) h r f (0 i + ) s f (1 i + ) h u f (0 j + ) v f (1 j + ) ¯ h R f (1 i − ) S f (0 i − ) ¯ h U f (1 j − ) V f (0 j − ) dh. For each f , the integral in (3.17) can be done using the Weingarten calculus (Theorem2.5). To do this, fix bijections for each f ∈ { a, b, c, d } I + f def = I + f,R ∪ I + f,R − ∪ I + f,w ∼ = [ k + (cid:96) + p f ] I − f def = I − f,R ∪ I − f,R − ∪ I − f,w ∼ = [ k + (cid:96) + p f ]such that I + f,w ∼ = [ k + (cid:96) + 1 , k + (cid:96) + p f ] , I − f,w ∼ = [ k + (cid:96) + 1 , k + (cid:96) + p f ]and I + f,R ∼ = [ k ] , I − f,R ∼ = [ k ] , I + f,R − ∼ = [ k + 1 , (cid:96) ] , I − f,R − ∼ = [ k + 1 , k + (cid:96) ] (3.18)correspond to the original numberings of I + f,R , I − f,R , I + f,R − , I − f,R − .Hence if σ f , τ f ∈ S k + (cid:96) + p f we view σ f , τ f : I + f → I − f by the above fixed bijections. Foreach f ∈ { a, b, c, d } we say ( a , r f , u f , R f , U f ) → σ f if for all i ∈ I + f , i (cid:48) ∈ I − f with σ f ( i ) = i (cid:48) ,we have [ r f (cid:116) u f (cid:116) a ](0 i ) = [ R f (cid:116) U f (cid:116) a ](1 i (cid:48) );here we wrote e.g. [ r f (cid:116) u f (cid:116) a ] for the function that a , r f , u f induce on { i : i ∈ I + f ). Similarlywe say ( a , s f , v f , S f , V f ) → τ f if for all i ∈ I + f , i (cid:48) ∈ I − f with τ f ( i ) = i (cid:48) we have[ s f (cid:116) v f (cid:116) a ](1 i ) = [ S f (cid:116) V f (cid:116) a ](0 i (cid:48) ) . Theorem 2.5 translates to (cid:90) h ∈ U ( n ) (cid:89) i ∈ I + f,w (cid:89) j ∈ I − f,w (cid:89) i + ∈ I + f,R (cid:89) i − ∈ I − f,R (cid:89) j + ∈ I + f,R − (cid:89) j − ∈ I − f,R − h a (0 i ) a (1 i ) ¯ h a (1 j ) a (0 j ) h r f (0 i + ) s f (1 i + ) h u f (0 j + ) v f (1 j + ) ¯ h R f (1 i − ) S f (0 i − ) ¯ h U f (1 j − ) V f (0 j − ) dh = (cid:88) σ f ,τ f ∈ S k + (cid:96) + pf Wg n,k + (cid:96) + p f ( σ f τ − f ) { ( a , r f , u f , R f , U f ) → σ f , ( a , s f , v f , S f , V f ) → τ f } ,
22o putting this into (3.17) gives J n ( w, [ µ, ν ])= (cid:88) σ f ,τ f ∈ S k + (cid:96) + pf (cid:89) f ∈{ a,b,c,d } Wg n,k + (cid:96) + p f ( σ f τ − f ) (cid:88) p i (cid:88) a ∈A ( w ) , r f , R f , V f , v f , U f , u f , s f , S f ( a , r f , u f , R f , U f ) → σ f ( a , s f , v f , S f , V f ) → τ f β V a p s a ¯ β U a p r a β V b p s b ¯ β U b p r b β u a p R a ¯ β v a p S a β u b p R b ¯ β v b p S b β V c p s c ¯ β U c p r c β V d p s d ¯ β U d p r d β u c p R c ¯ β v c p S c β u d p R d ¯ β v d p S d . Here we make our main improvement over the classical Weingarten calculus. We introducethe following beneficial property that the σ f , τ f possibly have. Forbidden matchings property:
For every f ∈ { a, b, c, d } the following hold: neither σ f nor τ f map any element of I + f,R to an element of I − f,R − , or map an element of I + f,R − to an element of I − f,R .We have the following key lemma. Lemma 3.4.
If for some f ∈ { a, b, c, d } , σ f and τ f do not have the forbidden matchings property, then for any choice of p , . . . , p (cid:88) a ∈A ( w ) , r f , R f , V f , v f , U f , u f , s f , S f ( a , r f , u f , R f , U f ) → σ f ( a , s f , v f , S f , V f ) → τ f (3.19) β V a p s a ¯ β U a p r a β V b p s b ¯ β U b p r b β u a p R a ¯ β v a p S a β u b p R b ¯ β v b p S b β V c p s c ¯ β U c p r c β V d p s d ¯ β U d p r d β u c p R c ¯ β v c p S c β u d p R d ¯ β v d p S d = 0 . Proof.
