Matrix Group Integrals, Surfaces, and Mapping Class Groups II: O(n) and Sp(n)
MMatrix Group Integrals, Surfaces, and Mapping Class Groups II:O ( n ) and Sp ( n ) Michael Magee and Doron Puder ∗ June 25, 2019
Abstract
Let w be a word in the free group on r generators. The expected value of the trace of theword in r independent Haar elements of O( n ) gives a function T r O w ( n ) of n . We show that T r O w ( n ) has a convergent Laurent expansion at n = ∞ involving maps on surfaces and L -Eulercharacteristics of mapping class groups associated to these maps. This can be compared toknown, by now classical, results for the GUE and GOE ensembles, and is similar to previousresults concerning U ( n ), yet with some surprising twists.A priori to our result, T r O w ( n ) does not change if w is replaced with α ( w ) where α is anautomorphism of the free group. One main feature of the Laurent expansion we obtain is thatits coefficients respect this symmetry under Aut( F r ).As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of T r O w ( n ) as n → ∞ , we generalize a formula of Frobenius and Schur, and we obtain a universalityresult on random orthogonal matrices sampled according to words in free groups, generalizing atheorem of Diaconis and Shahshahani.Our results are obtained more generally for a tuple of words w , . . . , w (cid:96) , leading to functions T r O w ,...,w (cid:96) . We also obtain all the analogous results for the compact symplectic groups Sp( n )through a rather mysterious duality formula. Contents α G . . . . . . . . . . . . . . . . . 81.4 Notation and paper organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Surfaces ∗ ( w , . . . , w (cid:96) ) . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Proof of Theorem 1.5 when Surfaces ∗ ( w , . . . , w (cid:96) ) is empty . . . . . . . . . . . . . 162.4 Incompressible and almost-incompressible maps . . . . . . . . . . . . . . . . . . . 17 T r Gw ,...,w (cid:96) ( n ) T r O w ,...,w (cid:96) . . . . . . . . . . . . . . . . . . . . . . . . 193.3 First Laurent series expansion at infinity . . . . . . . . . . . . . . . . . . . . . . . 203.4 Signed matchings and a new Laurent series expansion . . . . . . . . . . . . . . . 213.5 Proof of Corollary 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ∗ D. P. was supported by the Israel Science Foundation (grant No. 1071/16). a r X i v : . [ m a t h . G T ] J un The transverse map complex 25
MCG( f ) on the transverse map complex. 28 L -invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Proof of Theorem 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 The L -Euler characteristic is the usual one for almost-incompressible maps . . . 32 A Proof of Theorem 1.2: relationship between O and Sp Let F r be the free group on r generators with a fixed basis B = { x , . . . , x r } . For any word w ∈ F r and group H there is a word map w : H r → H defined by substitutions, for example, if r = 2 and w = x x − then w ( h , h ) = h h − . In thispaper we consider the case that H is a compact orthogonal or symplectic group. For n ∈ Z ≥ ,the orthogonal group O( n ) is the group of n × n real matrices A such that A T A = AA T = Id n .For n ∈ Z ≥ the symplectic group Sp( n ) is the group of 2 n × n complex matrices A such that A ∗ A = AA ∗ = Id n and AJ A T = J . Here A ∗ is the adjoint matrix A T and J is the matrix J def = (cid:18) n Id n − Id n n (cid:19) . (1.1)We fix words w , . . . , w (cid:96) ∈ F r . In this paper the main object of study is the following type of matrixintegral T r Gw ,...,w (cid:96) ( n ) def = (cid:90) G ( n ) r tr( w ( g , . . . , g r )) · · · tr( w (cid:96) ( g , . . . , g r )) dµ n ( g ) . . . dµ n ( g r )where G ( n ) is either the orthogonal group O( n ) or symplectic group Sp( n ), and µ n is the Haarprobability measure on the respective group. We write G = O or Sp respectively to refer to thetype of group being studied.These values, the expected value of the trace and product of traces, are natural objects ofstudy. First, the measure induced by a word w on O ( n ) or Sp ( n ) (see Remark 1.15 for the precisedefinition) is completely determined by these values when w , . . . , w (cid:96) are various (positive) powers of w . Second, these values are parallel to well-studied quantities in various models of random matricesand in free probability, and are related to questions on representation varieties and more. See theintroduction of [MP19b] for more details.Our paper is centered around writing the functions T r Gw ,...,w (cid:96) ( n ) as Laurent series in the variable( n + 1 − α G ) − with positive radii of convergence , where α O = 2 and α Sp = are called ‘Jackparameters’ in the literature [Nov17]. In § Of course, since ( n + 1 − α G ) − and n − are related by z (cid:55)→ z ( z +1 − α G ) , which fixes 0 and is a local biholomorphismthere, Laurent series in ( n + 1 − α G ) − give rise to Laurent series in n − .
2n fact, the functions T r Gw ,...,w (cid:96) ( n ) are not only meromorphic at ∞ , but actually given byrational functions of n , with integer coefficients, when n is sufficiently large. This is a reasonablystraightforward consequence of the Weingarten Calculus and is proved in Corollaries 3.5 and A.6below. See Table 1 for some examples.The ability to find a Laurent series at ∞ for T r Gw ,...,w (cid:96) ( n ) using diagrammatic expansions willnot be surprising to experts in Free Probability Theory and in particular, those who have workedwith the Weingarten Calculus. However, this is not the main point of this paper.
The integrals T r Gw ,...,w (cid:96) ( n ) have a priori symmetries under Aut( F r ), the automorphism group of F r , according tothe following lemma. Lemma 1.1. If α is an automorphism of F r , then T r Gw ,...,w (cid:96) ( n ) = T r Gα ( w ) ,...,α ( w (cid:96) ) ( n ) . This is proved in the case (cid:96) = 1 in [MP15, § (cid:96) can be proved usingthe same argument. In particular, if w ∈ F r then T r Gw ( n ) is a function of n that only depends on w up to automorphism; in other words, it reflects algebraic properties of w , not just combinatorialproperties. So a priori, any Laurent series expansion of T r Gw ,...,w (cid:96) ( n ) will involve Aut( F r )-invariantsof w , . . . , w (cid:96) and this brings us to the true goal of the paper: to find Laurent series expansions at ∞ for T r Gw ,...,w (cid:96) ( n ) in terms of Aut( F r ) -invariants of w , . . . , w (cid:96) . In fact, the expressions for G = O or Sp are closely related; we will prove Theorem 1.2.
There is N = N ( w , . . . , w (cid:96) ) ≥ such that when n ≥ N we have T r Sp w ,...,w (cid:96) ( n ) = ( − (cid:96) T r O w ,...,w (cid:96) ( − n ) . The quantity T r O w ,...,w (cid:96) ( − n ) is interpreted using the rational function form of T r O w ,...,w (cid:96) for n ≥ N . In particular, this identity relates the Laurent series at ∞ of T r Sp w ,...,w (cid:96) ( n ) and T r O w ,...,w (cid:96) ( n ) . This type of duality between O and Sp has been observed previously in various contexts, someof which we discuss in § Understanding the functions T r Gw ,...,w (cid:96) ( n ) involves studying maps from surfaces to the wedge of r circles, denoted by (cid:87) r S , as was also the case in [MP19b], where the corresponding theoremswere proved for the functions T r Uw ,...,w (cid:96) ( n ) that arise from compact unitary groups. One centraldifference that appears here is that for unitary groups, only orientable surfaces featured in thedescription of T r Uw ,...,w (cid:96) ( n ), whereas the description of T r Gw ,...,w (cid:96) ( n ) for G = O or Sp also involvesnon-orientable surfaces.We call the point in (cid:87) r S at which the circles are wedged together o . We fix an orientationof each circle that gives us an identification π ( (cid:87) r S , o ) ∼ = F r and we identify the generator x ∈ B with the loop that traverses the circle S x corresponding to x according to its given orientation. Definition 1.3 (Admissible maps) . Given w , . . . , w (cid:96) ∈ F r , consider pairs (Σ , f ) where • Σ is a compact surface, not necessarily connected, with (cid:96) ordered and oriented boundarycomponents δ , . . . , δ (cid:96) and a marked point v j on each δ j . Note the orientation of the j thboundary component specifies a generator [ δ j ] of π ( δ j , v j ). We also require that Σ has noclosed components. • f : Σ → (cid:87) r S is a continuous function such that f ( v j ) = o and f ∗ ([ δ j ]) = w j ∈ π ( (cid:87) r S , o ) ∼ = F r for all 1 ≤ j ≤ (cid:96) . We define N precisely in (3.2). w , . . . , w (cid:96) T r O w ,...,w (cid:96) ( n ) χ max Admissible maps with χ (Σ) = χ max x y n − P , , f ) with MCG ( f ) = { } x y n − P , , f ) with MCG ( f ) = { } [ x, y ] n + n − n − n ( n +2)( n − P , , f ) with MCG ( f ) = { } xy x − y − − P , , f ) with MCG ( f ) ∼ = Z xy x − y − n − P , , f ) with MCG ( f ) = { } xyx yx y n +2 n ( n +2)( n − − P , , f ) withMCG ( f ) = { } w, w for w = x y n + n +2 n +4 n ( n +2)( n − f ) = { } w, w for w = x y f ) = { } w, w, w for w = x y n +3 n − n +6 n +16)( n − n − n ( n +2)( n +4) − (cid:116) P , ) withMCG ( f ) = { } Table 1: Some examples of the rational expression for T r O w ,...,w (cid:96) ( n ). All these examples con-tain words in F with generators { x, y } . The notation [ x, y ] is for the commutator xyx − y − .We let χ max = χ max ( w , . . . , w (cid:96) ) denote the maximal Euler characteristic of a surface in Surfaces ∗ ( w , . . . , w (cid:96) ) – see Page 16. See Definitions 1.3 and 1.4 for the notions of admissiblemaps and MCG ( f ) appearing in the right column. In that column, P g,b denotes the non-orientablesurface of genus g with b boundary components (so χ ( P g ) = 2 − g − b ). We give more details inSection 6.3 and Table 2.We call such a pair an admissible map (for w , . . . , w (cid:96) ). Consider two admissible maps (Σ , f )and (Σ (cid:48) , f (cid:48) ) with boundary components δ , . . . , δ (cid:96) and δ (cid:48) , . . . , δ (cid:48) (cid:96) and marked points { v j } and { v (cid:48) j } respectively. We say (Σ , f ) and (Σ (cid:48) , f (cid:48) ) are equivalent and write (Σ , f ) ≈ (Σ (cid:48) , f (cid:48) ) if • there exists a homeomorphism F : Σ → Σ (cid:48) such that F ( v j ) = v (cid:48) j and F ∗ ([ δ j ]) = [ δ (cid:48) j ] for all1 ≤ j ≤ (cid:96) . Here F ∗ is the map induced on the relevant fundamental group. This conditionsays that F preserves the orientation of each boundary component. • there is a homotopy between f (cid:48) ◦ F and f on Σ. This homotopy is relative to the points { v j } .The set Surfaces ∗ ( w , . . . , w (cid:96) ) is defined to be the resulting collection of equivalence classes [(Σ , f )] Surfaces ∗ of admissible maps for w , . . . , w (cid:96) .Definition 1.3 modifies the definition of Surfaces ( w , . . . , w (cid:96) ) given in [MP19b, Def. 1.3], by drop-ping the requirement that Σ is orientable, and any compatibility between the orientations of bound-ary components of Σ. Hence Surfaces ( w , . . . , w (cid:96) ) ⊂ Surfaces ∗ ( w , . . . , w (cid:96) ). The set4igure 1.1: Schematic illustration of two admissible maps (Σ , f ) and (Σ , f ) when (cid:96) = 2 andthere are two words w , w . Base points v j are marked by diamonds, and an oriented boundarycomponent labeled by w j maps to w j under ( f i ) ∗ , for the relevant i = 1 , Surfaces ∗ ( w , . . . , w (cid:96) ) will index the summation in our Laurent series expansion of T r Ow ,...,w (cid:96) ( n ),but to understand the contribution of a given [(Σ , f )] we must take into account the internal sym-metries of this pair.If Σ is a compact surface, the mapping class group of Σ, denoted MCG(Σ), is the collection ofhomeomorphisms of Σ that fix the boundary pointwise, modulo homeomorphisms that are isotopicto the identity through homeomorphisms of this type. Note that for Σ fixed, MCG(Σ) acts onthe collection of homotopy classes [ f ] of f such that [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ). To take intoaccount the function f in our definition of symmetries, we make the following definition. Definition 1.4.
For [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ), we define MCG( f ) to be the stabilizer of [ f ] MCG( f )in MCG(Σ). This group is well-defined up to isomorphism.A certain integer invariant of MCG( f ), where [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ), appears in ourformula for T r Gw ,...,w (cid:96) ( n ). The invariant that appears is the L -Euler characteristic, denoted by χ (2) (MCG( f )) , and defined precisely in § . This is defined for a class of groups that we will provein § f ) where [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ). Moreover, for certain [(Σ , f )],including all those contributing to the ‘leading and second-leading order’ terms of the Laurent seriesexpansion of T r Gw ,...,w (cid:96) ( n ), we will prove in § χ (2) (MCG( f )) coincides with a much tamerinvariant: the Euler characteristic of a finite CW -complex that is an Eilenberg-Maclane space oftype K (MCG( f ) , , f )] is in Surfaces ∗ ( w , . . . , w (cid:96) ) we write χ (Σ) for the usual topological Euler characteristic of Σ. Clearlythis does not depend on the representative chosen for [(Σ , f )].5e can now state our main theorem. Theorem 1.5.
There is M = M ( w , . . . , w (cid:96) ) ≥ such that for n > M , T r O w ,...,w (cid:96) ( n ) is given bythe following absolutely convergent Laurent series in ( n − − : T r O w ,...,w (cid:96) ( n ) = (cid:88) [(Σ ,f )] ∈ Surfaces ∗ ( w ,...,w (cid:96) ) ( n − χ (Σ) χ (2) (MCG( f )) . This also gives a Laurent series expansion in ( n + ) − for T r Sp w ,...,w (cid:96) ( n ) in view of Theorem 1.2.Remark . For fixed (cid:96) , the possible values of χ (Σ) for [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) are boundedabove in terms of (cid:96) . Hence the powers of n − Remark . In the course of the proof of Theorem 1.5 we prove that for each fixed χ , there areonly finitely many [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) such that χ (Σ) = χ and χ (2) (MCG( f )) (cid:54) = 0. Wealso prove that each χ (2) (MCG( f )) ∈ Z , so the Laurent series has integer coefficients. Remark . Although Theorem 1.5 holds if some of the w i = 1, it simplifies the paper to assumethat all w i (cid:54) = 1, which we do from now on.Theorem 1.5 can be viewed as analogous to the genus expansions of GOE matrix integrals interms of Euler characteristics of mapping class groups obtained by Goulden-Jackson [GJ97] andGoulden, Harer, and Jackson [GHJ01]. These results extended previous results of Harer and Zagier[HZ86] on the GUE ensemble.The connection between O( n ) matrix integrals and not-necessarily-oriented surfaces was previ-ously pointed out by Mingo and Popa [MP13] and Redelmeier [Red15]. Theorem 1.5 has several important corollaries. Firstly, for a fixed word w ∈ F r the rate of decay of T r Gw ( n ) as n → ∞ can be bounded in terms of the square length of w and the commutator lengthof w . Definition 1.9.
The square length of w ∈ F r , denoted sql( w ), is the minimum number s such that w can be written as the product of s squares, or ∞ if it is not possible to write w as the productof squares. The commutator length of w , denoted cl( w ), is the minimum number g such that w can be written as the product of g commutators, or ∞ if w cannot be written as the product ofcommutators.One has the elementary inequality sql( w ) ≤ w ) + 1 . (1.2)This follows from the identities[ a, b ] = ( ab ) ( b − a − b ) ( b − ) , a [ b, c ] = ( a ba − ) ( ab − a − ca − ) ( ac − ) for any a, b, c ∈ F r . These identities also imply that when sql( w ) = ∞ , cl( w ) = ∞ . Corollary 1.10.