Indeed suppose σ a matches an element i ∈ I + a,R with j ∈ I − a,R − ; σ a ( i ) = j . With ourgiven fixed bijections (3.18), i corresponds to an element of [ k ] and j corresponds to an elementof [ k + 1 , k + (cid:96) ] . Without loss of generality in the argument suppose that i corresponds to 1and j corresponds to k + 1. The condition σ a ( i ) = j and ( a , r a , u a , R a , U a ) → σ f means thatas functions on [ k ] and [ k + 1 , k + (cid:96) ], r a (1) = U a ( k + 1). There are no other constraints onthese values.Then for all variables in (3.19) fixed apart from r a and U a , and all values of r a , U a fixedother than r a (1) and U a ( k + 1) the ensuing sum over r a , U a is (cid:88) r a (1)= U a ( k +1) β U a p r a . But recalling the contraction operators from (2.2), this sum is the coordinate of e r a (2) ⊗· · · · · · e r a ( k ) ⊗ ˇ e U a ( k +2) ⊗ · · · ⊗ ˇ e U a ( k + (cid:96) ) in c , ( v p ). But c , ( v p ) = 0 because v p ∈ ˙ T k,(cid:96)n . We henceforth write (cid:80) ∗ σ f ,τ f to mean the sum is restricted to σ f , τ f satisfying the forbid- en matchings property. Lemma 3.4 now implies J n ( w, [ µ, ν ])= ∗ (cid:88) σ f ,τ f ∈ S pf + k + (cid:96) (cid:89) f ∈{ a,b,c,d } Wg n,k + (cid:96) + p f ( σ f τ − f ) (cid:88) p i (cid:88) a ∈A ( w ) , r f , R f , V f , v f , U f , u f , s f , S f ( a , r f , u f , R f , U f ) → σ f ( a , s f , v f , S f , V f ) → τ f β V a p s a ¯ β U a p r a β V b p s b ¯ β U b p r b β u a p R a ¯ β v a p S a β u b p R b ¯ β v b p S b β V c p s c ¯ β U c p r c β V d p s d ¯ β U d p r d β u c p R c ¯ β v c p S c β u d p R d ¯ β v d p S d . (3.20)Moreover, we can significantly tidy up (3.20). For everything in (3.20) fixed except for e.g. p , the ensuing sum over p is (cid:88) p β V a p s a ¯ β U b p r b = (cid:88) p (cid:104) e U b r b , v p (cid:105)(cid:104) v p , e V a s a (cid:105) = (cid:104) q θ e U b r b , e V a s a (cid:105) . Therefore executing the sums over p i in (3.20) we replace the sum over p i and the productover β -terms by (cid:104) q θ e U b r b , e V a s a (cid:105)(cid:104) q θ e v a S a , e V b s b (cid:105)(cid:104) q θ e v b S b , e u a R a (cid:105)(cid:104) q θ e U c r c , e u b R b (cid:105)× (3.21) (cid:104) q θ e U d r d , e V c s c (cid:105)(cid:104) q θ e v c S c , e V d s d (cid:105)(cid:104) q θ e v d S d , e u c R c (cid:105)(cid:104) q θ e U a r a , e u d R d (cid:105) . By Proposition 3.2 we have e.g. (cid:104) q θ e U b r b , e V a s a (cid:105) = (cid:88) π ∈ S k + (cid:96) z θ ( π ) (cid:104) ϕ − [ ρ k,(cid:96)n ( π )] e U b r b , e V a s a (cid:105) Now recall from (3.12) that ϕ − [ ρ k + (cid:96)n ( π )] = (cid:88) I =( i ,...,i k ) ,J =( j k +1 ,...,j k + (cid:96) ) e JI (cid:48) ( I (cid:116) J ; π ) ⊗ ˇ e J (cid:48) ( I (cid:116) J ; π ) I . This means that (cid:104) ϕ − [ ρ k + (cid:96)n ( π )] e U b r b , e V a s a (cid:105) is either equal to 0 or 1 and (cid:104) ϕ − [ ρ k + (cid:96)n ( π )] e U b r b , e V a s a (cid:105) =1 if and only if, letting (3.18) induce identifications { i : i ∈ I + a,R } ∼ = [ k ] , { i : i ∈ I − b,R − } ∼ = [ k + 1 , k + (cid:96) ] , { i : i ∈ I + b,R } ∼ = [ k ] , { i : i ∈ I − a,R − } ∼ = [ k + 1 , k + (cid:96) ] , via their given indexing of intervals, we have [ s a (cid:116) U b ] ◦ π = [ r b (cid:116) V a ], where e.g. s a (cid:116) U b isthe function either on endpoints of intervals or on [ k + (cid:96) ] induced by the union of s a and U b .24ence, repeating this argument,(3.21) = (cid:88) π ,...,π ∈ S k + (cid:96) (cid:32) (cid:89) i =1 z θ ( π i ) (cid:33) { [ s a (cid:116) U b ] ◦ π = [ r b (cid:116) V a ] , [ s b (cid:116) v a ] ◦ π = [ S a (cid:116) V b ] , [ R a (cid:116) v b ] ◦ π = [ S b (cid:116) u a ] , [ R b (cid:116) U c ] ◦ π = [ r c (cid:116) u b ] , [ s c (cid:116) U d ] ◦ π = [ r d (cid:116) V c ] , [ s d (cid:116) v c ] ◦ π = [ S c (cid:116) V d ] , [ R c (cid:116) v d ] ◦ π = [ S d (cid:116) u c ] , [ R d (cid:116) U a ] ◦ π = [ r a (cid:116) u d ] } . Putting all these arguments together gives J n ( w, [ µ, ν ])= ∗ (cid:88) σ f ,τ f ∈ S pf + k + (cid:96) (cid:88) π ,...,π ∈ S k + (cid:96) (cid:89) f ∈{ a,b,c,d } Wg n,k + (cid:96) + p f ( σ f τ − f ) (cid:32) (cid:89) i =1 z θ ( π i ) (cid:33)(cid:88) p i (cid:88) a ∈A ( w ) , r f , R f , V f , v f , U f , u f , s f , S f ( a , r f , u f , R f , U f ) → σ f ( a , s f , v f , S f , V f ) → τ f { [ s a (cid:116) U b ] ◦ π = [ r b (cid:116) V a ] , [ s b (cid:116) v a ] ◦ π = [ S a (cid:116) V b ] , [ R a (cid:116) v b ] ◦ π = [ S b (cid:116) u a ] , [ R b (cid:116) U c ] ◦ π = [ r c (cid:116) u b ] , [ s c (cid:116) U d ] ◦ π = [ r d (cid:116) V c ] , [ s d (cid:116) v c ] ◦ π = [ S c (cid:116) V d ] , [ R c (cid:116) v d ] ◦ π = [ S d (cid:116) u c ] , [ R d (cid:116) U a ] ◦ π = [ r a (cid:116) u d ] } . This formula says that we can calculate J n ( w, [ µ, ν ]) by summing over some combinatorialdata of matchings (the σ f , τ f , π i ) a quantity that we can understand well times a count of thenumber of indices that satisfy the prescribed matchings. To formalize this point of view wemake the following definition. Definition 3.5. A matching datum of the triple ( w, k, (cid:96) ) is a pair ( σ f , τ f ) ∈ S k + (cid:96) + p f × S k + (cid:96) + p f as above, satisfying the forbidden matchings property for each f ∈ { a, b, c, d } , together with( π , . . . , π ) ∈ ( S k + (cid:96) ) . We write MATCH ( w, k, (cid:96) )for the finite collection of all matching data for ( w, k, (cid:96) ).Given a matching datum { σ f , τ f , π i } , we write N ( { σ f , τ f , π i } ) for the number of choicesof a ∈ A ( w ) , r f , R f , V f , v f , U f , u f , s f , S f such that( a , r f , u f , R f , U f ) → σ f , ( a , s f , v f , S f , V f ) → τ f , [ s a (cid:116) U b ] ◦ π = [ r b (cid:116) V a ] , [ s b (cid:116) v a ] ◦ π = [ S a (cid:116) V b ] , [ R a (cid:116) v b ] ◦ π = [ S b (cid:116) u a ] , [ R b (cid:116) U c ] ◦ π = [ r c (cid:116) u b ] , [ s c (cid:116) U d ] ◦ π = [ r d (cid:116) V c ] , [ s d (cid:116) v c ] ◦ π = [ S c (cid:116) V d ] , [ R c (cid:116) v d ] ◦ π = [ S d (cid:116) u c ] , [ R d (cid:116) U a ] ◦ π = [ r a (cid:116) u d ] . (3.22)With this notation, we have proved the following theorem.25 heorem 3.6. For k + (cid:96) > , µ (cid:96) k and ν (cid:96) (cid:96) , w ∈ [ F , F ] , we have J n ( w, [ µ, ν ]) = (cid:88) { σ f ,τ f ,π i }∈ MATCH ( w,k,(cid:96) ) (cid:32) (cid:89) i =1 z θ ( π i ) (cid:33) (cid:89) f ∈{ a,b,c,d } Wg n,k + (cid:96) + p f ( σ f τ − f ) N ( { σ f , τ f , π i } ) . (3.23)We conclude this section by bounding the terms z θ ( π i ) and Wg n,k + (cid:96) + p f ( σ f τ − f ) usingProposition 2.7 and Lemma 3.3. Note that (cid:80) f ∈{ a,b,c,d } p f = | w | . This yields Corollary 3.7.
For k + (cid:96) > , µ (cid:96) k and ν (cid:96) (cid:96) , w ∈ [ F , F ] , we have J n ( w, [ µ, ν ]) (cid:28) k,(cid:96),w n − k − (cid:96) − | w | (cid:88) { σ f ,τ f ,π i }∈ MATCH ( w,k,(cid:96) ) n − (cid:80) f | σ f τ − f |− (cid:80) i =1 (cid:107) π i (cid:107) k,(cid:96) N ( { σ f , τ f , π i } ) . (3.24)We will proceed in the next section to understand all the quantities in (3.24) in topologicalterms by constructing a surface from each { σ f , τ f , π i } . We now show how a datum in
MATCH ( w, k, (cid:96) ) can be used to construct a surface such thatthe terms appearing in (3.23) can be bounded by topological features of the surface. Thisconstruction is similar to the constructions of [MP19, MP15], but with the presence of ad-ditional π i adding a new aspect. We continue to assume g = 2 for simplicity. We can stillassume that γ ∈ [Γ , Γ ] and hence w ∈ [ F , F ]. Construction of the 1-skeleton π -intervals. The identifications of the previous section mean that we view π : { i : i ∈ I + b,R ∪ I − a,R − } → { i (cid:48) : i (cid:48) ∈ I + a,R ∪ I − b,R − } ,π : { i : i ∈ I − a,R ∪ I − b,R − } → { i (cid:48) : i (cid:48) ∈ I + b,R ∪ I + a,R − } ,π : { i : i ∈ I − b,R ∪ I + a,R − } → { i (cid:48) : i (cid:48) ∈ I − a,R ∪ I + b,R − } ,π : { i : i ∈ I + c,R ∪ I + b,R − } → { i (cid:48) : i (cid:48) ∈ I − b,R ∪ I − c,R − } ,π : { i : i ∈ I + d,R ∪ I − c,R − } → { i (cid:48) : i (cid:48) ∈ I + c,R ∪ I − d,R − } ,π : { i : i ∈ I − c,R ∪ I − d,R − } → { i (cid:48) : i (cid:48) ∈ I + d,R ∪ I + c,R − } ,π : { i : i ∈ I − d,R ∪ I + c,R − } → { i (cid:48) : i (cid:48) ∈ I − c,R ∪ I + d,R − } ,π : { i : i ∈ I + a,R ∪ I + d,R − } → { i (cid:48) : i (cid:48) ∈ I − d,R ∪ I − a,R − } . (4.1)We add an arc between any two interval endpoints that are mapped to one another by some π i . All the intervals added here are called π -intervals . The purpose of this construction is that26he conditions concerning π i in (3.22) correspond to the fact that two end-points of intervalsconnected by a π -interval are assigned the same value in [ n ] by the relevant functions out of r f , R f , V f , v f , U f , u f , s f , S f (at most one of these functions has any given interval endpointin its domain). The π -intervals together with the R -intervals and R − intervals form a collection of loopsthat we call R ± - π -loops. σ -arcs and τ -arcs. Recall from the previous sections that we view σ f , τ f : I + f → I − f . We add an arc between each 0 i and 1 i (cid:48) with σ f ( i ) = i (cid:48) and between each 1 i and 0 i (cid:48) with τ f ( i ) = i (cid:48) . These arcs are called σ f -arcs and τ f -arcs respectively. Any σ f -arc (resp. τ f -arc) isalso called a σ -arc (resp. τ -arc). Notice even though an arc is formally the same as an interval,we distinguish these types of objects. The only arcs that exist are σ -arcs and τ -arcs. Thepurpose of this construction is that the conditions pertaining to σ f , τ f in (3.22) are equivalentto the fact that two end-points of intervals connected by a σ -arc or τ -arc are assigned thesame value in [ n ] by the relevant functions out of a , r f , R f , V f , v f , U f , u f , s f , S f . After adding these arcs, every endpoint of an interval has exactly one arc emanating fromit. We have therefore now constructed a trivalent graph G ( { σ f , τ f , π i } ) . The number of vertices of this graph is the twice the total number of w -intervals, R -intervals,and R − -intervals which is 2( | w | + 8( k + (cid:96) )). Therefore we have χ ( G ( { σ f , τ f , π i } )) = − ( | w | + 8( k + (cid:96) )) . (4.2)(For general g , we have χ ( G ( { σ f , τ f , π i } )) = − ( | w | + 4 g ( k + (cid:96) )).) Moreover, the conditions in(3.22) are now interpreted purely in terms of the combinatorics of this graph. Gluing in discs
There are two types of cycles in G ( { σ f , τ f , π i } ) that we wish to consider: • Cycles that alternate between following either a w -intermediate interval or a π -intervaland then either a σ -arc or a τ -arc. These cycles are disjoint from one another, and every σ or τ -arc is contained in exactly one such cycle. We call these cycles type-I cycles. Forevery type-I cycle, we glue a disc to G ( { σ f , τ f , π i } ) along its boundary, following thecycle. These discs will be called type-I discs. (These are analogous to the o -discs of[MP19].) • Cycles that alternate between following either a w -interval, an R -interval, or an R − -interval and then either a σ -arc or a τ -arc. Again, these cycles are disjoint, and every σ or τ -arc is contained in exactly one such cycle. We call these cycles type-II cycles. For every type-II cycle, we glue a disc to G ( { σ f , τ f , π i } ) identifying the boundary of thedisc with the cycle. These discs will be called type-II discs. (These are similar to the z -discs of [MP19].) 27ecause every interior of an interval meets exactly one of the glued-in discs, and every archas two boundary segments of discs glued to it, the object resulting from gluing in these discsis a decorated topological surface that we denote byΣ( { σ f , τ f , π i } ) . The boundary components of Σ( { σ f , τ f , π i } ) consist of the w -loop and the R ± - π -loops. Itis not hard to check that Σ( { σ f , τ f , π i } ) is orientable with an orientation compatible withthe fixed orientations of the boundary loops corresponding to traversing every w -interval or R ± -interval from 0 to 1.We view the given CW-complex structure, and the assigned labelings and directions ofthe intervals that now pave ∂ Σ as part of the data of Σ( { σ f , τ f , π i } ). The number of discs ofΣ( { σ f , τ f , π i } ) is connected to the quantities appearing in Proposition 3.6 as follows. Lemma 4.1. N ( { σ f , τ f , π i } ) = n { type-I discs of Σ( { σ f , τ f , π i } ) } .Proof. The constraints on the functions a , r f , R f , V f , v f , U f , u f , s f , S f in (3.22) now corre-spond to the fact that altogether, they assign the same value in [ n ] to every interval end-pointin the same type-I-cycle, and there are no other constraints between them.The quantities | σ f τ − f | in (3.24) can also be related to Σ( { σ f , τ f , π i } ) as follows. Lemma 4.2.