For G = O or Sp , we have T r Gw ( n ) = O (cid:16) n − min(sql( w ) , w )) (cid:17) as n → ∞ . We interpret the right hand side as when sql( w ) = ∞ . See (3.9) for the precise definition of M . emark . The inequality (1.2) implies that T r Gw ( n ) = O ( n − sql( w ) ) unless (1.2) is an equality.In that case, sql( w ) = 2cl( w ) + 1 and T r Gw ( n ) = O ( n − sql( w ) ). An analogous result holds for G = U the unitary group: T r U w ( n ) = O (cid:0) n − w ) (cid:1) [MP19b, Corollary 1.8]. Yet another analogousresult, albeit with quite a different flavor, holds in the case of the symmetric group. Let T r Sym w ( n )denote the trace of the ( n − n ). Then T r Sym w ( n ) = θ (cid:0) n − π ( w ) (cid:1) , where π ( w ) is the smallest rank of a subgroup of F r containing w as anon-free-generator and θ means there is equality up to positive multiplicative constants for largeenough n [PP15, Theorem 1.8]. See also [MP19a, Theorem 1.11] for similar results in the case ofgeneralized symmetric groups. Remark . Although we have stated Corollary 1.10 as a corollary of Theorem 1.5, the proof issignificantly simpler and does not require any discussion of L -invariants. We explain the proof ofCorollary 1.10 in § Remark . The consequence of Corollary 1.10 that T r Gw ( n ) = 0 when w cannot be written as theproduct of squares can also be proved directly from the definition of T r Gw ( n ), by Lemma 2.6 (seealso Lemma 2.3). Remark . When w = x x . . . x s ∈ F r , r ≥ s , a result of Frobenius and Schur [FS06] (for s = 1)and a straightforward generalization (see [MP19a, Section 2] or [PS14, Proposition 3.1(3)]) gives T r O w ( n ) = 1 n s − , T r Sp w ( n ) = ( − s (2 n ) s − . (1.3)In fact analogs of these formulas hold for any compact group G . Thus Theorem 1.5 and Corollary1.10 can be viewed as a generalization of these formulas to arbitrary words in F r . However, thisgeneralization is not as simple as it might appear on the surface. For example, when w = x x ,combining Theorem 1.5 with (1.3) gives for large nn − = (cid:88) [(Σ ,f )] ∈ Surfaces ∗ ( x x ) ( n − χ (Σ) χ (2) (MCG( f )) . Since n − does not agree with any Laurent polynomial of n − n , this implies that thereare infinitely many different χ such that there is [(Σ , f )] in Surfaces ∗ ( x x ) with χ (Σ) = χ and χ (2) (MCG( f )) (cid:54) = 0. We analyze these [(Σ , f )] in Example 6.2. Remark . In another direction, in [MP19a] we show using in part Corollary 1.10 that the word x . . . x s is uniquely determined, up to automorphisms, by the ‘word measures’ induced by the wordon compact groups, that is, the pushforwards of Haar measures on G r under the word map w . Corollary 1.16.
If all w , . . . , w (cid:96) are not equal to , the limit lim n →∞ T r O w ,...,w (cid:96) ( n ) exists, and is aninteger that counts the number of pairs [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) such that all the connectedcomponents of Σ are annuli or M¨obius bands.In algebraic terms, this integer is the weighted number of ways to partition w , . . . , w (cid:96) intosingletons and pairs, such that the word in every singleton is a square, and every pair { w, w (cid:48) } hasthe property that w (cid:48) is conjugate to either w or w − . The weight given to such a partition is (cid:89) { w,w (cid:48) } max { d ≥ w = u d } where { w, w (cid:48) } run over the pairs of the partition. We prove Corollary 1.16 in § orollary 1.17. Suppose that w (cid:54) = 1 and w = u d with d ≥ such that u ∈ F r is not a properpower of any other element of F r . Then for all (cid:96) ≥ and j , . . . , j (cid:96) ∈ Z , the limit lim n →∞ T r O w j ,...,w j(cid:96) ( n ) exists and only depends on d and the j k , not on u . Moreover, this collection of limits determines d .The same result holds with O replaced by Sp . The phenomenon observed in Corollary 1.17 is also known to be present for other families ofgroups including unitary groups U ( n ) [M´SS07, R˘ad06], symmetric groups S n [Nic94] (complementedin [LP10]), and GL n ( F q ) [PW]. See also [DP19, Theorem 1.3] for a similar phenomenon for Ginibreensembles. We will prove Corollary 1.17 in § r = 1, so that there is only one letter x = x , and w k = x j k for k = 1 , . . . , (cid:96), we have T r O x j ,...,x j(cid:96) ( n ) = (cid:90) O( n ) tr (cid:0) g j (cid:1) · · · tr (cid:0) g j (cid:96) (cid:1) dµ n ( g ) . Diaconis and Shahshahani [DS94, Thm. 4] prove that in this case, T r O x j ,...,x j(cid:96) ( n ) is exactly theinteger described in Corollary 1.16 for any n sufficiently large, and give a closed formula for thisinteger. Moreover in (ibid.) Diaconis and Shahshahani use this fact to prove that for j ∈ N , thecollection of random variables tr( g ) , tr (cid:0) g (cid:1) . . . , tr (cid:0) g j (cid:1) , where g is chosen according to Haar measureon O( n ), converge in probability as n → ∞ to independent normal variables with different centersand variances. Using the method of moments, Corollary 1.17 implies that one has the same resultif x is replaced by any non-trivial word w that is not a proper power. More precisely, we have thefollowing result. Corollary 1.18 (Universality for traces of non-powers) . Let w ∈ F r , w (cid:54) = 1 , and w not aproper power of another element in F r . For fixed (cid:96) ≥ , consider the real-valued random vari-ables T n (cid:0) w j (cid:1) def = tr (cid:0) w ( g , . . . , g r ) j (cid:1) on the probability space (O( n ) r , µ rn ) for j = 1 , . . . , (cid:96) . The T n (cid:0) w j (cid:1) converge in probability as n → ∞ : (cid:16) T n ( w ) , T n (cid:0) w (cid:1) , . . . , T n (cid:16) w (cid:96) (cid:17)(cid:17) n →∞ −−−−−−→ probability ( Z , Z , . . . , Z (cid:96) ) , where Z , . . . , Z (cid:96) are independent real normal random variables, such that when j is odd, Z j hasmean 0 and variance j , and when j is even, Z j has mean and variance j + 1 . Some related results were previously obtained by Mingo and Popa in [MP13] where they obtain‘real second order freeness’ of independent Haar elements of O( n ). This concept does not seem toimply the explicit statement of Corollary 1.18. Sp and O , and the parameters α G Duality between Sp and OThe formula that appears in Theorem 1.2 was stated by Deligne [Del16] in a private communica-tion to the second named author of this paper, without the explicit calculation of the sign ( − (cid:96) .Deligne’s reasoning is that one has the identities of ‘supergroups’O( − n ) = O(0 | n ) = Sp( n ) . We do not know how to make this into a rigorous concise proof of Theorem 1.2 at the moment. Theproof we give in the Appendix relies on a technical combinatorial comparison of the terms arisingin T r O w ,...,w (cid:96) and T r Sp w ,...,w (cid:96) from the Weingarten Calculus.8ormulas similar to Theorem 1.2 were observed by Mkrtchyan [Mkr81] in the early 1980s in thesetting of O vs Sp gauge theory. The duality also shows up as a duality between the GOE and GSEensembles [MW03]. The introduction to (ibid.) also contains an overview of what was known aboutthe O-Sp duality at that time. The O( − n ) = Sp( n ) formula has more recently been interpreted ina different way, in terms of Casimir operators, by Mkrtchyan and Veselov in [MV11]. The parameters α G In this section we mention other occurrences of the parameters α G in random matrix theory. Wehave observed in [MP19b] and the current paper that it is most natural to expand T r Gw ,...,w (cid:96) ( n ) asa Laurent series in ( n + 1 − α G ) − , where α G = 2 , , for G = O , U , Sp respectively. It is morecommon in the literature that the parameter β G def = 2 α G appears. Hence β G = 1 , , G = O , U , Sp respectively. One sees that 1 , , β G .One historical role that these parameters play in random matrix theory is that they give unifiedexpressions for the joint eigenvalue distributions of Dyson’s Orthogonal, Unitary, and SymplecticCircular Ensembles (COE/CUE/CSE) introduced by Dyson in [Dys62]. Results of Weyl [Wey39](for the CUE) and Dyson [Dys62, Theorem 8] (for the COE and CSE) say that the joint eigenvaluedensity of a matrix in one of these ensembles is given by C β (cid:89) k (cid:54) = l | exp( iθ k ) − exp( iθ l ) | β where exp( iθ k ) are the eigenvalues of the random matrix, C β is a normalizing constant, and β = 1for COE, β = 2 for CUE, and β = 4 for CSE.More recently, and in a context more closely related to the current paper, the parameters α G appear in the work of Novaes [Nov17] who calculates Laurent series in ( n +1 − α G ) − for Weingartenfunctions on G ( n ) with G = U , O , Sp. The origin of the shifted parameter in ( loc. cit. ) is its useof the following result of Forrester [For06, eq. 3.10]: if B is sampled from U( n ) , O( n ) , or Sp( n )according to Haar measure, then the density function of the top left m × m submatrix A of g isgiven by a determinant involving A raised to an exponent that is an explicit function of α G . Thisis very different to the appearance of α G in the current paper.Indeed, the origin of α G in the current paper is topological and based on the following observa-tions: • There are exactly two types of connected surfaces with boundary that have trivial mappingclass group: a disc, and a M¨obius band (cf. Lemma 5.6 and Proposition 5.8). • As a result, when we compute the terms χ (2) (MCG( f )) that appear in Theorem 1.5, we needto enumerate surfaces that are formed by gluing together discs and M¨obius bands. • On the other hand, using the Weingarten calculus to expand T r O w ,...,w (cid:96) ( n ) leads to a formulathat involves enumerating surfaces that are formed only by gluing discs (i.e., given as CW -complexes). This formula is given in Proposition 3.6. The COE of dimension n is the space of symmetric unitary matrices. This can be identified with U( n ) / O( n ) andas such, has a natural probability measure coming from Haar measure on U( n ). The CUE of dimension n is U( n )with its Haar measure. The CSE of dimension 2 n is the space of self-dual unitary matrices, that can be identifiedwith U(2 n ) / Sp( n ) and hence given the probability measure coming from Haar measure on U(2 n ). CW -complexes toa formula involving surfaces formed by gluing together discs and M¨obius bands. This is somewhatsurprisingly accomplished simply by replacing the parameter n by n − G = O, and isgiven by Proposition 3.13. For n ∈ N , we use the notation [ n ] for the set { , , . . . , n } . If f is a map between topological spaceswith base points, then f ∗ is the induced map between the fundamental groups of the spaces. Wewrite ∅ for the empty set. If w ∈ F r we write | w | for the word length of w in reduced form. Wewrite log for the natural logarithm (base e ).The paper is organized as follows. Section 2 describes how one can construct admissible mapsin Surfaces ∗ ( w , . . . , w (cid:96) ) from sets of matchings of the letters of w , . . . , w (cid:96) , and gives some basicdefinitions and facts about general elements of Surfaces ∗ ( w , . . . , w (cid:96) ). In Section 3 we discuss theWeingarten calculus, give a combinatorial formula for T r O w ,...,w (cid:96) ( n ) (Theorem 3.4), derive twodifferent Laurent series expansions of T r O w ,...,w (cid:96) ( n ) (Propositions 3.6 and 3.13), and reduce ourmain theorem, Theorem 1.5, to a theorem about a single admissible map in Surfaces ∗ ( w , . . . , w (cid:96) )(Theorem 3.16). Section 4 introduces the complex of transverse maps associated with some [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) and proves it is contractible, following closely with the analogous result in[MP19b]. In Section 5 we finish the proof of Theorems 3.16 and 1.5, and in Section 6 we proveCorollaries 1.16, 1.17 and 1.18, and discuss some concrete examples. Finally, Appendix A gives acombinatorial proof of Theorem 1.2. Acknowledgements
We thank Pierre Deligne for the formulation of Theorem 1.2. We also thank Marcel Novaes andSasha Veselov for helpful discussions related to this work.
In the rest of this paper, we view w , . . . , w (cid:96) ∈ F r as fixed. For a given word w k ∈ F r we may write w k = x ε k j k x ε k j k . . . x ε k | wk | j k | wk | , ε ku ∈ {± } , j ku ∈ [ r ] , (2.1)where if j ku = j ku +1 , then ε ku = ε ku +1 . In other words, we write each w k in reduced form. Recall that B is the basis x , . . . , x r . We define the total unsigned exponent of a generator x j ∈ B in w , . . . , w (cid:96) totalunsignedexponentto be (cid:96) (cid:88) k =1 { ≤ u ≤ | w k | : j ku = j } . In this section we assume that the total unsigned exponent of x in w , . . . , w (cid:96) is even for each x ∈ B ,and write 2 L x for this quantity. L x A matching of the set [2 k ] = { , . . . , k } is a partition of [2 k ] into pairs. The collection of matchingsof [2 k ] is denoted by M k . We use two ways to identify a matching in M k with a permutation in M k k . In the first we identify a matching m with a permutation whose cycle decomposition consistsof disjoint transpositions given by the matched pairs of m . We call the resulting permutation π m . π m In the second, given a matching m ∈ M k we canonically view m as an ordered list of orderedpairs (( m (1) , m (2) ) , ( m (3) , m (4) ) , . . . , ( m (2 k − , m (2 k ) )) with m (1) < m (3) < . . . < m (2 k − , m (2 r − < m (2 r ) , r ∈ [ k ] . (2.2)As such, we have an embedding M k → S k , m (cid:55)→ σ m by sending the matching m to the permutation σ m : i (cid:55)→ m ( i ) . σ m Following Collins and ´Sniady [C´S06] we introduce a metric on M k as follows. For a permutation σ , write | σ | for the minimum number of transpositions that σ can be written as a product of. Formatchings m, m (cid:48) ∈ M k , we define ρρ ( m, m (cid:48) ) = | π m π m (cid:48) | . (2.3)Since both permutations π m and π m (cid:48) have the same sign, ρ ( m, m (cid:48) ) ∈ Z ≥ . (cid:87) r S We now describe certain markings of (cid:87) r S that will be used in our construction of maps on surfaces.For any given tuple of positive integers { κ x } x ∈ B we will mark additional points on the circles of (cid:87) r S as follows. On the circle corresponding to the each generator x ∈ B we mark κ x + 1 distinctordered points ( x, , . . . , ( x, κ x ), placed along the circle according to its orientation, and disjointfrom o . This is illustrated in the bottom part of Figure 2.1 for the case B = { x, y } and κ x = κ y = 1. Recall our ongoing assumptions that all words are (cid:54) = 1. For each word w (cid:54) = 1 we construct anoriented marked circle C ( w ). Write w = x ε j x ε j . . . x ε | w | j | w | (2.4)in reduced form (as in (2.1)). Begin with | w | disjoint copies of [0 , u th copy by[0 , u . Give each interval the orientation from 0 to 1. On each interval choose arbitrarily a map γ u : [0 , u → (cid:87) r S such that γ u (0) = γ u (1) = o , γ u : (0 , u → S x ju − { o } is a diffeomorphism, and the loop in (cid:87) r S parameterized by γ u , based at o , corresponds to x ε u j u ∈ F r at the level of the fundamental group.Now cyclically concatenate all the intervals and maps together to obtain a circle C ( w ) and a map γ w : C ( w ) → (cid:87) r S .Let v w be the initial point 0 ∈ [0 , of this circle C ( w ). The map γ w has the property that γ w ( v w ) = o and ( γ w ) ∗ maps a generator of π ( C ( w ) , v w ) to w ∈ F r . We give C ( w ) the orientationsuch that the order of the intervals read, beginning at v w and following the orientation, matchesthe left to right order of (2.4). As such, the intervals of C ( w ) are in one-to-one correspondence withthe letters of w .To clarify and summarize, by definition w ∈ F r = π ( o, (cid:87) r S ) corresponds to a homotopyclass of a loop based at o . What we have done here is pick a particular representative, γ w , of thishomotopy class such that the sequence of circles traversed by the loop is prescribed by w through(2.4), and each traversal of a circle is monotone.11 .1.4 Construction of a map on a surface from words and matchings In this section we will describe a construction that takes in the following
Input. • A tuple κ = { κ x } x ∈ B of non-negative integers. • A collection m = { ( m x, , . . . , m x,κ x ) } x ∈ B that contains for each generator x ∈ B , an ordered( κ x + 1)-tuple ( m x, , . . . , m x,κ x ) of elements in M L x , where 2 L x is the unsigned exponent of x in w , . . . , w (cid:96) . We denote by MATCH κ = MATCH κ ( w , . . . , w (cid:96) ) the collection of all possiblesuch m , for fixed κ . We write MATCH ∗ = ∪ κ ∈ Z B ≥ MATCH κ . If m ∈ MATCH κ we will say κ ( m ) = κ . Warning: our notation MATCH in this paper is not exactly the same as in[MP19b] a . a In [MP19b] it was assumed that the matchings m x,i were subordinate to a partition of [2 L x ] into two blocks,corresponding to the occurrences of x +1 and x − in w , . . . , w (cid:96) , whereas here our matchings are arbitrary matchingsof [2 L x ]. The output of our construction is the following
Output. • A surface Σ m with (cid:96) oriented boundary components C ( w ) , . . . , C ( w (cid:96) ), with a given CW -complex structure, and a marked point v k on each boundary component C ( w k ). • A continuous function f m : Σ m → (cid:87) r S such that f m | C ( w k ) is the function γ w k constructedin § f m ( v j ) = o for each 1 ≤ i ≤ (cid:96) and ( f m ) ∗ maps the generator of π ( C ( w k ) , v k ) specified by the orientation of C ( w k ) to w k .Note that the resulting (Σ m , f m ) is an admissible map, as in Definition 1.3, for w , . . . , w (cid:96) .The construction is essentially the same as in [MP19b, § § I. The one-skeleton.