We have (cid:89) f ∈{ a,b,c,d } n −| σ f τ − f | = n − k + (cid:96) ) − | w | n { type-II discs of Σ( { σ f , τ f , π i } ) } . Proof.
Recalling the definition of | σ f τ − f | from Proposition 2.7, we can also write | σ f τ − f | = k + (cid:96) + p f − { cycles of σ f τ − f } .The cycles of { σ f τ − f : f ∈ { a, b, c, d }} are in 1:1 correspondence with the type-II cycles ofΣ( { σ f , τ f , π i } ) and hence also the type-II discs. Therefore (cid:89) f ∈{ a,b,c,d } n −| σ f τ − f | = n − k + (cid:96) ) n (cid:80) f ∈{ a,b,c,d } ( − p f + { cycles of σ f τ − f } ) = n − k + (cid:96) ) − | w | n { type-II discs of Σ( { σ f , τ f , π i } ) } . We are now able to prove the following.
Theorem 4.3.
For k + (cid:96) > , µ (cid:96) k and ν (cid:96) (cid:96) , w ∈ [ F , F ] , we have J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) (cid:88) { σ f ,τ f ,π i }∈ MATCH ( w,k,(cid:96) ) n − (cid:80) i =1 (cid:107) π i (cid:107) k,(cid:96) n χ (Σ( { σ f ,τ f ,π i } )) . Proof.
Combining Lemmas 4.1 and 4.2 with Corollary 3.7 gives J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) n − k − (cid:96) −| w | (cid:88) { σ f ,τ f ,π i }∈ MATCH ( w,k,(cid:96) ) n − (cid:80) i =1 (cid:107) π i (cid:107) k,(cid:96) n { discs of Σ( { σ f , τ f , π i } ) } . J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) (cid:88) { σ f ,τ f ,π i }∈ MATCH ( w,k,(cid:96) ) n − (cid:80) i =1 (cid:107) π i (cid:107) k,(cid:96) n χ ( G ( { σ f ,τ f ,π i } ))+ { discs of Σ( { σ f , τ f , π i } ) } = (cid:88) { σ f ,τ f ,π i }∈ MATCH ( w,k,(cid:96) ) n − (cid:80) i =1 (cid:107) π i (cid:107) k,(cid:96) n χ (Σ( { σ f ,τ f ,π i } )) . Theorem 4.3 suggests that we now bound χ (Σ( { σ f , τ f , π i } )) − (cid:88) i =1 (cid:107) π i (cid:107) k,(cid:96) for all { σ f , τ f , π i } ∈ MATCH ( w, k, (cid:96) ). To do this, we make some observations that simplifythe task. If C is a simple closed curve in a surface S , then compressing S along C means thatwe cut S along C and then glue discs to cap off any new boundary components created bythe cut.Suppose that we are given { σ f , τ f , π i } ∈ MATCH ( w, k, (cid:96) ). Then { σ f , σ f , π i } is also in MATCH ( w, k, (cid:96) ) (the forbidden matching property continues to hold). It is not hard to seethat χ (Σ( { σ f , σ f , π i } )) ≥ χ (Σ( { σ f , τ f , π i } )) . Indeed, the τ f arcs can be replaced by σ f -parallel arcs inside the type-II discs of Σ( { σ f , τ f , π i } ).The resulting surface’s arcs may not cut the surface into discs, but this can be fixed by(possibly repeatedly) compressing the surface along simple closed curves disjoint from thearcs, leaving the combinatorial data of the arcs unchanged but only potentially increasing theEuler characteristic.It remains to deal with the sum (cid:80) i =1 (cid:107) π i (cid:107) k,(cid:96) .Suppose again that an arbitrary { σ f , τ f , π i } ∈ MATCH ( w, k, (cid:96) ) is given. For each i ∈ [8]write π i = π ∗ i σ i where π ∗ i ∈ S k × S (cid:96) , σ i = ( π ∗ i ) − π i ∈ S k + (cid:96) , and | σ i | = (cid:107) π i (cid:107) k,(cid:96) . Let X = Σ( { σ f , τ f , π i } ).Take Σ( { σ f , τ f , π i } ) and add to it all the π ∗ i -intervals that would have been added if π i wasreplaced by π ∗ i for each i ∈ [8] in its construction. The resulting object X is the decoratedsurface X together with a collection of π ∗ i -intervals with endpoints in the boundary of X ,and interiors disjoint from X . This adds 8( k + (cid:96) ) edges to X and hence χ ( X ) = χ (Σ( { σ f , τ f , π i } )) − k + (cid:96) ) . Now we consider all cycles that for any fixed i ∈ [8], alternate between π i -intervals and π ∗ i -intervals. The number of these cycles is the total number of cycles of the permutations { ( π ∗ i ) − π i : i ∈ [8] } . On the other hand, the number of cycles of ( π ∗ i ) − π i is k + (cid:96) − | ( π ∗ i ) − π i | = k + (cid:96) − | σ i | = k + (cid:96) − (cid:107) π i (cid:107) k,(cid:96)
29o in total there are 8( k + (cid:96) ) − (cid:80) i (cid:107) π i (cid:107) k,(cid:96) of these cycles. For every such cycle, we glue a discalong its boundary to the cycle. The resulting object is denoted X . Now, X is a topologicalsurface, and we added 8( k + (cid:96) ) − (cid:80) i (cid:107) π i (cid:107) k,(cid:96) discs to X to form X , so χ ( X ) = χ ( X ) + 8( k + (cid:96) ) − (cid:88) i (cid:107) π i (cid:107) k,(cid:96) = χ (Σ( { σ f , τ f , π i } )) − (cid:88) i (cid:107) π i (cid:107) k,(cid:96) . Now ‘forget’ all the original π i -intervals from X to form X . The surface X is a decoratedsurface in the same sense as X , except the connected components of X − { arcs } may notbe discs. Similarly to before, by sequentially compressing X along non-nullhomotopic simpleclosed curves disjoint from arcs, if they exist, we obtain a new decorated surface X . Moreover,and this is the main point, X is the same as Σ( { σ f , τ f , π ∗ i } ) in the sense that they are relatedby a decoration-respecting cellular homeomorphism. Compression can only increase the Eulercharacteristic, so we obtain χ (Σ( { σ f , τ f , π ∗ i } )) ≥ χ ( X ) = χ ( X ) = χ (Σ( { σ f , τ f , π i } ) − (cid:88) i (cid:107) π i (cid:107) k,(cid:96) . Combining these two arguments proves the following proposition.
Proposition 4.4.
For any given { σ f , τ f , π i } , there exist π ∗ i ∈ S k × S (cid:96) for i ∈ [8] such that χ (Σ( { σ f , σ f , π ∗ i } )) − (cid:88) i =1 (cid:107) π ∗ i (cid:107) k,(cid:96) = χ (Σ( { σ f , σ f , π ∗ i } )) ≥ χ (Σ( { σ f , τ f , π i } )) − (cid:88) i =1 (cid:107) π i (cid:107) k,(cid:96) . This has the following immediate corollary when combined with Theorem 4.3. Let
MATCH ∗ ( w, k, (cid:96) )denote the subset of MATCH ( w, k, (cid:96) ) consisting of { σ f , σ f , π i } (i.e. σ f = τ f for each f ∈{ a, b, c, d } ) with π i ∈ S k × S (cid:96) for each i ∈ [8]. Corollary 4.5.
For k + (cid:96) > , µ (cid:96) k and ν (cid:96) (cid:96) , w ∈ [ F , F ] , we have J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) n max { σf ,σf ,πi }∈ MATCH ∗ ( w,k,(cid:96) ) χ (Σ( { σ f ,σ f ,π i } )) . The benefit to having π i ∈ S k × S (cid:96) for i ∈ [8] is the following. Suppose now that { σ f , σ f , π i } ∈ MATCH ∗ ( w, k, (cid:96) ). Recall that the boundary loops of Σ( { σ f , σ f , π i } ) consistof one w -loop and some number of R ± - π -loops. The condition that each π i ∈ S k × S (cid:96) meansthat no π -interval ever connects an endpoint of a R -interval with an endpoint of an R − -interval. So every boundary component of Σ( { σ f , σ f , π i } ) that is not the w -loop containseither only R -intervals or only R − -intervals, and in fact, when following the boundary com-ponent and reading the directions and labels of the intervals according to traversing each from0 to 1, reads out a positive power of R (in the former case of only R -intervals) or a negativepower of R − (in the latter case of only R − -intervals). The sum of the positive powers of R inboundary loops is k , and the sum of the negative powers of R is − (cid:96) . Knowing this boundarystructure is extremely important for the arguments in the next sections.30 .3 A topological result that proves Theorem 3.1 Here, in the spirit of Culler [Cul81], we explain another way to think about the surfacesΣ( { σ f , σ f , π i } ) for { σ f , σ f , π i } ∈ MATCH ∗ ( w, k, (cid:96) ) that is easier to work with than the con-struction we gave. At this point we also show how things work for general g ≥
2. An arc ina surface Σ is a properly embedded interval in Σ with endpoints in the boundary ∂ Σ. Definition 4.6.
For w ∈ F g , we define surfaces ( w, k, (cid:96) ) to be the set of all decorated surfacesΣ ∗ as follows. A decorated surface Σ ∗ ∈ surfaces ( w, k, (cid:96) ) is an oriented surface with bound-ary, with compatibly oriented boundary components, together with a collection of disjointembedded arcs that cut Σ ∗ into topological discs. One boundary component is assigned tobe a w -loop, and every other boundary component is assigned to be either a R -loop or an R − -loop. Each arc is assigned a transverse direction and a label in { a , b , . . . , a g , b g } . Everyarc-endpoint in ∂ Σ ∗ inherits a transverse direction and label from the assigned direction andlabel of its arc. We require that Σ ∗ satisfy the following properties. P1 When one follows the w -loop according to its assigned orientation, and reads f when an f -labeled arc-endpoint is traversed in its given direction, and f − when an f -labeledarc-endpoint is traversed counter to its given direction, one reads a cyclic rotation of w in reduced form, depending on where one begins to read. P2 When one follows any R -loop according to its assigned orientation in the same way asbefore, one reads (a cyclic rotation) of some positive power of R g in reduced form. Thesum of these positive powers over all R -loops is k . P3 When one follows any R − -loop according to its assigned orientation in the same way asbefore, one reads (a cyclic rotation) of some negative power of R g in reduced form. Thesum of these negative powers over all R − -loops is − (cid:96) . P4 No arc connects an R -loop to an R − -loop.Given a surface Σ( { σ f , σ f , π i } ) with { σ f , σ f , π i } ∈ MATCH ∗ ( w, k, (cid:96) ), all the type-II discs ofthe surface are rectangles. Hence, by collapsing each w -interval, R -interval, and R − -intervalto a point, and collapsing every type-II rectangle to an arc, we obtain a CW-complex thatis a surface with boundary, cut into discs by arcs. Every arc inherits a transverse directionand label from the compatible assigned directions and labels of the intervals in the boundaryof its originating type-II rectangle. We call this modified surface Σ ∗ = Σ ∗ ( { σ f , π i } ). Itclearly satisfies P1-P3 and P4 follows from the forbidden matchings property. (Of course,when g = 2, we identify { a, b, c, d } with { a , b , a , b } .) We also have χ (Σ( { σ f , σ f , π i } )) = χ (Σ ∗ ( { σ f , π i } )). With Definition 4.6 and the remarks proceeding it, we can now state afurther consequence of Corollary 4.5 as it extends to general g ≥ Corollary 4.7.