We first perform the construction of § w k . Thisgives us a collection of oriented based circles ( C ( w k ) , v w k ) that are respectively subdivided into | w k | intervals, and maps γ w k : C ( w k ) → (cid:87) r S . These oriented circles will form the oriented boundary δ Σ m def = (cid:96) (cid:91) k =1 C ( w k )of Σ m and the map γ def = ∪ (cid:96)k =1 γ w k : δ Σ m → (cid:87) r S will be the restriction of f m to the boundary of Σ m . We write v k = v w k in the sequel for the markedpoints on the C ( w k ).For each x ∈ B , the preimage γ − ( S x − { o } )is a union of 2 L x open subintervals. This collection of intervals is denoted by I x = I x ( w , . . . , w (cid:96) ). We identify I x with [2 L x ] in some arbitrary but fixed way. Moreover, each I x element of I x corresponds to a letter of a unique w k .Next we add to the one-skeleton of Σ m by gluing some arcs by their endpoints to δ Σ m . We willcall these arcs matching arcs. For each x ∈ B we mark points ( x, , . . . , ( x, κ x ) on S x as in § γ maps each interval I ∈ I x monotonically to S x , there are uniquely specified distinct points p I (0) , . . . , p I ( κ x ) in I such that γ ( p I ( k )) = ( x, k ) for 0 ≤ k ≤ κ x . p I ( k )Now, for every pair of intervals I and J that are matched by m x,k , glue a matching arc to δ Σ m with endpoints at p I ( k ) and p J ( k ). Carrying out this process for all generators x , now every p I ( k )point in δ Σ m is the endpoint of a unique arc. Call the resulting one dimensional CW -complexΣ (1) m . It consists of δ Σ m together with the matching-arcs described in the current paragraph. Moreprecisely, we consider Σ (1) m to have the following 0 and 1-cells: • The 0-cells (vertices) are just the points p I ( k ). Since there are 2 L x intervals in I x labeled by x and κ x + 1 points p I ( k ) in each such interval, there are 2 (cid:80) x ∈ B L x ( κ x + 1) vertices in the0-skeleton. • The 1-cells (edges) are of two different types. Firstly, there is an edge in δ Σ m for each con-nected component of δ Σ m −{ p I ( k ) } . Hence there are 2 (cid:80) x ∈ B L x ( κ x +1) such edges. Secondly,there is an edge for each matching-arc. Since the matching arcs give a matching of the points p I ( k ), there are (cid:80) x ∈ B L x ( κ x + 1) such edges. Hence in total there are 3 (cid:80) x ∈ B L x ( κ x + 1)edges.Additionally, each component C ( w k ) of δ Σ m contains a marked point v k . These are not part ofthe CW -complex structure. We define f (1) m on Σ (1) m by f (1) m | δ Σ m = γ and by requiring that f (1) m isconstant on all matching arcs. This completely specifies f (1) m , since the endpoints of matching arcsare in δ Σ m , so the value of f (1) m on matching arcs is specified by γ , and by construction of thematching arcs, the two endpoints of any two matching arcs have the same value under γ . II. The two-skeleton.
Next we complete the construction of Σ m by gluing in discs that willbe the 2-cells of the CW -complex. We glue in different types of discs as follows. Type-( x, k ) discs. For fixed x , if 0 ≤ k < κ x , let R x,k be the open interval in S x − { ( x, i ) } − { o } that abuts the points ( x, k ) and ( x, k + 1). Let R x,k be the closure of this component. The preimage( f (1) m ) − ( R x,k ) is a collection of disjoint cycles in Σ (1) m and for each of these cycles we glue theboundary of a disc simply along the cycle. These discs are called type- ( x, k ) discs.Type-o discs. Let R o be the star-like connected component of (cid:87) r S − { ( x, i ) } that containsthe point o and abuts points of the form ( x,
0) and ( x, κ x ) (for all x ). Let R o be the closure ofthis component. The preimage ( f (1) m ) − ( R o ) is a one-dimensional subcomplex in Σ (1) m . We considersimple cycles c in ( f (1) m ) − ( R o ) with the property that f (1) m | c never traverses a point ( x,
0) or ( x, κ x ).Namely, when going along c , whenever f (1) m ( c ) reaches some ( x,
0) or ( x, κ x ), it then stays at thispoint for a while and then leaves in a backtracking move. For such a cycle we glue the boundary ofa disc simply along the cycle. We call the disc a type-o disc. The resulting (unoriented) surface obtained by gluing in these discs is Σ m . Its CW-complexstructure is the CW-complex we described for Σ (1) m together with the glued discs as two-cells. Note that each disc D of Σ m has its boundary mapped to a nullhomotopic curve in (cid:87) r S ; fortype-( x, k ) discs this is because the boundary is mapped into an interval, and for type- o discs this isdue to our condition that f (1) m never traverses a point ( x, k ) when restricted to the boundary of thedisc. Hence we can extend f (1) m to a continuous function from the disc to R o or R x,k respectively,such that the only points in the preimage of { ( x, k ) } are in Σ (1) m . In other words, the extendedfunction maps the interior of the disc to either R o or some R x,k . We pick such an extension for eachdisc and this defines f m on all of Σ m . Note that the extension at every disk, and therefore f m ingeneral, is unique up to homotopy. This concludes the construction of the pair (Σ m , f m ) . m , f m ) in the case that r = 2, (cid:96) = 1, and w = x y . We write x, y for the generators of F . Here κ x = κ y = 1, so there are two matchings per generator. The topleft picture shows the matching arcs attached to C ( w ). The top right picture shows the constructedΣ m . The dotted line in both top pictures follows along the matching arcs to show how the singletype- o disc should be glued to Σ (1) m . We have called the boundary intervals I, J, K, H . The colors ofthe points marked in ∨ S in the bottom picture match with the matching arcs in their preimageunder f m . Σ m In this section we calculate the Euler characteristic of Σ m . Recall the definition of ρ from (2.3) andthat m ∈ MATCH κ ( w , . . . , w (cid:96) ) is a collection of matchings. Lemma 2.1.
We have χ (Σ m ) = − (cid:88) x ∈ B L x + { type- o discs of Σ m } − (cid:88) x ∈ B, ≤ k<κ x ρ ( m x,k , m x,k +1 ) . (2.5) Proof.
We have χ (Σ m ) = V − E + F = 2 (cid:88) x ∈ B L x ( κ x + 1) − (cid:88) x ∈ B L x ( κ x + 1) + { type- o discs of Σ m } + (cid:88) x ∈ B, ≤ k<κ x { type-( x, k ) discs of Σ m } = − (cid:88) x ∈ B L x ( κ x + 1) + { type- o discs of Σ m } + (cid:88) x ∈ B, ≤ k<κ x { type-( x, k ) discs of Σ m } . (2.6)14ecall from § m ∈ M k a permutation π m in S k all of whose cycleshave length 2. We now make the observation that every type-( x, k ) disc corresponds to exactlytwo cycles of π m x,k π m x,k +1 , and every cycle of π m x,k π m x,k +1 corresponds a to unique type-( x, k ) disc.Since for any σ ∈ S L x we have | σ | = 2 L x − { cycles of σ } , we have ρ ( m x,k , m x,k +1 ) = (cid:12)(cid:12) π m x,k π m x,k +1 (cid:12)(cid:12) L x − { cycles of π m x,k π m x,k +1 } L x − { type-( x, k ) discs of Σ m } . Therefore { type-( x, k ) discs of Σ m } = L x − ρ ( m x,k , m x,k +1 ). Hence from (2.6) χ (Σ m ) = − (cid:88) x ∈ B L x ( κ x + 1) + { type- o discs of Σ m } + (cid:88) x ∈ B, ≤ k<κ x L x − ρ ( m x,k , m x,k +1 )= − (cid:88) x ∈ B L x + { type- o discs of Σ m } − (cid:88) x ∈ B, ≤ k<κ x ρ ( m x,k , m x,k +1 ) . (2.7) Surfaces ∗ ( w , . . . , w (cid:96) ) The passage between the algebraic and topological view on
Surfaces ∗ ( w , . . . , w (cid:96) ) stems from thefollowing result of Culler [Cul81, 1.1]: Lemma 2.2 (Culler) . An element w ∈ F r is a product of g commutators (resp. s squares) if andonly if there exists an admissible map (Σ , f ) for w with Σ a genus g orientable surface (resp. aconnected sum of s copies of R P ) with a disc removed. We first describe when
Surfaces ∗ ( w , . . . , w (cid:96) ) is empty. Lemma 2.3.
The following are equivalent:1.
Surfaces ∗ ( w , . . . , w (cid:96) ) is non-empty.2. The unsigned exponent of each x ∈ B in w , . . . , w (cid:96) is even.3. w w · · · w (cid:96) can be written as a product of squares in F r .Proof. This is very close to standard facts, in particular the results appearing in Culler [Cul81], butwe give a proof here for completeness.We first prove that the first statement implies the second. Consider the map h : π ( (cid:87) r S , o ) → H ( (cid:87) r S , Z / Z )which induces a map H : F r → ( Z / Z ) | B | . Then H maps x i ∈ B to the corresponding stan-dard generator e i of ( Z / Z ) | B | . If [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) and δ , . . . , δ (cid:96) are the boundarycomponents of Σ then we have 0 = (cid:96) (cid:88) i =1 [ δ i ] ∈ H (Σ , Z / Z ) . Applying the map induced by f on homology to this equation implies (cid:80) (cid:96)i =1 H ( w i ) = 0, which meansthe unsigned exponent of each x ∈ B in w , . . . , w (cid:96) is even.Conversely, if the unsigned exponent of each x ∈ B in w , . . . , w (cid:96) is even, then the construction of § Surfaces ∗ ( w , . . . , w (cid:96) ) is non-empty. This proves that the second statement impliesthe first. 15he second statement is easily seen to be equivalent to the statement that the unsigned ex-ponent of each x ∈ B in w w . . . w (cid:96) is even. By what we have proved, this is equivalent to Surfaces ∗ ( w w . . . w (cid:96) ) (cid:54) = ∅ . Then by Lemma 2.2, this is equivalent to w w . . . w (cid:96) being either theproduct of commutators or the product of squares. But since any commutator is a product ofsquares (see § w , . . . , w (cid:96) , let χ max χ max ( w , . . . , w (cid:96) ) def = max { χ : [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) , χ (Σ) = χ } , or −∞ if Surfaces ∗ ( w , . . . , w (cid:96) ) is empty. Note that if Surfaces ∗ ( w , . . . , w (cid:96) ) is non-empty the max-imum clearly exists. Indeed, any connected component of Σ with [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) )must contain a boundary component, so Σ has at most (cid:96) components. Moreover, any connectedcomponent of Σ has Euler characteristic at most 1, so χ (Σ) ≤ (cid:96) . Lemma 2.4.
If all w j (cid:54) = 1 , then χ max ( w , . . . , w (cid:96) ) ≤ . Moreover, [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) with χ (Σ) = 0 if and only if all the connected components of Σ are annuli or M¨obius bands.Proof. As mentioned above, if [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) then writing Σ = ∪ mk =1 Σ k as a unionof connected components, we have m ≤ (cid:96) , each Σ k has at least one boundary component, and χ (Σ k ) ≤
1. However, χ (Σ k ) is equal to 1 only if Σ k is a disc, which can only happen if the w j thatmarks the boundary component of the disc is w j = 1. This is because the boundary component ofthe disc is nullhomotopic in the disc. So given all w j (cid:54) = 1, we have χ (Σ k ) ≤
0. Given χ (Σ) = 0, thisimplies χ (Σ k ) = 0 for 1 ≤ k ≤ m . Finally, by the classification of surfaces, we have χ (Σ k ) = 0 onlyif Σ k is an annulus or a M¨obius band.Conversely, if all the connected components of Σ are annuli or M¨obius bands then obviously χ (Σ) = 0. Lemma 2.5. If w ∈ F r then χ max ( w ) = 1 − min(sql( w ) , w )) .Proof. Since any [(Σ , f )] ∈ Surfaces ∗ ( w ) has Σ connected, Culler’s Lemma 2.2 is easily seen toimply the result, since the Euler characteristic of a genus g orientable surface with one boundarycomponent is 1 − g and the Euler characteristic of a connected sum of s copies of R P with a discremoved is 1 − s . Surfaces ∗ ( w , . . . , w (cid:96) ) is empty Lemma 2.6.
If the total unsigned exponent of some x ∈ B in w , . . . , w (cid:96) is not even, then T r Gw ,...,w (cid:96) ( n ) = 0 for any n ≥ and for G = O , Sp .Proof. Suppose for example that x = x . Let u be the (odd) total unsigned exponent of x in w , . . . , w (cid:96) . We give the proof for G = O. Note that minus the identity, − I n , is in the center ofO( n ). We have T r O w ,...,w (cid:96) ( n ) = (cid:90) O( n ) r tr( w ( g , . . . , g r )) . . . tr( w (cid:96) ( g , . . . , g r )) dµ n ( g ) . . . dµ n ( g r )= (cid:90) O( n ) r tr( w ( − I n g , . . . , g r )) . . . tr( w (cid:96) ( − I n g , . . . , g r )) dµ n ( g ) . . . dµ n ( g r )= ( − u (cid:90) O( n ) r tr( w ( g , . . . , g r )) . . . tr( w (cid:96) ( g , . . . , g r )) dµ n ( g ) . . . dµ n ( g r )= − (cid:90) O( n ) r tr( w ( g , . . . , g r )) . . . tr( w (cid:96) ( g , . . . , g r )) dµ n ( g ) . . . dµ n ( g r )= −T r O w ,...,w (cid:96) ( n ) . The proof for G = Sp is the same, using that − I n ∈ Sp( n ).16 roof of Theorem 1.5 when Surfaces ∗ ( w , . . . , w (cid:96) ) is empty. If Surfaces ∗ ( w , . . . , w (cid:96) ) is empty, thenby Lemma 2.3, the total unsigned exponent of some x ∈ B in w , . . . , w (cid:96) is not even. Therefore byLemma 2.6, T r O w ,...,w (cid:96) ( n ) = 0 for any n ≥ Certain types of admissible maps (Σ , f ) will pay a special role later in the paper (see § incompressible and almost-incompressible maps. Definition 2.7.
Let (Σ , f ) be an admissible map. We say that (Σ , f ) is compressible if there isan essential simple closed curve c ⊂ Σ such that f ( c ) is nullhomotopic in (cid:87) r S . We call c a compressing curve. We say (Σ , f ) is incompressible if it is not compressible. If every compressingcurve is non-generic in the sense of Definition 5.7 below, namely, if every essential compressing curveis either one-sided or bounds a M¨obius band, we say that (Σ , f ) is almost-incompressible .It is clear that if (Σ , f ) is (almost-) incompressible and (Σ (cid:48) , f (cid:48) ) ≈ (Σ , f ) then (Σ (cid:48) , f (cid:48) ) is also(almost-) incompressible. Therefore there is a well-defined notion of an element[(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) being (almost-) incompressible. Lemma 2.8.
Let [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) . If χ (Σ) = χ max ( w , . . . , w (cid:96) ) then [(Σ , f )] isincompressible, and if χ (Σ) = χ max ( w , . . . , w (cid:96) ) − then [(Σ , f )] is almost-incompressible.Proof. Denote χ max def = χ max ( w , . . . , w (cid:96) ) and assume that χ (Σ) ≥ χ max −
1. If (Σ , f ) is compress-ible, let c ⊂ Σ be a compressing curve. We can cut Σ along c to obtain a new surface Σ cut witheither b = 1 or b = 2 new boundary components. Then we can cap discs on any new boundary com-ponents of Σ cut to obtain a new surface Σ (cid:48) . Moreover, since f ( c ) was nullhomotopic, it is possibleto extend f across these discs to obtain a new pair (Σ (cid:48) , f (cid:48) ) with χ (Σ (cid:48) ) = χ (Σ) + b .If b = 1, then (Σ (cid:48) , f (cid:48) ) is admissible, since we must have cut along a one-sided curve and hence didnot create any new connected components. Since χ (Σ (cid:48) ) = χ (Σ) + 1, we must have χ (Σ) = χ max − b = 2 and every connected component of Σ (cid:48) has boundary, then the pair (Σ (cid:48) , f (cid:48) ) is admissibleand χ (Σ (cid:48) ) = χ (Σ) + 2, in contradiction to χ (Σ) ≥ χ max −
1. However, in Σ (cid:48) we might have createdat most one closed component S , in which case the pair (Σ (cid:48) , f (cid:48) ) is not admissible. In this case wecan simply delete S to obtain an admissible map (Σ (cid:48)(cid:48) , f (cid:48)(cid:48) ). Since c was essential, S is not a sphere,hence χ ( S ) ≤ χ (Σ (cid:48)(cid:48) ) = χ (Σ (cid:48) ) − χ ( S ) ≥ χ (Σ (cid:48) ) − χ (Σ) + 1 and therefore χ (Σ) ≤ χ max − c was generic, S is also not R P , and then χ ( S ) ≤ χ (Σ) ≤ χ max −
2, acontradiction. T r Gw ,...,w (cid:96) ( n ) Since we have proved Theorem 1.5 when
Surfaces ∗ ( w , . . . , w (cid:96) ) = ∅ , we assume the contrary for therest of the paper, and hence in view of Lemma 2.3 we assume that the total unsigned exponent ofeach x in w , . . . , w (cid:96) is even. We write 2 L x for the total unsigned exponent of x in w , . . . , w (cid:96) . L x Throughout the paper (cid:104)• , •(cid:105) = (cid:104)• , •(cid:105) O will denote the standard Hermitian form on C n . We willwrite (cid:104)• , •(cid:105) Sp for the symplectic form on C n given by (cid:104) v, w (cid:105) Sp def = v T J w.
We let { e i } be the standard basis of C n or C n . The Weingarten calculus allows one to interpret e i Essential means not homotopic to a point. G ( n ), for G = O , Sp, in terms of matchings. Thesource of the Weingarten calculus is Brauer-Schur-Weyl duality between G and an appropriateBrauer algebra. Let Z ( n ) denote the ring of rational functions of n with coefficients in Z . The following Theoremwas proved for G = O by Collins and ´Sniady in [C´S06, Cor. 3.4 ]. The corresponding theorem for G = Sp had its proof outlined by Collins and ´Sniady in [C´S06, Thm. 4.1 and following discussion].The theorem was subsequently precisely stated and proved by Collins and Stolz in [CS08, Prop.3.2]; see also Matsumoto [Mat13, Thm. 2.4]. Theorem 3.1.
For G = O , Sp , there are unique, computable, functions Wg Gk : M k × M k → Z ( n ) with the following properties. For m , m ∈ M k let us write Wg Gk ( m , m ; n ) def = Wg Gk ( m , m ) | n forthe evaluation of the rational function at n . Assume that n ≥ k if G = O and n ≥ k if G = Sp .We have (cid:90) G ( n ) g i j . . . g i k j k dµ n ( g ) = (cid:88) m ,m ∈ M k δ G i ,m δ G j ,m Wg Gk ( m , m ; n ) . Here i = ( i , . . . , i k ) , j = ( j , . . . , j k ) , and δ G i ,m def = k (cid:89) r =1 (cid:104) e i m (2 r − , e i m (2 r ) (cid:105) G where m ( j ) are as in (2.2). The integral of the product of an odd number of matrix coefficients is . The function Wg Gk is called the Weingarten function of G . Although here we take Theorem3.1 as the definition of the Weingarten functions, it is certainly worth pointing out that the papers[C´S06, Mat13] contain explicit formulas for the Weingarten functions. It follows from these formulasthat for any m , m ∈ M k , the rational function Wg Gk ( m , m ) can be computed by an (explicit)algorithm in a finite number of steps.The following lemma of Matsumoto [Mat13, § Lemma 3.2.
For n ∈ Z ≥ with n ≥ k and m , m ∈ M k Wg Sp k ( m , m ; n ) = ( − k sign( σ m σ − m )Wg O k ( m , m ; − n ) . The following theorem of Collins and ´Sniady gives the full Laurent expansion for Wg O k ( m , m )at n = ∞ and estimates the order of vanishing. In view of Lemma 3.2, one has analogous resultsfor G = Sp. Theorem 3.3.
Fix m , m ∈ M k .1. [C´S06, Lemma 3.12] We have Wg O k ( m , m ; n ) = n − k (cid:88) l ≥ (cid:88) m = m (cid:48) ,m (cid:48) ,...,m (cid:48) l = m ∈ M k m (cid:48) i (cid:54) = m (cid:48) i +1 ( − l n − ρ ( m (cid:48) ,m (cid:48) ) − ... − ρ ( m (cid:48) l − ,m (cid:48) l ) . The sum is absolutely convergent for n ≥ k .2. [C´S06, Thm. 3.13] Wg O k ( m , m ; n ) = O n →∞ ( n − k − ρ ( m ,m ) ) . .2 A rational function form of T r O w ,...,w (cid:96) We wish to find a rational function of n which agrees with the integral T r O w ,...,w (cid:96) ( n ) = (cid:90) O( n ) r tr( w ( g , . . . , g r )) . . . tr( w (cid:96) ( g , . . . , g r )) dµ n ( g ) . . . dµ n ( g r ) (3.1)for sufficiently large n , depending only on w , . . . , w (cid:96) . We denote by κ ≡ x ∈ B . We now define NN = N ( w , . . . , w (cid:96) ) def = max { L x : x ∈ B } . (3.2) Theorem 3.4.
For n ≥ N , we have T r O w ,...,w (cid:96) ( n ) = (cid:88) m ∈ MATCH κ ≡ n { type- o discs of Σ m } (cid:89) x ∈ B Wg O L x ( m x, , m x, ; n ) . (3.3) Proof.
Assume n ≥ N . We assume each w k = x ε k j k x ε k j k . . . x ε k | wk | j k | wk | is written as a reduced word, as in(2.1), and aim to evaluate (3.1). Hence we havetr( w k ( g , . . . , g r )) = (cid:88) q k ,...,q k | wk | ( g ε k j k ) q k q k ( g ε k j k ) q k q k . . . ( g ε k | wk | j k | wk | ) q k | wk | q k . (3.4)It will be helpful to think about the indices appearing in the above expression in the followingalternative way. Recall from § (cid:83) (cid:96)k =1 C ( w k )and a map γ : (cid:83) (cid:96)k =1 C ( w k ) → (cid:87) r S . As in § x,
0) and ( x,
1) on the circlein (cid:87) r S corresponding to x ∈ B for each such x . As in § p I (0) and p I (1) on (cid:83) (cid:96)k =1 C ( w k ). Recall the collection of intervals I x = I x ( w , . . . , w (cid:96) ), and that we have identified I x with [2 L x ] for each x ∈ B . Let I = ∪ x ∈ B I x . For x ∈ B we will let I ± x be the collection of intervalsthat correspond to occurrences of x ± in w , . . . , w (cid:96) . We denote I ± = ∪ x ∈ B I ± x . I , I ± x , I ± For each x and k ∈ { , } we accordingly identify the points { p I ( k ) : I ∈ I x } with the set [2 L x ].This allows us to think of the choices of q k , . . . , q k | w k | over the various w k as an assignment a : { p I ( k ) : k = 0 , } → [ n ]with the property that two immediately adjacent marked points p, q in (cid:83) (cid:96)k =1 C ( w k ) that are notof the form { p, q } = { p I (0) , p I (1) } (i.e., internal to some interval) must have a ( p ) = a ( q ). Write AA = A ( w , . . . , w (cid:96) ) for the collection of all such assignments a .To each interval I ∈ I x we attach the group element g ( I ) = g i where x = x i ∈ B . Using that g − = g T , we can rewrite the product over k of the expressions in (3.4) as (cid:96) (cid:89) k =1 tr( w k ( g , . . . , g r )) = (cid:88) a ∈A ( w ) (cid:89) x ∈ B (cid:89) I ∈I x g ( I ) a ( p I (0)) a ( p I (1)) . (3.5)For a ∈ A and m = { ( m x, , m x, ) } x ∈ B ∈ MATCH κ ≡ a collection of matchings, we say a (cid:96) m if a (cid:96) m whenever m x,i matches p I ( i ) and p J ( i ), these points are assigned the same index by a . In this casewe also say that m x, and m x, respect a .If x = x i , a direct consequence of Theorem 3.1 is (cid:90) O( n ) (cid:89) I ∈I x g ( I ) a ( p I (0)) a ( p I (1)) dµ n ( g i ) = (cid:88) matchings m x, of { p I (0): I ∈ I x } that respect a matchings m x, of { p I (1): I ∈ I x } that respect a Wg O L x ( m x, , m x, ; n ) . (cid:90) O( n ) r (cid:89) x ∈ B (cid:89) I ∈I x g ( I ) a ( p I (0)) a ( p I (1)) dµ n ( g ) . . . dµ n ( g r ) = (cid:88) m ∈ MATCH κ ≡ : a (cid:96) m (cid:89) x ∈ B Wg O L x ( m x, , m x, ; n ) . (3.6)Reordering summation and integration, we obtain T r O w ,...,w (cid:96) ( n ) = (cid:88) a ∈A (cid:88) m ∈ MATCH κ ≡ : a (cid:96) m (cid:89) x ∈ B Wg O L x ( m x, , m x, ; n ) . = (cid:88) m ∈ MATCH κ ≡ { a ∈ A : a (cid:96) m } (cid:89) x ∈ B Wg O L x ( m x, , m x, ; n ) . (3.7)For fixed m , the condition a (cid:96) m holds if and only if for every type- o disc of Σ m , a is constant onthe set of p I ( k ) that meet the boundary of that disc. Hence { a ∈ A : a (cid:96) m } = n { type- o discs of Σ m } . This proves the theorem.Theorem 3.4 has the following easy corollary that was stated in the Introduction.
Corollary 3.5.
There is a computable rational function T r O w ,...,w (cid:96) ∈ Z ( n ) such that for n ≥ N , T r O w ,...,w (cid:96) ( n ) is given by evaluating T r O w ,...,w (cid:96) at n .Proof. The formula given in Theorem 3.4 expresses T r O w ,...,w (cid:96) as a finite sum of computable rationalfunctions since MATCH κ ≡ is finite. Due to Theorem 1.2, we only need to discuss T r O w ,...,w (cid:96) ( n ) throughout the rest of the paper. Thefull Laurent series expansion of T r O w ,...,w (cid:96) ( n ) at n = ∞ involves elements of MATCH ∗ ( w , . . . , w (cid:96) )with extra restrictions. Recall that an element of MATCH ∗ = MATCH ∗ ( w , . . . , w (cid:96) ) is, for some MATCH ∗ κ = { κ x } x ∈ B ∈ Z B ≥ , a collection m = { ( m x, , . . . , m x,κ x ) } x ∈ B of tuples of matchings, where m x,i is a matching of [2 L x ]. For any κ we write MATCH κ = MATCH κ ( w , . . . , w (cid:96) ) for the elements m MATCH κ of MATCH κ with the additional constraint that m x,i (cid:54) = m x,i +1 for each 0 ≤ i < κ x , and similarlydefine MATCH ∗ . We will also write | κ | = (cid:80) x ∈ B κ x . MATCH ∗ | κ | Proposition 3.6.
For n ≥ N , T r O w ,...,w (cid:96) ( n ) is given by the following absolutely convergent series: T r O w ,...,w (cid:96) ( n ) = (cid:88) m ∈ MATCH ∗ ( − | κ ( m ) | n χ (Σ m ) . (3.8) Proof.
Putting the power series expansion in n − for the orthogonal Weingarten function given inTheorem 3.3 into Theorem 3.4 gives T r O w ,...,w (cid:96) ( n ) = (cid:88) m = { ( m x, ,m x, ) } x ∈ B n { type- o discs of Σ m } (cid:89) x ∈ B n − L x (cid:88) κ x ≥ (cid:88) m x, = m (cid:48) x, ,m (cid:48) x, ,...,m (cid:48) x,κx = m x, ∈ M Lx m (cid:48) x,i (cid:54) = m (cid:48) x,i +1 ( − κ x n − ρ ( m (cid:48) x, ,m (cid:48) x, ) − ... − ρ ( m (cid:48) x,κx − ,m (cid:48) x,κx )= n − (cid:80) x ∈ B L x (cid:88) ( κ x ) x ∈ B ∈ Z B ≥ ( − (cid:80) x ∈ B κ x (cid:88) m (cid:48) ∈ MATCH κ n { type- o discs of Σ m (cid:48) } (cid:89) x ∈ B n − ρ ( m (cid:48) x, ,m (cid:48) x, ) − ... − ρ ( m (cid:48) x,κx − ,m (cid:48) x,κx ) . o discs of Σ m (cid:48) are the same as those of Σ m , since theyonly depend on the outer matchings ( m (cid:48) x, , m (cid:48) x,κ x ) x ∈ B , that are the same as in m . Also note thateach of the expressions for Wg O are absolutely convergent when n ≥ N , and we only used finitelymany such expressions, corresponding to the finitely many choices of m ∈ MATCH κ ≡ ( w , . . . , w (cid:96) ).Using Lemma 2.1, the above can be rewritten as (cid:88) m (cid:48) ∈ MATCH ∗ ( − | κ ( m (cid:48) ) | n χ (Σ m (cid:48) ) . This gives the result.