For k + (cid:96) > , µ (cid:96) k , ν (cid:96) (cid:96) , w ∈ [ F g , F g ] , as n → ∞J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) n max { χ (Σ ∗ ) : Σ ∗ ∈ surfaces ( w,k,(cid:96) ) } . In order for Corollary 4.7 to give us strong enough results it needs to be combined withthe following non-trivial topological bound.
Proposition 4.8. If w ∈ [ F g , F g ] is a shortest element representing the conjugacy class of γ ∈ Γ g , w (cid:54) = id , and Σ ∗ ∈ surfaces ( w, k, (cid:96) ) then χ (Σ ∗ ) ≤ − ( k + (cid:96) ) . emark . Proposition 4.8 is by no means a trivial statement and one has to use that w is a shortest element representing the conjugacy class of some element of Γ g . For example, if w = R g , then w represents the conjugacy class of id Γ g , but for k = 0 and (cid:96) = 1 there is an‘obvious’ annulus in surfaces ( w, , χ = 0 > − ( k + (cid:96) ) = −
1. Proposition 4.8 alsorequires w (cid:54) = id; if w = id then for k = 0 and (cid:96) = 1 one can take a disc with no arcs as a validelement of surfaces (id , , χ = 1 > − ( k + (cid:96) ) = 0. In fact this disc is ultimatelyresponsible for E g,n [tr id ] = n .The proof of Proposition 4.8 is self-contained and given in §§ Proof of Theorem 3.1 given Proposition 4.8.
Since Theorem 3.1 was proved when k = (cid:96) = 0in §§ k + (cid:96) >
0. Then combining Corollary 4.7 and Proposition 4.8 gives J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) n − ( k + (cid:96) ) . On the other hand, D [ µ,ν ] ( n ) = O (cid:0) n k + (cid:96) (cid:1) from (2.1). Therefore D [ µ,ν ] ( n ) J n ( w, [ µ, ν ]) (cid:28) w,k,(cid:96) As we mentioned in §§ w ∈ [ F g , F g ]is a shortest element representing the conjugacy class of γ ∈ Γ g . We use a combinatorialcharacterization of such words that stems from Dehn’s algorithm [Deh12] for solving theproblem of whether a given word represents the identity in Γ g . The ideas of Dehn’s algorithmwere refined by Birman and Series in [BS87]. In [MP20], the author and Puder used Birmanand Series’ results (alongside other methods) to obtain the analog of Theorem 1.2 when thefamily of groups SU ( n ) is replaced by the family of symmetric groups S n . Similar consequencesof the work of Dehn, Birman, and Series that we used in (ibid.) will be used here.We now follow the language of [MP20] to state the results we need in this paper. We stressone more time that these results are simple and direct consequences of the work of Birmanand Series.We view the universal cover of Σ g as a disc tiled by 4 g -gons that we call U . We assumeevery edge of this tiling is directed and labeled by some element of { a , b , . . . , a g , b g } suchthat when we read counter-clockwise along the boundary of any octagon we read the reducedcyclic word [ a , b ] · · · [ a g , b g ]. By fixing a basepoint u ∈ U we obtain a free cellular action ofΓ g on U that respects the labels and directions of edges and identifies the quotient Γ g \ U withΣ g ; this gives a description of Σ g as a 4 g -gon with glued sides as is typical.Now suppose that γ ∈ Γ is not the identity. The quotient A γ def = (cid:104) γ (cid:105)\ U of U by the cyclicgroup generated by γ is an open annulus tiled by infinitely many 4 g -gons. The edges of A γ inherit directions and labels from those of the edges of U . The point u ∈ U maps to somepoint denoted by x ∈ A γ .Now let w ∈ F g be an element that represents γ , and identify w with a combinatorialword by writing w in reduced form. Beginning at x , and following the path spelled outby w beginning at x , we obtain an oriented closed loop L w in the one-skeleton of A γ . If w is a shortest element representing the conjugacy class of γ , then this loop L w must nothave self-intersections. In this case, that we from now assume, L w is therefore a topologically32igure 4.1: Illustration of a piece P of ˆ L w in the case when the reduced form of w contains a g a − b − as a subword. The edges of L w are in bold. The piece is indicated by the dottedlines. This piece P has e ( P ) = 2, he ( P ) = 7, and χ ( P ) = 1. Note that a piece may also runalong the other side of L w .embedded circle in the annulus A γ that is non-nullhomotopic and cuts A γ into two annuli A ± γ .Every vertex of A γ has 4 g incident half-edges each of which has an orientation and directiongiven by the edge they are in. Going clockwise, the cyclic order of the half-edges incident atany vertex is‘ a -outgoing, b -incoming, a -incoming, b -outgoing, ... , a g -outgoing, b g -incoming, a g -incoming, b g -outgoing’.We define ˆ L w to be the loop L w with all incident half edges in A γ attached. We call thenew half-edges added hanging half-edges. Moreover, we thicken up ˆ L w by viewing each edge of L w as a rectangle, each hanginghalf-edge as a half-rectangle, and each vertex replaced by a disc. In other words, we take asmall neighborhood of ˆ L w in A γ . We now think of ˆ L w as the thickened version. This is atopological annulus, where the hanging half-edges have become stubs hanging off. A piece of ˆ L w is a contiguous collection of hanging half-rectangles and rectangle sides following edges of L w in the boundary of ˆ L w . Such a piece is in either A + γ or A − γ . Given a piece P of ˆ L w wewrite e ( P ) for the number of rectangle sides following edges of L w , and he ( P ) for the numberof hanging-half edges in P . We say that a piece P has Euler characteristic χ ( P ) = 0 if itfollows an entire boundary component of ˆ L w , and χ ( P ) = 1 otherwise as we view it as aninterval running along the rectangle sides and around the sides of the hanging half-rectangles.See Figure 4.1 for an illustration of a piece of ˆ L w .Birman and Series prove in [BS87, Thm. 2.12(a)] that if w is a shortest element repre-senting the conjugacy class of γ ∈ Γ g then there are strong restrictions on the pieces of ˆ L w that can appear. This has the following consequence which is given by [MP20, Proof of Lem.4.20]. Lemma 4.10. If w is a shortest element representing the conjugacy class of γ ∈ Γ g , and both and hence w are non-identity, then for any piece P of ˆ L w , we have e ( P ) ≤ (2 g − he ( P ) + 2 gχ ( P ) . Proof.