In the section we modify our previous definitions to get a new combinatorial Laurent expansion for T r Ow ,...,w (cid:96) ( n ) in the shifted parameter ( n − − , or equivalently, an expansion for T r Ow ,...,w (cid:96) ( n + 1)in the parameter n − . The reason for doing this is that we want to add into our Laurent expansionadditional surfaces that are constructed from discs and M¨obius bands. The resulting marked surfacesare the ones that are not stabilized by any element of the mapping class group of the surface; seeLemma 5.9 for the precise statement. The introduction of these extra surfaces is essential in allowingus to give clean expressions for the coefficients as in Theorem 1.5. We formalize this as follows. Definition 3.7.
Let SMATCH ∗ = SMATCH ∗ ( w , . . . , w (cid:96) ) be the collection of pairs ( m , ε ) where SMATCH ∗ • m ∈ MATCH ∗ ( w , . . . , w (cid:96) ), • ε is a function from the 2-cells of Σ m to {− , } , • if m x,i = m x,i +1 then at least one type-( x, i ) disc of Σ m must be assigned − ε .Let κ ( m , ε ) def = κ ( m ). We call a pair ( m , ε ) a signed matching. Definition 3.8.
Given ( m , ε ) ∈ SMATCH ∗ , we construct a new pair (Σ m ,ε , f m ,ε ) where Σ m ,ε is asurface and f m ,ε : Σ m ,ε → (cid:87) r S as follows: • Let Σ m ,ε be the surface obtained by connected summing a real projective plane R P ontoeach 2-cell of Σ m that is assigned − ε . • On the neighborhood of each disc that was cut out to perform a connected sum, homotope f m to be a constant other than { o } ∪ { ( x, k ) } , while maintaining the property that the onlypoints in the preimage of { ( x, k ) } are in Σ (1) m . This is possible because f m maps each 2-cell ofΣ m to a contractible piece of (cid:87) r S . Now extend the function by the constant to the added R P . Performing this homotopy then extension for each R P added to Σ m yields f m ,ε .The resulting pair (Σ m ,ε , f m ,ε ) is an admissible map in the sense of Definition 1.3. Remark . Note that Σ m ,ε is no longer a CW -complex; rather it is a CW -complex with some2-cells replaced by M¨obius bands.The following Lemma is an obvious consequence of the fact χ ( R P ) = 1. Lemma 3.10.
The Euler characteristic of Σ m ,ε is χ (Σ m ,ε ) = χ (Σ m ) − | ε − ( {− } ) | . m is the same matching data as from Figure 2.1. On the left is Σ m with2-cells D , D , D labeled. Here, ε is given by ε ( D ) = ε ( D ) = − ε ( D ) = 1. The resultingΣ m ,ε is drawn on the right, where we draw a ⊗ on the surface to mean an R P has been connectedsummed there. Definition 3.11.
There is a map forget : SMATCH ∗ → MATCH ∗ as follows. Given ( m , ε ), let forget ( m , ε ) ∈ MATCH ∗ be the set of matching data obtained by repeatedly replacing pairs of theform m x,i = m x,i +1 by m x,i , and re-indexing, until there are no consecutive duplicate matchings.Note that because of this removal of duplicates, forget does not respect the Z B gradings of SMATCH ∗ and MATCH ∗ by κ .We now define MM = M ( w , . . . , w (cid:96) ) = max { L x : x ∈ B } log 2 = N log 2 (3.9)where log denotes the natural logarithm. This M is the quantity that appears in Theorem 1.5.Note that M > N . The reason for this choice of parameter will be explained in the proof of thenext lemma.
Lemma 3.12.
For n > M , for any m ∈ MATCH ∗ we have ( − | κ ( m ) | ( n + 1) χ (Σ m ) = (cid:88) ( m (cid:48) ,ε ): forget ( m (cid:48) ,ε )= m ( − | κ ( m (cid:48) ,ε ) | n χ (Σ m (cid:48) ,ε ) , (3.10) where the right hand side is absolutely convergent.Proof. Write m = { ( m x, , . . . , m x,κ x ) } x ∈ B . To obtain ( m (cid:48) , ε ) as in the right hand side of (3.10)from m we make the following choices, that we split into two types. A. For each x ∈ B and 0 ≤ k ≤ κ x we choose d ( x, k ) ∈ Z ≥ and replace m x,k with d ( x, k ) + 1repeats of m x,k . Let m (cid:48) = { ( m (cid:48) x, , . . . , m (cid:48) x,κ (cid:48) x ) } x ∈ B be the resulting new tuples of matchings. Let J x,k be the collection of k (cid:48) such that m (cid:48) x,k (cid:48) = m (cid:48) x,k (cid:48) +1 and m x,k (cid:48) was formed by duplicating m x,k . Hence | J x,k | = d ( x, k ). For each k (cid:48) ∈ J x,k we furthermore have to choose ε x,k (cid:48) on the type-( x, k (cid:48) ) discs ofΣ m (cid:48) , such that ε x,k (cid:48) assigns − x, k (cid:48) ) discsare rectangles, and there are L x of them. B. Independently of the above, we choose some ε on the 2-cells of Σ m , since these correspondbijectively to the two cells of Σ m (cid:48) that were not created by the previous step.Consider the generating function G ( n ) = (cid:88) ( m (cid:48) ,ε ): forget ( m (cid:48) ,ε )= m ( − | κ ( m (cid:48) ,ε ) |−| κ ( m ) | n χ (Σ m (cid:48) ,ε ) − χ (Σ m ) . (3.11)22hatever choices we make in the two steps above, they affect both χ (Σ m (cid:48) ,ε ) − χ (Σ m ) and | κ ( m (cid:48) , ε ) |−| κ ( m ) | independently of one another. Therefore the generating function G splits as a product over m x,k (type A above) and the discs of Σ m (type B above).We explain the contribution from the choices of type A. Since the effect of the choice made fora given m x,k is to contribute d ( x, k ) to | κ ( m (cid:48) , ε ) | − | κ ( m ) | , and for each k (cid:48) ∈ J x,k the contributionof ε x,k (cid:48) to χ (Σ m (cid:48) ,ε ) − χ (Σ m ) is −| ε − x,k (cid:48) ( {− } ) | , the multiplicative contribution from a fixed m x,k to G ( n ) is (cid:88) d ( x,k ) ≥ ( − d ( x,k ) (cid:89) k (cid:48) ∈ J x,k (cid:88) ε x,k (cid:48) (cid:54)≡ n −| ε − x,k (cid:48) ( {− } ) | = (cid:88) d ( x,k ) ≥ ( − d ( x,k ) (cid:89) k (cid:48) ∈ J x,k (cid:0) (1 + n − ) L x − (cid:1) = (cid:88) d ( x,k ) ≥ ( − d ( x,k ) (cid:0) (1 + n − ) L x − (cid:1) d ( x,k ) = 11 + (1 + n − ) L x − n − ) L x . All the sums are absolutely convergent when (cid:12)(cid:12) (1 + n − ) L x − (cid:12)(cid:12) < n > M (this isthe reason for the choice of M ). Multiplying all these contributions together over all m x,k the totalmultiplicative contribution is (cid:89) x ∈ B (cid:89) ≤ k ≤ κ x n − ) L x = 1(1 + n − ) (cid:80) x ∈ B ( κ x +1) L x . (3.12)Now we explain the contribution from choices of type B. The choices made contribute 0 to | κ ( m (cid:48) , ε ) | − | κ ( m ) | and −| ε − ( {− } ) | to χ (Σ m (cid:48) ,ε ) − χ (Σ m ). Hence the multiplicative contributionof these choices to G ( n ) is more simply (cid:88) ε n −| ε − ( { } ) | = (1 + n − ) { discs of Σ m } . (3.13)Multiplying (3.12) and (3.13) together and using (2.6) we obtain G ( n ) = (1 + n − ) { discs of Σ m }− (cid:80) x ∈ B ( κ x +1) L x = (1 + n − ) χ (Σ m ) = (cid:18) n + 1 n (cid:19) χ (Σ m ) . (3.14)Equating (3.11) and (3.14) and rearranging gives the result. Proposition 3.13.
For n > M , T r O w ,...,w (cid:96) ( n + 1) is given by the following absolutely convergentseries: T r O w ,...,w (cid:96) ( n + 1) = (cid:88) ( m ,ε ) ∈ SMATCH ∗ ( − | κ ( m ,ε ) | n χ (Σ m ,ε ) . (3.15) Proof.
Assume n > M . Since
M > N , we have by Proposition 3.6 and Lemma 3.12 T r O w ,...,w (cid:96) ( n + 1) = (cid:88) m ∈ MATCH ∗ ( − | κ ( m ) | ( n + 1) χ (Σ m ) = (cid:88) m ∈ MATCH ∗ (cid:88) ( m (cid:48) ,ε ): forget ( m (cid:48) ,ε )= m ( − | κ ( m (cid:48) ,ε ) | n χ (Σ m (cid:48) ,ε ) , where the right hand side is absolutely convergent. This clearly gives (3.15).23 orollary 3.14. For fixed w , . . . , w (cid:96) and fixed χ ∈ Z , there are only finitely many elements ( m , ε ) of SMATCH ∗ ( w , . . . , w (cid:96) ) with χ (Σ m ,ε ) = χ .Proof. This could be proven by a direct combinatorial argument similarly to [MP19b, Claim 2.10].It is also a direct consequence of the sum (3.15) in Proposition 3.13 being absolutely convergent for n > M . Indeed, if there were infinitely many m ∈ SMATCH ∗ with χ (Σ m ) = χ then there wouldbe infinitely many summands in (3.15) with absolute value n χ .Since every ( m , ε ) ∈ SMATCH ∗ gives rise to an admissible map (Σ m ,ε , f m ,ε ), it makes sense topartition elements of SMATCH ∗ according to the equivalence class of (Σ m ,ε , f m ,ε )in Surfaces ∗ ( w , . . . , w (cid:96) ). Thus, given [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) we defineSMATCH ∗ ( w , . . . , w (cid:96) ; Σ , f ) def = { ( m , ε ) ∈ SMATCH ∗ ( w , . . . , w (cid:96) ) : (Σ m ,ε , f m ,ε ) ≈ (Σ , f ) } . Then we can rewrite Proposition 3.13 as,
Corollary 3.15.
For n > M , T r O w ,...,w (cid:96) ( n + 1) = (cid:88) [(Σ ,f )] ∈ Surfaces ∗ ( w ,...,w (cid:96) ) n χ (Σ) (cid:88) ( m ,ε ) ∈ SMATCH ∗ ( w ,...,w (cid:96) ;Σ ,f ) ( − | κ ( m ,ε ) | . Corollary 3.15 reduces the proof of our main theorem (Theorem 1.5), when all unsigned expo-nents of x ∈ B in w , . . . , w (cid:96) are even, to the following. Theorem 3.16.
For [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) , the L -Euler characteristic χ (2) (MCG( f )) is well-defined, and given by χ (2) (MCG( f )) = (cid:88) ( m ,ε ) ∈ SMATCH ∗ ( w ,...,w (cid:96) ;Σ ,f ) ( − | κ ( m ,ε ) | . (3.16)The proof of Theorem 3.16 will be the subject of § §
5. Henceforth, we will writeSMATCH ∗ (Σ , f ) = SMATCH ∗ ( w , . . . , w (cid:96) ; Σ , f ). SMATCH ∗ (Σ ,f ) Proof of Corollary 1.10.
By Lemma 2.5, if 1 − min(sql( w ) , w )) = −∞ , then Surfaces ∗ ( w ) isempty, and Lemmas 2.3 and 2.6 tell us T r O w ( n ) = 0 for all n , which gives the result.If 1 − min(sql( w ) , w )) is finite, then by Proposition 3.6, (3.8) is a convergent Laurent seriesin n − with positive radius of convergence, and the order of the zero at ∞ is at least − max m ∈ MATCH ∗ ( w ,...,w (cid:96) ) χ (Σ m ) . Note that we only have a bound here, since there could be cancellations between the coefficients of n χ for any given χ . On the other hand, every m ∈ MATCH ∗ gives an admissible map (Σ m , f m ) somax m ∈ MATCH ∗ ( w ,...,w (cid:96) ) χ (Σ m ) ≤ χ max ( w , . . . , w (cid:96) ) . This implies T r O w ( n ) = O ( n χ max ( w ) )as n → ∞ . Finally, Lemma 2.5 tells us we can replace this by O ( n − min(sql( w ) , w )) ) as stated inCorollary 1.10. 24 emark . In fact, every admissible map (Σ , f ) ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) of maximal Euler charac-teristic (so χ (Σ) = χ max ( w , . . . , w (cid:96) )) can be constructed from a suitable m ∈ MATCH ∗ ( w , . . . , w (cid:96) ),so we have max m ∈ MATCH ∗ ( w ,...,w (cid:96) ) χ (Σ m ) = χ max ( w , . . . , w (cid:96) ) . Indeed, this is a simple generalization of [Cul81, Theorem 1.5]. The leading exponent of T r Ow ,...,w (cid:96) ( n )is strictly smaller than χ max ( w , . . . , w (cid:96) ) only if the coefficients χ (2) (MCG ( f )) of the maps [Σ , f ] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) with χ (Σ) = χ max ( w , . . . , w (cid:96) ) sum up to zero. In this section we follow [MP19b, §
3] closely. Our goal here is to extend the results of ( loc. cit. )to surfaces that might be non-orientable. Since our aim is to prove Theorem 3.16, we now fix anadmissible map (Σ , f ) for w , . . . , w (cid:96) . It will be useful to mark an additional set of points V o on theboundary of Σ with the following properties: • V o contains the original marked points { v j } ⊂ δ Σ. • V o ⊂ f − ( { o } ), and V o is finite. • | δ j ∩ V o | = | w j | and so δ j − V o consists of | w j | intervals. Ordering these intervals according tothe orientation of δ j , beginning at v j , the u th interval, directed according to δ j , maps under f to a loop in (cid:87) r S that corresponds to x ε ku j ku ∈ π ( (cid:87) r S , o ) ∼ = F r .We fix this choice of V o henceforth. We use the terminology arc to refer to an embedding of a closed interval in a compact surface suchthat the endpoints of the arc are in the boundary of the surface and these are the only points ofthe arc in the boundary. We use the terminology curve to refer to an embedding of a circle in asurface, disjoint from the boundary of the surface. Note that our notion of curve is what is usuallyreferred to as a simple closed curve.
Definition 4.1.
Let Σ be a compact surface. A continuous function f : Σ → (cid:87) r S is said to be transverse to a point p ∈ (cid:87) r S − { o } if f − ( { p } ) is a disjoint union of arcs and curves, and everyarc or curve in the preimage has a tubular neighborhood that is cut into two halves by the arc orcurve, and the two halves map under f to two different (local) sides of p in (cid:87) r S .Note that this definition prevents ‘one-sided’ curves in f − ( { p } ). A curve is one-sided if athickening of the curve is a M¨obius band, or equivalently, cutting along the curve results in asurface with only one new boundary component. Definition 4.2. A transverse map on Σ is a tuple κ = { κ x } x ∈ B , a choice for each x ∈ B of κ x + 1distinct transversion points ( x, , . . . , ( x, κ x ) in S x , ordered according to the orientation of S x , anda continuous function g : Σ → (cid:87) r S that is transverse to all the points { ( x, j ) : x ∈ B, ≤ j ≤ κ x } ,such that g − { o } ∩ δ Σ = V o .We say that a transverse map g realizes (Σ , f ) if g is homotopic to f relative to V o .Let J x,j be the connected component of (cid:87) r S − { ( x, j ) : x ∈ B, ≤ j ≤ κ x } that is boundedby the points ( x, j ) and ( x, j + 1). Let U o be the connected component that contains o . We call aconnected component of g − ( U o ) an o -zone of g and a connected component of g − ( J x,j ) an ( x, j )-zone of g , or if we do not care about x and j , simply an x -zone. We say that a transverse map is25 lling if all its zones are topological discs. We say the map is almost-filling if all its zones are discsor M¨obius bands.Two transverse maps g and g on Σ are said to be isotopic if they are homotopic throughtransverse maps with the same parameters κ . In this homotopy, the points ( x, j ) are allowed tovary continuously in S x − { o } .We refer to transverse maps realizing (Σ , f ) as simply transverse maps. As in [MP19b, § x, j )-colored arcs and curves are the components of f − { ( x, j ) } andthe normal direction to the curve is given by the order in which the two local sides of the arc andcurve map to two local sides of ( x, j ) in S x , with the order coming from the fixed orientation of S x .The following definition is the same as in [MP19b, § Definition 4.3 (Loose and strict transverse maps) . We say a transverse map g is loose if it satisfies Restriction 1
There are no o -zones or z -zones containing no element of V o with the property thatall the bounding arcs and curves of the zone are pointing inwards, or all pointing outwards,and all the bounding arcs and curves have the same color. Note this rules out the possibilitythat there is a zone that is bounded by one curve e.g. a disc, or a M¨obius band. Restriction 2
Any segment of the boundary of Σ that is bounded by two same colored endpointsof arcs, that are both directed inwards or both outwards, must contain an element of V o andhence be part of an o -zone.The transverse map g is called strict if it also satisfies Restriction 3
For every x ∈ B and 0 ≤ j < κ x there must be an ( x, j )-zone that is neither arectangle (bounded by two arcs and two boundary segments) nor an annulus bounded by twocurves. Remark . Note that
Restriction 2 implies that if g is a transverse map with parameters { κ x } x ∈ B ,any connected component subinterval of δ Σ − V o contains for some x ∈ B exactly κ x + 1 points thatfor some order of the subinterval map to ( x, , . . . , ( x, κ x ) respectively. Example 4.5.
Consider the pairs (Σ m , f m ) constructed in § f m is a loose transverse map on Σ m , and it is, furthermore, strict, if and only if m ∈ MATCH ∗ . In this case, V o are the endpoints of the intervals used in § . .
3. The zones of f m are the 2-cells of Σ m , hence f m is filling. Example 4.6.
Consider now the pairs (Σ m ,ε , f m ,ε ) constructed in Definition 3.8. Each of these areadmissible maps, and f m ,ε is a strict transverse map on Σ m ,ε . Now, the zones of f m ,ε may be eitherdiscs or M¨obius bands, depending on ε . In this case, f m ,ε is almost-filling. Definition 4.7.
The poset of transverse maps realizing (Σ , f ) , denoted ( T , (cid:22) ), has underlying set T = T (Σ , f ) of isotopy classes [ g ] of strict transverse maps g realizing (Σ , f ). The partial order (cid:22) is defined by [ g ] (cid:22) [ g ] if g is obtained from g by forgetting transversion points. (After we forgettransversion points we re-index the remaining ( x, i ) , . . . , ( x, i r ) (cid:55)→ ( x, , . . . , ( x, r ).)As in [MP19b, §
3] we have the following lemmas.
Lemma 4.8. If g is a strict transverse map realizing (Σ , f ) and g (cid:48) is a transverse map obtainedfrom g by forgetting points of transversion then g (cid:48) is also a strict transverse map realizing (Σ , f ) . roof. Same as [MP19b, Lemma 3.7].
Lemma 4.9. T = T (Σ , f ) is not empty.Proof. Same as [MP19b, Lemma 3.8].A polysimplex is a subset of R k of the form ∆ k × ∆ k × . . . × ∆ k r where (cid:80) rj =1 k j = k and the∆ k j are standard simplices in R k j . The polysimplex ∆ k × ∆ k × . . . × ∆ k r has dimension k . A polysimplicial complex is the natural generalization of a simplicial complex that allows cells to bepolysimplices. Definition 4.10 (Complex of transverse maps) . The complex of transverse maps realizing (Σ , f )is the polysimplicial complex with a polysimplex poly([ g ]) ∼ = (cid:81) x ∈ B ∆ κ x for each element [ g ] of T (Σ , f ) with associated parameters { κ x } x ∈ B . The faces of poly([ g ]) are poly([ g (cid:48) ]) where [ g (cid:48) ] (cid:22) [ g ].The resulting polysimplicial complex is denoted |T | poly = |T (Σ , f ) | poly . It can be naturally identifiedwith a closed subset of Euclidean space and is given the subspace topology. Remark . Lemma 4.8 implies that the face relations of |T | poly make sense: the property ofbeing a strict transverse map is preserved under passing to subfaces, and it is obvious that if g isa transverse map realizing (Σ , f ) and g is obtained from g by forgetting transversion points, then g and g have the same underlying map and hence g realizes (Σ , f ).Also note that Restriction 3 implies that any minimal element [ g ] of T corresponds to exactlyone vertex of any given polysimplex of |T | poly containing [ g ], so |T | poly is really a polysimplicialcomplex.The poset ( T , (cid:22) ) also gives rise to a simplicial complex called the order complex and denotedby |T | . The k − simplices of |T | are chains[ g ] (cid:22) [ g ] (cid:22) . . . (cid:22) [ g k ]in ( T , (cid:22) ), and passing to subfaces corresponds to deleting elements from chains. Fact 4.12. [MP19b, Claim 3.10] |T | is the barycentric subdivision of |T | poly . In particular, |T | and |T | poly are homeomorphic.
The proof of this fact has nothing to do with the issue of whether Σ is orientable, so carries overto the current situation.
Lemma 4.13.
The complex |T | poly is finite dimensional with dim( |T | poly ) ≤ (cid:96) − χ (Σ) .Proof. The proof is along the same lines as the proof of [MP19b, Lemma 3.12]. The point of theproof is that given a transverse map gχ (Σ) = (cid:88) Σ (cid:48) (cid:18) χ (Σ (cid:48) ) −
12 { arcs of f in the boundary of Σ (cid:48) } (cid:19) (4.1)where the sum is over zones of g and an arc is counted twice for Σ (cid:48) if it meets Σ (cid:48) on both sides.First we note that because of Restriction 1 χ (Σ (cid:48) ) − { arcs of f in the boundary of Σ (cid:48) } ispositive only when Σ (cid:48) is a disc that meets exactly one arc, and on one side. This is still true afterdropping the assumption that Σ is orientable, using the classification of surfaces. Each such zonemust be an o -zone containing a point v j , and this can only happen if w is not cyclically reduced.Thus each of these zones contributes 1 / χ (Σ) isat most (cid:96)/ (cid:48) that contribute 0 to χ (Σ) include annuli bounded by two curvesand rectangles. As Σ is not necessarily orientable, there is now the extra possibility of a M¨obiusband bounded by a curve. However, this is forbidden by Restriction 1. x -zone Σ (cid:48) not considered thus far contributes at most − χ (Σ). Indeed, χ (Σ (cid:48) ) − { arcs of f in the boundary of Σ (cid:48) } is an integer, since every x -zone meets an even number of arcs,and we have classified the zones that contribute ≥
0. Moreover, for each x ∈ B and 0 ≤ k < κ x ,there is an ( x, k )-zone contributing at most − Restriction 3 . Hence dim([ g ]) = (cid:80) x ∈ B κ x ≤ (cid:96)/ − χ (Σ).The main goal of this § Theorem 4.14.
The polysimplicial complex |T | poly is contractible.
The motivation for this theorem is that |T | poly carries an action of MCG( f ) that will allow usto calculate χ (2) (MCG( f )) in terms of the orbits of MCG( f ) on T . On the other hand, these orbitscan be related to the terms in (3.16) (see Lemma 5.11).The proof of Theorem 4.14 is the same as the proof of [MP19b, Theorem 3.14]. However, thereis one minor point that needs adjusting. Here we refer to terminology of [MP19b] to explain theadjustment for the sake of completeness. In the classification of maximal null-arc systems on pages32-33 of (ibid.) , it is argued that any component of the complement of a maximal system Ω ofnull-arcs that has one boundary component consisting of a closed null-arc, contains a pair of pantsdisjoint from the curves of g , where g is an auxiliary transverse map with κ x = 0 for all x andsuch that the arcs and curves of g are disjoint from Ω. This should be replaced by the followinganalysis. Let Σ (cid:48) be a component of the complement of Ω that is bounded by a closed null-arc. IfΣ (cid:48) is orientable then it contains a pair of pants disjoint from the curves of g , and this contradictsthe maximality of Ω as in [MP19b, pg. 33]. If Σ (cid:48) is not orientable, then Σ (cid:48) contains a simple closedcurve γ that bounds a M¨obius band, both of which are disjoint from the arcs and curves of g . Onthe M¨obius band consider the waist curve. We can add a new null-arc that is disjoint from the oldones and essentially crosses the waist curve of the M¨obius band and is hence not homotopic to anynull-arc in Ω. This contradicts the maximality of Ω. MCG( f ) on the transverse map complex. L -invariants For a discrete group G , the L -Euler characteristic χ (2) ( G ) is defined as follows. First of all, wemake the following definition. Definition 5.1.
We say that X is a G - CW -complex if X is a CW -complex, with a cellular actionof G , such that if g ∈ G preserves an open cell of X , then g acts as the identity on that cell.For a discrete group G , the group von Neumann algebra N ( G ) is the algebra of G -equivariantbounded operators on (cid:96) ( G ). Let X be a G - CW -complex, and let C ∗ ( X ) be the singular chaincomplex of X . Since C ∗ ( X ) is a complex of left Z [ G ]-modules, we can form the chain complex . . . → N ( G ) ⊗ Z [ G ] C p +1 ( X ) d p +1 −−−→ N ( G ) ⊗ Z [ G ] C p ( X ) d p −→ N ( G ) ⊗ Z [ G ] C p − ( X ) → . . . . This is a complex of left Hilbert N ( G )-modules, following [L¨uc02, Def. 1.15], and the boundarymaps are bounded G -equivariant operators between Hilbert spaces. We define H (2) p ( X ; G ) def = ker( d p )closure(image( d p +1 )) . Each of these homology groups is also a Hilbert N ( G )-module and hence has an associated vonNeumann dimension [L¨uc02, Def. 6.20] (2) p ( X ; G ) def = dim N ( G ) H (2) p ( X ; G ) ∈ [0 , ∞ ] . Following [L¨uc02, Def. 6.79], let χ (2) ( X ; G ) def = (cid:88) p ∈ Z ≥ ( − p b (2) p ( X, G )if the sum is absolutely convergent. Note this assumes at the very least that all the b (2) p ( X ; G )are finite. If EG is a contractible G - CW -complex with a free action of G , and the sum defining χ (2) ( EG ; G ) is absolutely convergent, then we define b (2) p ( G ) def = b (2) p ( EG ; G ) , χ (2) ( G ) def = χ (2) ( EG ; G ) . These quantities do not depend on EG , so give invariants of G (when they are defined). The reasonfor this is that b (2) p ( EG ; G ) is invariant under G -equivariant homotopy equivalence of EG [L¨uc02,Thm. 6.54], and EG always exists and is unique up to such homotopy equivalences [tD72, tD87].In this paper we will calculate χ (2) (MCG( f )), for [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ), by othermeans, which make use of the following definition. Definition 5.2.
Let B ∞ be the class of discrete groups such that all b (2) p ( G ) are defined and equalto 0 for all p ∈ Z ≥ .If G is a discrete group and X a G - CW -complex, and σ is a cell of X , then we define the isotropygroup G σ to be the stabilizer of σ in G . We use the convention that | G σ | = 0 if G σ is infinite. Wewill use the following theorem to calculate χ (2) (MCG( f )). Theorem 5.3.
Let G be a discrete group, and X be a G - CW -complex with the following properties • X is acyclic. • All the isotropy groups of G σ are either infinite and in the class B ∞ , or finite. • We have (cid:88) [ σ ] ∈ G \ X | G σ | < ∞ . (5.1) Then χ (2) ( G ) is well-defined and given by χ (2) ( G ) = (cid:88) [ σ ] ∈ G \ X ( − dim( σ ) | G σ | . Theorem 5.3 is a synthesis of results in L¨uck [L¨uc02, Thm. 6.80(1), Ex. 6.20].As in [MP19b, Section 4], we use the following theorem, essentially due to Cheeger and Gromov(cf. [CG86, Corollary 0.6]), as a source of groups lying in B ∞ . The precise statement we need canbe deduced from [L¨uc02, Theorem 7.2, items (1) and (2)]. Recall that a discrete group is called amenable if it has a finitely additive left invariant probability measure. Theorem 5.4 (Cheeger-Gromov) . If G is a discrete group containing a normal infinite amenablesubgroup then G ∈ B ∞ . .2 Proof of Theorem 3.16 When we apply Theorem 5.3 to prove Theorem 3.16, we will take X = |T | poly . Recall that T = T (Σ , f ) for [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ). Let Γ = MCG( f ). We now prepare the necessaryinputs for Theorem 5.3 in the following lemmas. Lemma 5.5.
The action of Γ on T makes |T | poly into a Γ - CW -complex.Proof. The same as the proof of [MP19b, Lemma 4.5].Analogously to [MP19b, § T ∞ , (cid:22) ) denote the subposet of T consisting of isotopy T ∞ classes of transverse maps that are not almost-filling. Recall a transverse map is almost-filling ifits zones are discs or M¨obius bands. This definition marks an essential departure from [MP19b],where the presence of M¨obius bands is not possible due to the surfaces under consideration beingorientable. In fact, this difference is what is responsible for the shift by the Jack parameter in ourmain theorem (Theorem 1.5). The reason discs and M¨obius bands are singled out here is becausethese are precisely the type of zones that have trivial mapping class group: Lemma 5.6.
The mapping class group of a disc or a M¨obius band is trivial.Proof. The statement for a disc is the Alexander Lemma [FM12, Lemma 2.1]. The statement for aM¨obius band can be found in Epstein [Eps66, Theorem 3.4].
Definition 5.7.
We say that a two-sided simple closed curve in Σ is generic if it is not homotopicto a boundary component, and does not bound either a disc or a M¨obius band.We also need the following proposition that appears in [Stu06, Prop. 4.4].
Proposition 5.8. If Σ is any surface with boundary, and c , . . . , c r are a collection of disjoint,pairwise non-isotopic, generic two-sided simple closed curves in Σ , then the Dehn twists in c , . . . , c r generate a subgroup of MCG(Σ) that is isomorphic to Z r . Lemma 5.9.