Since w is a shortest element representing some non-identity conjugacy class in Γ g , inthe language of [MP20], L w is a boundary reduced tiled surface. Then the proof of [MP20,Lem. 4.20] contains the result stated in the lemma.This inequality plays a crucial role in the next section. Suppose that g ≥ w ∈ [ F g , F g ] is a non-identity shortest element representing theconjugacy class of γ ∈ Γ g . In particular, w is cyclically reduced. We let R = R g . Now fix k, (cid:96) ∈ N and suppose Σ ∗ ∈ surfaces ( w, k, (cid:96) ). The arcs of Σ ∗ are of three different types: WR An arc with one endpoint in the w -loop and one endpoint in an R or R − -loop. RR An arc with both endpoints in R or R − loops. By property P4, the endpoints of suchan arc are both in R -loops or both in R − -loops. WW An arc with both endpoints in the w -loop.The boundary of any disc of Σ ∗ alternates between segments of ∂ Σ ∗ and arcs. A disc is a pre-piece disc if its boundary contains exactly one segment of the w -loop. A disc is called a junction disc if it is not a pre-piece disc. We say that a junction disc is piece-adjacent if itmeets a WR-arc-side.To be precise, we view all discs as open discs, and hence not containing any arcs. A discmeets certain arc-sides along its boundary; it is possible for a disc to meet both sides of thesame arc and we view this scenario as the disc meeting two separate arc-sides. We say anarc-side has the same type WR/RR/WW as its corresponding arc.Note that any pre-piece disc cannot meet any WW-arc-side: if it did, the disc could onlymeet this one arc-side together with one segment of the w -loop and this would contradict thefact that w is cyclically reduced since the arc matches a letter f with a cyclically adjacentletter f − of w . It is also clear that any pre-piece disc meets exactly 2 WR-arc-sides: the onesthat emanate from the sole segment of the w -loop. So in light of P4 a pre-piece disc takesone of the forms shown in Figure 4.2.We define a piece of Σ ∗ to be a connected component of { pre-piece discs } ∪ { WR-arcs } . A piece of Σ ∗ is therefore either a contiguous collection of pre-piece discs that meet only alongWR-arcs, or a single WR-arc. If P is a piece of Σ ∗ , either χ ( P ) = 1, or χ ( P ) = 0, in whichcase P meets the entire w -loop and is the unique piece.We now have two definitions of pieces; pieces of ˆ L w and pieces of Σ ∗ . These are, as thenames suggest, closely related, and this is the key observation in the proof of Proposition 4.8.Indeed, the reader should carefully consider Figure 4.3 that leads to the following lemma. Inanalogy to pieces of ˆ L w , if P is any piece of Σ ∗ , we write e ( P ) for the number of WR-arcs34igure 4.2: Possible forms of pre-piece discs. The number of R -loop segments or R − -loopsegments is at least 1 and bounded given k and (cid:96) . The arrows denote the orientations of theboundary loops.Figure 4.3: Given a segment of the w -loop corresponding to a juncture between letters a − b − in w , if this segment is part of a pre-piece disc then some possible forms of that disc are shownabove. This juncture between letters of w corresponds to a vertex in L w . The right handillustration shows the neighborhood of this vertex in the annulus A γ , where the bold linescorrespond to half-edges of L w . The right hand picture actually almost determines the lefthand pictures. Indeed, given the a arc on the top-left, the next arc has to be a b g arc withthe given direction, since only b − g cyclically precedes a in R g or any power of R g . Then thenext arc a g with its direction is determined since only a g cyclically precedes b g in R g . Thiscontinues until an arc labeled by b and with an incoming direction is reached, as in the rightarc of the top-left picture. At this point, the boundary of the disc may close up. (This isanalogous to what happens in the bottom picture, where an analogous pattern occurs.) Theonly indeterminacy is that after reaching a b arc with an incoming direction for the firsttime, the entire pattern shown in the right hand picture may repeat any number of times, aslong as k and (cid:96) allow it. The upshot of this is that any pre-piece disc has at least as manyincident RR-arc-sides as there are hanging half-edges on the corresponding side of L w , at thecorresponding vertex. 35n P , and he ( P ) for the number of RR-arc sides that meet P (this is zero if P is a singleWR-arc). Lemma 4.11. If w is a shortest element representing the conjugacy class of γ ∈ Γ g , k, (cid:96) ∈ N ,and Σ ∗ ∈ surfaces ( w, k, (cid:96) ) then for any piece P of Σ ∗ , we have e ( P ) ≤ (2 g − he ( P ) + 2 gχ ( P ) . Proof.
Given any piece P of Σ ∗ , it contains a consecutive (possibly cyclic) series of WR-arcsthat correspond to a contiguous collection of edges in the loop L w . The discs of P correspondto certain vertices of L w ; each of these vertices has two emanating half-edges belonging tothe edges defined by WR-arcs of P . The piece P can either meet only R -loops or meet only R − -loops.We define a piece P (cid:48) of ˆ L w corresponding to P as follows. If P meets R -loops, then P (cid:48) consists of rectangle sides along the edges of L w corresponding to the WR-arcs of P togetherwith all hanging half-edges at vertices corresponding to discs of P that are on the left of L w as it is traversed in its assigned orientation (corresponding to reading w along L w ). If P (cid:48) meets R − -loops, then P (cid:48) is defined similarly with the modification that we include insteadhanging half-edges on the right of L w . Figure 4.3 together with its captioned discussion nowshows that he ( P (cid:48) ) ≤ he ( P ) , and e ( P ) = e ( P (cid:48) ) by construction. We also have χ ( P (cid:48) ) = χ ( P ). Therefore Lemma 4.10applied to P (cid:48) implies e ( P ) = e ( P (cid:48) ) ≤ (2 g − he ( P (cid:48) ) + 2 gχ ( P (cid:48) ) ≤ (2 g − he ( P ) + 2 gχ ( P ) . Let N RR be the number of RR-arcs, N W R the number of WR-arcs, and N W W the numberof WW-arcs in Σ ∗ . In the following we refer to discs of Σ ∗ simply as discs. Since there are4 g ( k + (cid:96) ) incidences between arcs and R -loops or R − loops we have2 N RR + N W R = 4 g ( k + (cid:96) ) . (4.3)Let Σ be the surface formed by cutting Σ ∗ along all RR-arcs. We have χ (Σ ) = (cid:88) discs D (cid:18) − d (cid:48) ( D )2 (cid:19) where d (cid:48) ( D ) is the number of arc-sides meeting D that are not of type RR. (In other words,36 (cid:48) ( D ) is the degree of D in the dual graph of Σ .) We partition this sum according to χ (Σ ) = S + S + S ,S = (cid:88) pre-piece discs D (cid:18) − d (cid:48) ( D )2 (cid:19) ,S = (cid:88) piece-adjacent junction discs D (cid:18) − d (cid:48) ( D )2 (cid:19) ,S = (cid:88) not piece-adjacent junction discs D (cid:18) − d (cid:48) ( D )2 (cid:19) . Note first that a pre-piece disc has d (cid:48) ( D ) = 2 (cf. Fig 4.2). Hence S = 0. We deal with S next. For a disc D of Σ ∗ , let d W R ( D ) denote the number of WR-arc-sides meeting D . Notethat a piece-adjacent junction disc D has d W R ( D ) > S as S = (cid:88) piece-adjacent junction discs D (cid:18) − d (cid:48) ( D )2 (cid:19) d W R ( D ) (cid:88) incidences between D and WR-arc-sides (cid:88) pieces P (cid:88) incidences between P and some junction disc D along WR-arc d W R ( D ) (cid:18) − d (cid:48) ( D )2 (cid:19) = (cid:88) pieces P (cid:88) incidences between P and some junction disc D along WR-arc Q ( D ) (4.4)where for a piece-adjacent junction disc DQ ( D ) def = 1 d W R ( D ) (cid:18) − d (cid:48) ( D )2 (cid:19) . Suppose that D is a piece-adjacent junction disc. By parity considerations, d W R ( D ) iseven. We estimate Q ( D ) by splitting into two cases. If d W R ( D ) = 2 then d (cid:48) ( D ) ≥ D would meet only 2 WR arc-sides and other RR arc-sides, hence be a pre-piecedisc and not be a junction disc. In this case Q ( D ) = 12 (cid:18) − d (cid:48) ( D )2 (cid:19) ≤ (cid:18) − (cid:19) = − . Otherwise, d W R ( D ) ≥ d (cid:48) ( D ) ≥ d W R ( D ), we have Q ( D ) ≤ d W R ( D ) (cid:18) − d W R ( D )2 (cid:19) = 1 d W R ( D ) − ≤ −
12 = − . So we have proved that for all piece-adjacent junction discs D , Q ( D ) ≤ − . Putting this into(4.4) gives S ≤ − (cid:88) pieces P (cid:88) incidences between P and some junction disc D along WR-arc − (cid:88) pieces P χ ( P ) = − (cid:88) pieces P χ ( P ) . (4.5)37e now turn to S . Here is the key moment where w (cid:54) = id is used . Since w (cid:54) = id, any discmust meet an arc. Indeed, the only other possibility is that the boundary of the disc is anentire boundary loop that has no emanating arcs. This hypothetical boundary loop cannotbe an R or R − -loop, so it has to be the w -loop. But this would entail w = id.Hence any disc contributing to S meets no WR-arc-side, but meets some arc-side. There-fore it meets only WW-arcs or only RR-arcs. Every disc D contributing to S meeting onlyWW-arcs gives a non-positive contribution since w is cyclically reduced hence d (cid:48) ( D ) ≥ D contributing to S meeting only RR-arcs, which we will call an RR-disc , has d (cid:48) ( D ) = 0 and hence contributes 1 to S .This shows S ≤ { RR-discs } . (4.6)In total combining S = 0 with (4.5) and (4.6) we get χ (Σ ) ≤ { RR-discs } − (cid:88) pieces P of Σ ∗ χ ( P ) . To obtain Σ ∗ from Σ we have to glue all cut RR-arcs, of which there are N RR . Each gluingdecreases χ by 1 so χ (Σ ∗ ) ≤ { RR-discs } − N RR − (cid:88) pieces P of Σ ∗ χ ( P ) . Using Lemma 4.11 with the above gives χ (Σ ∗ ) ≤ { RR-discs } − N RR − (cid:88) pieces P of Σ ∗ χ ( P ) ≤ { RR-discs } − N RR − g (cid:88) pieces P of Σ ∗ e ( P ) + (2 g − g (cid:88) pieces P of Σ ∗ he ( P )= { RR-discs } − N RR − N W R g + (2 g − g (cid:88) pieces P of Σ ∗ he ( P ) . (4.7)Let he (cid:48) (Σ ∗ ) denote the total number of RR-arc-sides meeting RR-discs. Every RR-dischas to meet at least 4 g arc-sides; this observation is similar to the reasoning in Figure 4.3.Therefore he (cid:48) (Σ ∗ ) ≥ g { RR-discs } . (4.8)Every RR-arc-side either meets a piece P and contributes to he ( P ) or a disc meeting onlyRR-arc-sides and contributes to he (cid:48) (Σ ∗ ). Hence he (cid:48) (Σ ∗ ) + (cid:88) pieces P of Σ ∗ he ( P ) = 2 N RR . (4.9) Although technically, w (cid:54) = id was used to define L w and pieces etc, if w is the identity the proof ofProposition 4.8 could, a priori, circumvent these definitions. χ (Σ ∗ ) (4.8) ≤ he (cid:48) (Σ ∗ )4 g − N RR − N W R g + (2 g − g (cid:88) pieces P of Σ ∗ he ( P ) (4.9) = he (cid:48) (Σ ∗ )4 g − N RR − N W R g + (2 g − N RR g − (2 g − g he (cid:48) (Σ ∗ )= − g (2 N RR + N W R ) − g − g he (cid:48) (Σ ∗ ) ≤ − g (2 N RR + N W R ) (4.3) = − g ( k + (cid:96) )4 g = − ( k + (cid:96) ) . This completes the proof of Proposition 4.8. (cid:3)
Proof of Theorem 1.2.
Assume γ ∈ [Γ g , Γ g ] is not the identity and that w ∈ [ F g , F g ] isa shortest element representing the conjugacy class of γ , hence also not the identity. ByCorollary 2.10 we have E g,n [tr γ ] = ζ (2 g − n ) − (cid:88) ( µ, ν ) ∈ ˜Ω D [ µ,ν ] ( n ) J n ( w, [ µ, ν ]) + O w,g (cid:18) n (cid:19) , where ˜Ω is a finite collection of pairs of Young diagrams. We know lim n →∞ ζ (2 g − n ) = 1from (2.11) and for each fixed ( µ, ν ), D [ µ,ν ] ( n ) J n ( w, [ µ, ν ]) = O w,µ,ν (1) by Theorem 3.1. Hence E g,n [tr γ ] = O γ (1) as n → ∞ as required. References [AB83] M. F. Atiyah and R. Bott. The Yang-Mills Equations over Riemann Surfaces.
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