The isotropy groups Γ [ g ] of the action of Γ on T can be classified as follows • Γ [ g ] = { id } if [ g ] ∈ T − T ∞ , • Γ [ g ] is infinite and in the class B ∞ if [ g ] ∈ T ∞ .Proof. The proof of the first statement (when [ g ] is almost-filling) is similar to the proof of [MP19b,Lemma 4.7], but incorporating Lemma 5.6 instead of simply the Alexander Lemma.The proof of the statement given when [ g ] ∈ T ∞ is similar to the proof of [MP19b, Lemma 4.8].Given [ g ] ∈ T ∞ , we create a list of simple closed curves c , . . . , c k as follows. For every boundarycomponent of any zone of g , add to the list the simple closed curve that follows close to the boundarycomponent inside the zone. After doing so, remove repeats of isotopic curves (e.g. if a zone of g isbounded by a simple closed curve, then in the previous step isotopic curves were created on bothsides). Also remove any curves that are not generic.Note that by construction the c i are pairwise non-isotopic, disjoint, two-sided, and generic. Weshould check that the collection of c i is not empty. Indeed, since [ g ] ∈ T ∞ , some zone Z of g is nota M¨obius band or a disc. Hence Z must have a boundary component that gave rise to a c i that isgeneric.Any Dehn twist D c i in one of the c i is in Γ [ g ] , since c i is disjoint from the arcs and curves of g , and [ g ] is determined by these. Since mapping classes in Γ [ g ] have representatives that respectthe zones of g , elements of Γ [ g ] permute the isotopy classes of the c i . Hence the group generated bythe D c i (we choose one Dehn twist for each c i ) is a normal subgroup of Γ [ g ] . By Proposition 5.8,this subgroup is isomorphic to Z d with d ≥
1. Since Z d is amenable by a result of von Neumann[von29], we deduce from Theorem 5.4 that Γ [ g ] is in B ∞ . Mapping classes fix the boundary pointwise. ∗ = SMATCH ∗ ( w , . . . , w (cid:96) ) from § g ] ∈T − T ∞ . We will now describe how [ g ] naturally defines an element ( m ([ g ]) , ε ([ g ])) of SMATCH ∗ .The points V o cut δ Σ into intervals, which by design, are naturally identified with the subintervalsof ∪ (cid:96)k =1 C ( w k ) that were used in their construction. Let κ = { κ x } x ∈ B be the parameters of [ g ].Consider, for each x ∈ B and 0 ≤ j ≤ κ x , the collection A ( x, j ) of arcs of g that are in the preimageof ( x, j ). These arcs naturally give a matching m x,j ( g ) of I x ( w , . . . , w (cid:96) ). This matching does notchange under isotopy of g . Hence reading off all the matchings as x and j vary, we obtain a tupleof matchings m ([ g ]) = { ( m x, ( g ) , . . . , m x,κ x ( g )) } x ∈ B ∈ MATCH ∗ with κ ( m ([ g ])) = κ . Note that Restriction 3, together with [ g ] ∈ T − T ∞ implies that if m x,i = m x,i +1 then at least one of the ( x, i )-zones of [ g ] is a M¨obius band.By Restriction 1 , together with [ g ] ∈ T − T ∞ , we have that g − ( { ( x, i ) } ) contains no curvesbut only arcs. By construction, the matching arcs of Σ m corresponding to m x,i are in one-to-onecorrespondence with the connected components of g − ( { ( x, i ) } ), and any zone of g corresponds to a2-cell of Σ m by matching up the matching arcs on the boundary of the 2-cell. However, a zone of g that is a M¨obius band may correspond to a 2-cell of Σ m that is a disc. To record this discrepancy,we define ε = ε ([ g ]) to assign 1 to each 2-cell of Σ m that corresponds to a zone of g that is a disc,and define ε to assign − m that corresponds to a zone of g that is a M¨obius band.Thus we have defined a map ( m , ε ) : T − T ∞ → SMATCH ∗ . Lemma 5.10.
Let [(Σ i , f i )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) and [ g i ] ∈ T (Σ i , f i ) for i = 1 , . Then ( m ([ g ]) , ε ([ g ])) = ( m ([ g ]) , ε ([ g ])) if and only if there is a homeomorphism φ : Σ → Σ that respects all markings of the boundaries of the two surfaces and such that [ g ◦ φ ] = [ g ] (as isotopy classes of transverse maps).Proof. First suppose that ( m ([ g ]) , ε ([ g ])) = ( m ([ g ]) , ε ([ g ])). By design of the map ( m , ε ), thereare homeomorphisms φ i : Σ m ([ g i ]) ,ε ([ g i ]) → Σ i such that for i = 1 , f m ([ g i ]) ,ε ([ g i ]) ◦ φ − i is a transverse map on Σ i that is isotopic to g i , and the φ i respect the boundary markings of the surfaces. The map φ = φ ◦ φ − satisfies [ g ◦ φ ] = [ g ].On the other hand, if φ is as in the statement of the lemma, then it is not hard to see that( m ([ g ]) , ε ([ g ])) = ( m ([ g ]) , ε ([ g ])). Lemma 5.11.
The map ( m , ε ) has the following properties:1. ( m , ε ) is invariant under Γ , that is, for γ ∈ Γ = MCG( f ) and [ g ] ∈ T − T ∞ , ( m ( γ [ g ]) , ε ( γ [ g ])) = ( m ([ g ]) , ε ([ g ])) .2. The image of ( m , ε ) is SMATCH ∗ (Σ , f ) .3. ( m , ε ) descends to a bijection (cid:92) ( m , ε ) : Γ \ ( T − T ∞ ) → SMATCH ∗ (Σ , f ) that respects the Z B ≥ gradings of the two sets given by the two incarnations of κ (on transverse maps and signedmatchings). roof. Part 1. This is a special case of one of the implications of Lemma 5.10.
Part 2.
This follows from the fact that given any ( m , ε ) ∈ SMATCH ∗ (Σ , f ), by definition(Σ m ,ε , f m ,ε ) ≈ (Σ , f ). So there is a homeomorphism h : Σ m ,ε → Σ such that f m ,ε ◦ h − is homotopicto f relative to V o . On the other hand, f m ,ε ◦ h − is a strict transverse map realizing (Σ , f ), all ofwhose zones are discs or M¨obius bands by construction. Hence [ f m ,ε ◦ h − ] ∈ T (Σ , f ) − T ∞ (Σ , f )with m ([ f m,ε ◦ h − ]) = m and ε ( f m ,ε ◦ h − ) = ε (by Part 1). Part 3.
The fact that ( m , ε ) descends to a surjective map (cid:92) ( m , ε ) : Γ \ ( T −T ∞ ) → SMATCH ∗ (Σ , f )follows from Parts 1 and 2. We need to prove (cid:92) ( m , ε ) is injective, in other words, if ( m ([ g ]) , ε ([ g ])) =( m ([ g ]) , ε ([ g ])) then there is some γ ∈ MCG( f ) such that γ ([ g ]) = [ g ]. The needed γ is furnishedby Lemma 5.10 (taking Σ = Σ = Σ). Corollary 5.12. Γ \ ( T − T ∞ ) is finite.Proof. By Lemma 5.11, Part 3, Γ \ ( T − T ∞ ) has the same cardinality as SMATCH ∗ (Σ , f ), which isfinite by Corollary 3.14, applied with χ = χ (Σ). Proof of Theorem 3.16.
Combining Theorems 4.14 and 5.3 (with X = |T | poly and G = Γ) togetherwith Lemmas 5.5 and 5.9 and Corollary 5.12 shows that χ (2) (Γ) is well-defined and given by χ (2) (Γ) = (cid:88) [[ g ]] ∈ Γ \ ( T −T ∞ ) ( − | κ ([ g ]) | . Finally, Lemma 5.11, Part 3, shows that the above sum can be replaced by χ (2) (Γ) = (cid:88) ( m ,ε ) ∈ SMATCH ∗ (Σ ,f ) ( − | κ ( m ,ε ) | that gives the formula stated in Theorem 3.16. L -Euler characteristic is the usual one for almost-incompressible maps Recall Definition 2.7 of incompressible and almost-incompressible maps.
Theorem 5.13. If [(Σ , f )] is an almost-incompressible element of Surfaces ∗ ( w , . . . , w (cid:96) ) then thereexists a finite CW -complex X ( f ) such that X ( f ) is an Eilenberg-Maclane space of type K (MCG( f ) , and χ (2) (MCG( f )) = χ ( X ( f )) where the right hand side is the usual topological Euler characteristic.Remark . Note that by Lemma 2.8, Theorem 5.13 applies to all [(Σ , f )] with χ (Σ) ≥ χ max ( w , . . . , w (cid:96) ) − Remark . Since such X ( f ) are unique up to weak homotopy equivalence, χ ( X ( f )) is an invariantof MCG( f ) usually simply denoted by χ (MCG( f )).The proof of Theorem 5.13 relies on the following lemma. Lemma 5.16.
Let [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) be almost-incompressible and T = T (Σ , f ) .Then all [ g ] ∈ T are almost-filling, namely, T ∞ (Σ , f ) is empty.Proof. Assume that [ g ] ∈ T is not almost-filling. Then there must be a generic simple closed curve c in Σ that is disjoint from the arcs and curves of g . Indeed, if g contains any curves then we can take c to be parallel to one of these curves, and c would then be generic by Restriction 1 . Otherwise,if there is a zone of g that is not a topological disc nor a M¨obius band, then we can take c to beany generic simple closed curve in this zone. Since c lives in only one zone of g , g ( c ) is confined32o a contractible region of (cid:87) r S , hence is nullhomotopic. Hence f ( c ) is also nullhomotopic, sinceby assumption g is homotopic to f . This contradicts our assumption, hence all [ g ] ∈ T (Σ , f ) arealmost-filling. Proof of Theorem 5.13.
Let X ( f ) = MCG( f ) \|T | poly . This is a finite CW -complex by Corollary5.12 and Lemma 5.16. Since by Lemma 5.9 the action of MCG( f ) on |T | poly is free and by Theorem4.14 |T | poly is contractible, standard arguments show that X ( f ) is connected, π ( X ( f )) ∼ = MCG( f ),and the higher homotopy groups π k ( X ( f )) = 0 for k ≥
2. This is the statement that X ( f ) is anEilenberg-Maclane space of type K (MCG( f ) , χ (2) (MCG( f )) = (cid:88) [ σ ] ∈ MCG( f ) \T ( − dim( σ ) = (cid:88) τ a cell of X ( f ) ( − dim( τ ) = χ ( X ( f )) . Note that the use of Theorem 5.3 is valid since the sum in (5.1) is finite by finiteness of X ( f ). Corollary 5.17.
For fixed w , . . . , w (cid:96) , there are only finitely many almost-incompressible elementsin Surfaces ∗ ( w , . . . , w (cid:96) ) .Proof. Suppose [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) is almost-incompressible. By Lemmas 4.9 and 5.16, T (Σ , f ) is non-empty, and all its elements are almost-filling. By Lemma 4.8, there is [ g ] ∈ T (Σ , f )with κ ([ g ]) = (0 , . . . , m , ε ) from § m ([ g ]) , ε ([ g ])) can only be obtained under the map ( m , ε ) on T (Σ , f ) for one particular[(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ). Therefore, the cardinality of the almost-incompressible maps in Surfaces ∗ ( w , . . . , w (cid:96) ) is at most the cardinality of the set { ( m , ε ) ∈ SMATCH ∗ ( w , . . . , w (cid:96) ) | κ ( m ) = (0 , , . . . , } . The latter set is clearly finite, since its elements are obtained by choosing a finite number of match-ings m of a finite set and then a finite number of possible maps ε from the discs of Σ m to {± } . Remark . The counting yielding the upper bound in the proof of the last corollary is muchredundant. First, if ( m , ε ) ∈ SMATCH ∗ ( w , . . . , w (cid:96) ) and (Σ m ,ε , f m ,ε ) is almost-filling, then ε assigns+1 to each 2-cell in Σ m except for, possibly, at most one 2-cell in every connected component of Σ m .Indeed, if there were two zones in the same connected component of Σ m ,ε which are M¨obius bands,then a simple closed curve tracing the boundary of one of these zones, continuing to the second zone,tracing its boundary and going back to the first zone along a parallel path (thus creating a kind ofa barbell-shape) would be a generic, compressing curve. Second, it is not hard to see that movinga single M¨obius band from one disc of Σ m to another disc in the same connected component, doesnot alter the element in Surfaces ∗ ( w , . . . , w (cid:96) ). Therefore, a given matching m corresponds to atmost 2 m almost-incompressible maps in Surfaces ∗ ( w , . . . , w (cid:96) ). We begin with the following lemma.
Lemma 6.1.
If all w j (cid:54) = 1 , [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) , and all the connected components of Σ are annuli or M¨obius bands, then MCG ( f ) is trivial and, thus, χ (2) (MCG( f )) = 1 . roof. Note that MCG( f ) is the product of the mapping class groups of f restricted to the variousconnected components of Σ , so it is sufficient to prove this when (cid:96) = 2 and Σ is an annulus or (cid:96) = 1and Σ is a M¨obius band. In the latter case, the whole mapping class group is trivial as stated inLemma 5.6 (due to [Eps66, Theorem 3.4]), and so, in particular, MCG( f ) = { } .Finally, suppose that Σ is an annulus. The mapping class group of Σ is isomorphic to Z andgenerated by a Dehn twist D in a curve parallel to the boundary. Consider a directed arc β connecting v to v ( v i is the marked point on the boundary component δ i ). Then f ( β ) is a loop in (cid:87) r S based at o , and we write f ∗ ( β ) ∈ π ( (cid:87) r S , o ) = F r for the class of this loop. For every n ∈ Z , D n ( β ) is also an arc in A with the same endpoints. If one boundary component of Σ is labeled w ,then f ∗ ( D n ( β )) = w ± n f ∗ ( β ) (cid:54) = f ∗ ( β ) for all 0 (cid:54) = n ∈ Z since w (cid:54) = 1. Hence D n / ∈ MCG( f ) and soMCG( f ) = { } . Lemma 6.2.
Let w and w be two words in F r , both (cid:54) = 1 . Let d be the maximal integer suchthat w = u d with u ∈ F r . The number of elements [(Σ , f )] ∈ Surfaces ∗ ( w , w ) such that Σ is anannulus is (cid:40) d if w is conjugate to either w or w − , . Proof.
Assume that [(Σ , f )] ∈ Surfaces ∗ ( w , w ) is an annulus. Then, f : Σ → (cid:87) r S defines afree homotopy between f ( δ ) and f ( δ ). Since free homotopy classes of oriented curves in (cid:87) r S correspond to conjugacy classes in F r , this shows that w must be conjugate to either w or w − ,depending on whether the orientations of the two boundary components of Σ agree or not.Since no non-identity element of F r is conjugate to its inverse, w cannot be conjugate to both w and w − . Without loss of generality, we assume from now on that w is conjugate to w . Let β be a directed arc as in the proof of Lemma 6.1, connecting v to v . Denote b = f ∗ ( β ) ∈ F r , andnote that w = bw b − . Also note that as β cuts Σ into a disc, the map f is completely determined,up to homotopy, by f ∗ ( β ). As in the proof of Lemma 6.1, { ([ f ] ◦ [ ρ ]) ∗ ( β ) | [ ρ ] ∈ MCG (Σ) } = { w n f ∗ ( β ) } n ∈ Z . But as the centralizer of w in F r is (cid:104) u (cid:105) , with u ∈ F r the d -th root of w as in the statement of thelemma, we have (cid:8) c ∈ F r (cid:12)(cid:12) w = cw c − (cid:9) = { u n b | n ∈ Z } . Thus, there are exactly d distinct orbitsof possible values of f ∗ ( β ) under the action of MCG (Σ), and therefore exactly d classes of annuliin Surfaces ∗ ( w , w ). Lemma 6.3.
Let w (cid:54) = 1 , w ∈ F r . The number of elements [(Σ , f )] ∈ Surfaces ∗ ( w ) such that Σ is aM¨obius band is 0 if w is not a square and if w is a square in F r .Proof. If w is not a square in F r then there are no [(Σ , f )] ∈ Surfaces ∗ ( w ) with Σ a M¨obius bandby Lemma 2.2. So suppose that w is a square in F r . Then by Lemma 2.2, there is at least one[(Σ , f )] ∈ Surfaces ∗ ( w ) with Σ a M¨obius band. Let [(Σ , f )] be of this form. Then up to homotopy,there is a unique arc α (Σ) in Σ joining v to itself and not separating Σ [Eps66, Proof of Thm. 3.4].We have f ∗ ( a ) = w , which uniquely specifies f ∗ ( α ). Let u be the unique solution of u = w in F r .Let ( M , f ) and ( M , f ) be admissible maps for w with the M i M¨obius bands. For i = 1 , α i be an embedded directed arc from v to itself in M i that does not separate M i . Since by theprevious paragraph ( f ) ∗ ( α ) = ( f ) ∗ ( α ) = u , the homeomorphism h from M to M that preservesthe markings on boundaries and maps α to α , has the property that f ◦ h is homotopic to f andthis shows [( M , f )] = [( M , f )] . Hence there is exactly one element [(Σ , f )] ∈ Surfaces ∗ ( w ) with Σ a M¨obius band.Proof of Corollary 1.16. We assume all w k (cid:54) = 1 and examine the expansion given in Theorem1.5. The limit lim n →∞ T r O w ,...,w (cid:96) ( n ) exists, since there are no [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) with34 (Σ) > Surfaces ∗ ( w , . . . , w (cid:96) ) def = { [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) : χ (Σ) = 0 } consists precisely of surfaces all the connected components of which are annuli or M¨obius bands.Thus lim n →∞ T r Ow ,...,w (cid:96) ( n ) = (cid:88) [(Σ ,f )] ∈ Surfaces ∗ ( w ,...,w (cid:96) ) χ (2) (MCG( f )) = (cid:88) [(Σ ,f )] ∈ Surfaces ∗ ( w ,...,w (cid:96) ) | Surfaces ∗ ( w , . . . , w (cid:96) ) | , the second equality following from Lemma 6.1. This proves the first statement of the corollary.Every admissible pair [(Σ , f )] ∈ Surfaces ∗ ( w , . . . , w (cid:96) ) induces a partition on { w , . . . , w (cid:96) } whereevery block consists of the words associated with one connected component of Σ. The algebraiccharacterization given in the statement of Corollary 1.16 follows from the geometric part using thesepartitions and Lemmas 6.2 and 6.3. Proof of Corollary 1.17.
Assume w = u d , where u (cid:54) = 1 is not a proper power. The first statement ofthis corollary follows readily from the algebraic characterization of lim n →∞ T r O w ,...,w (cid:96) ( n ) in Corol-lary 1.16: the valid partitions of the words w j , . . . , w j (cid:96) depend only on j , . . . , j (cid:96) , and the weightof every partition depends only on d and not on u . The collection of limits determines d using, forexample, the following two equalities:lim n →∞ T r O w ( n ) = (cid:40) d is even , d is odd , and lim n →∞ T r O w,w ( n ) = (cid:40) d + 1 if d is even ,d if d is odd . The analogous result for Sp ( n ) now follows from Theorem 1.2. Proof of Corollary 1.18.
Corollary 1.17 shows that the joint moments of T n ( w ) , . . . , T n (cid:0) w (cid:96) (cid:1) con-verge to the same values as the joint moments of T n ( x ) , . . . , T n (cid:0) x (cid:96) (cid:1) for some x ∈ B , as long as w (cid:54) = 1 and is not a proper power. By the method of moments one can now deduce the corollary,using that Diaconis and Shahshahani have shown in [DS94, Section 3] that these limits of momentsare precisely those of the multivariate normal distribution described in the statement. Fix s ≥
1. Let w s = x · · · x s ∈ F s . Here we analyze Surfaces ∗ ( w s ). Note that χ max ( w s ) =1 − sql ( w s ) = 1 − s , so there is no admissible map (Σ , f ) ∈ Surfaces ∗ ( w s ) with χ (Σ) > − s . Claim . Let t ∈ Z ≥ be a non-negative integer. Then there is exactly one [(Σ t , f t )] ∈ Surfaces ∗ ( w s ) with χ (Σ t ) = 1 − s − t that can be realized by an almost-filling strict transverse map. In particular,there is at most one [(Σ t , f t )] ∈ Surfaces ∗ ( w s ) with χ (2) (MCG( f t )) (cid:54) = 0 and χ (Σ t ) = 1 − s − t . Infact, χ (2) (MCG ( f t )) = (cid:40) δ t, if s = 1 , ( − t (cid:0) t + s − s − (cid:1) if s ≥ , (6.1) where δ t, is the Kronecker delta. Note that for all t ≥
0, Σ t is the non-orientable surface of genus s + t with one boundarycomponent (namely, the connected sum of s + t copies of R P , with a disc removed). When t = 0,MCG ( f ) is trivial and χ (2) (MCG ( f )) = χ (MCG ( f )) = 1. When t = 1, a simple analysis givesthat MCG ( f ) ∼ = π (Σ ) ∼ = F s , in which case χ (2) (MCG ( f )) = χ (MCG ( f )) = χ ( F s ) = 1 − s .This agrees with the t = 1 case in (6.1). It intrigues us to wonder whether MCG( f t ) is related tosome well-known group when t ≥
2. 35 roof of Claim 6.4.
If [(Σ , f )] ∈ Surfaces ∗ ( w s ) satisfies χ (2) (MCG ( f )) (cid:54) = 0, then it must be realizedby some almost-filling strict transverse map, by Theorem 3.16 and Lemma 5.11. So the secondstatement of the claim follows from the first one.Now fix t ≥ , f )] ∈ Surfaces ∗ ( w s ) satisfies χ (Σ) = 1 − s − t and isrealized by some almost-filling strict transverse map g . As g is almost-filling, it has only arcs andno curves. Note that each letter x i appears exactly twice in w s = x · · · x s , so there is only onepossible matching for every letter x i , matching the two occurrences of x i in w s . Therefore, there isa single ( x i , j )-zone for every 1 ≤ i ≤ s and 0 ≤ j < κ x i ([ g ]) which must be a M¨obius band. It iseasy to check that there is one o -zone in g , which may be a disc or a M¨obius band. A simple Eulercharacteristic calculation shows that exactly t of the zones of g are M¨obius bands. In particular, | κ ([ g ]) | ∈ { t − , t } .Now we modify [ g ] by forgetting all points of transversion ( x, i ) with i ≥
1. Let [ h ] ∈ T (Σ , f )be the resulting transverse map. This [ h ] has exactly one zone, it is an o -zone, and by the previousparagraph, this zone was obtained by gluing t M¨obius bands together, and is, thus, necessarily thenon-orientable surface of genus t with one boundary component. Because the matchings in [ h ] aredictated and so is the topological type of its sole zone, any [ h (cid:48) ] obtained in the same way fromsome other [(Σ (cid:48) , f (cid:48) )] with the same properties, would be equivalent to [ h ]. Namely, we could find[ h (cid:48) ] ∈ T (Σ (cid:48) , f (cid:48) ) with [ h (cid:48) ] = [ h ◦ φ ] for φ : Σ (cid:48) → Σ a homeomorphism respecting boundary markings.The same φ shows (Σ (cid:48) , f (cid:48) ) ≈ (Σ , f ). Hence there is exactly one [(Σ t , f t )] ∈ Surfaces ∗ ( w s ) with χ (Σ t ) = 1 − s − t which can be realized by an almost-filling strict transverse map. It is left to prove the equality (6.1). We prove it in two different ways. First, as mentionedabove, any almost-filling [ g ] ∈ T (Σ t , f t ) satisfies | κ ([ g ]) | ∈ { t − , t } , and the same analysis showsthat there is exactly one MCG ( f )-orbit of almost-filling strict transverse maps in T (Σ t , f t ) forevery valid choice of κ . There are (cid:0) t + s − t (cid:1) possible κ ∈ ( Z ≥ ) s with | κ | = t , each contributing ( − t to (3.16), and (cid:0) t − s − t − (cid:1) possible κ ∈ ( Z ≥ ) s with | κ | = t −
1, each contributing ( − t − to (3.16).The total sum is precisely the one specified in (6.1).The second proof uses the Frobenius-Schur type formula (1.3), by which for s = 1 T r O x ( n + 1) = 1 , and for s ≥ T r O w s ( n + 1) = 1( n + 1) s − = 1 n s − (cid:18) − n + 1 n − . . . (cid:19) s − = 1 n s − ∞ (cid:88) t =0 ( − t (cid:18) t + s − s − (cid:19) n t . Combining these two expressions with Theorem 1.5, we see that χ (2) (MCG ( f t )) is given by (6.1). We give here some more details about the examples from Table 1. We elaborate on the exactcontributions, in the language of Theorem 1.5, to the two leading terms with exponents χ max and χ max −
1. The data is summarized in Table 2. The analysis of these examples was carried outwith the help of a SageMath script, and using various observations and considerations. We do notdescribe the analysis here as we do not see it as crucial – we only aim to give a sense of how ourmain theorem plays out in concrete examples.The fourth column of Table 2 specifies the rational expressions for T r O w ,...,w (cid:96) ( n + 1) (unlike theexpression in Table 1 which gave the expression for T r O w ,...,w (cid:96) ( n )), as well as the coefficients of n χ max and of n χ max − in the Laurent expansion. The fifth column is the same as the fifth one inTable 1, while the sixth column lists the equivalence classes of maps [(Σ , f )] in Surfaces ∗ ( w , . . . , w (cid:96) )with χ (Σ) = χ max −
1. Note that by Lemma 2.8, when χ (Σ) = χ max the maps are incompress-ible, and when χ (Σ) = χ max −
1, the maps are always almost-incompressible and sometimes evenincompressible (in the table we point out specifically the cases where the stronger condition holds).36 w , . . . , w (cid:96) χ max T r O w ,...,w (cid:96) ( n + 1) andtwo leading terms Admissiblemaps with χ (Σ) = χ max Admissible maps with χ (Σ) = χ max − x y − n +1 = n − n + . . . one P , w. MCG( f )= { } one P , w. MCG ( f ) ∼ = F x y − n +1 = n − n + . . . one P , w. MCG( f )= { } two incompr. P , w.MCG ( f ) ∼ = Z ; one P , w. MCG ( f ) ∼ = F [ x, y ] n +4 n +3 n − n +1)( n +3) n = 1 + n + . . . one P , w. MCG( f )= { } one P , w. MCG ( f ) ∼ = Z xy x − y − − n + n + . . . one P , w. MCG( f ) ∼ = Z one P , w. MCG( f ) ∼ = Z × F xy x − y − − n +1 = n − n + . . . one P , w. MCG( f )= { } three incompr. P , w. MCG( f ) ∼ = Z ; one P , w. MCG( f ) ∼ = F xyx yx y − n +5( n +1)( n +3) n = n − n + ... three P , w. MCG( f )= { } one P , w. MCG ( f ) ∼ = F ;one P , w. MCG ( f ) ∼ = F w, w for w = x y n +4 n +7 n +8( n +1)( n +3) n = 1 + n + . . . one A w. MCG( f )= { } one P , w. MCG ( f ) ∼ = Z w, w for w = x y n + . . . one A w. MCG( f )= { } one P , w. MCG ( f ) ∼ = Z w, w, w for w = x y − n +7 n +13 n +15 n +24)( n − n ( n +1)( n +3)( n +5) = n − n + . . . three A (cid:116) P , w. MCG( f )= { } three A (cid:116) P , w. MCG( f ) ∼ = F ;three P , (cid:116) P , w. MCG( f ) ∼ = Z Table 2: This table gives more details about the examples from Table 1. Here A denotes an annulus,and as in Table 1, P g,b denotes the non-orientable surface of genus g with b boundary components(so χ ( P g,b ) = 2 − g − b ).Moreover, by Corollary 5.17, there are finitely many such equivalence classes in Surfaces ∗ ( w , . . . , w (cid:96) ), so we can indeed list them all. By Theorem 5.13, in all these cases, weget concrete, finite CW -complexes of type K (MCG ( f ) ,
1) for these maps, which means we canunderstand the groups pretty well. Indeed, we were able to compute the exact isomorphism typeof the groups MCG ( f ) in all cases mentioned in the table. The fact all groups but one are free isprobably only due to the fact that the words in these examples are rather short, which means thecomplexes associated with them tend to have low dimensions.37 Proof of Theorem 1.2: relationship between O and Sp Here we prove Theorem 1.2. Throughout this section, fix n with 2 n ≥ N ( w , . . . , w (cid:96) ), the latterdefined in (3.2). For i ∈ [2 n ] denoteˆ i def = (cid:40) i + n if 1 ≤ i ≤ n,i − n if n + 1 ≤ i ≤ n, and ξ ( i ) def = sign (cid:18) n + 12 − i (cid:19) = (cid:40) ≤ i ≤ n, − n + 1 ≤ i ≤ n. Recall that we think of Sp ( n ) as a subgroup of GL n ( C ), and that the matrix J was defined in(1.1). The following lemma follows easily from the fact that A − = J A T J T for A ∈ Sp ( n ). Lemma A.1. If A ∈ Sp ( n ) and i, j ∈ [2 n ] , then (cid:0) A − (cid:1) i,j = ξ ( i ) ξ ( j ) A ˆ j, ˆ i . (A.1)Our first goal is to obtain an analog of Theorem 3.4 for Sp ( n ), namely, to obtain a formula for T r Sp w ,...,w (cid:96) ( n ) as a finite sum over systems of matchings, only with an additional sign associatedwith every such system; see Proposition A.5 for the precise statement.We recall some of the notation we use here. Let 2 L = 2 (cid:80) x ∈ B L x = (cid:80) (cid:96)k =1 | w k | denotethe total number of letters in w , . . . , w (cid:96) . The k th boundary component of every surface in Surfaces ∗ ( w , . . . , w (cid:96) ) is subdivided to | w k | intervals corresponding to the letters of w k , and wedenoted by I , I + , I − the sets of all 2 L intervals, the subset of L intervals corresponding to positiveletters and its complement, respectively. Likewise, we denote by I x , I + x , I − x the analogous sets ofintervals corresponding to the instances of x ∈ B . We again identify I x with the set [2 L x ], for each x ∈ B , in the same way as in Sections 2.1.4 and 2.1.4. Similarly to the notation from Section 2.1.4,we denote by A = A ( w , . . . , w (cid:96) ) the set of index assignments a : { p I ( k ) | I ∈ I , k ∈ { , }} → [2 n ] , where for every two immediately adjacent marked points p, q in ∪ (cid:96)k =1 C ( w k ) that belong to differentintervals in I we have a ( p ) = a ( q ). (Note the range here is [2 n ] and not [ n ] as in Section 2.1.4).Given a ∈ A , let ˆ a be the assignment obtained after applying (A.1), namely,ˆ a ( p I ( i )) = (cid:40) a ( p I ( i )) if I ∈ I + (cid:92) a ( p I ( i )) if I ∈ I − .As we shall use Theorem 3.1 for evaluating T r Sp w ,...,w (cid:96) ( n ), we need the following expression whichgathers the total sign contribution for a given system of matchings m = { ( m x, , m x, ) } x ∈ B ∈ MATCH κ ≡ and an assignment a . Recall the notation δ Sp i ,m from Theorem 3.1. We let∆ ( a , m ) def = (cid:89) I ∈I − ξ ( a ( p I (0))) ξ ( a ( p I (1))) · (cid:89) x ∈ Bj = 0 , δ Spˆ a |{ pI ( j ) | I ∈I x } ,m x,j = (cid:89) I ∈I − ξ ( a ( p I (0))) ξ ( a ( p I (1))) · (cid:89) x ∈ Bj = 0 , (cid:89) ( p I ( j ) ,p J ( j ))matched by m x,j (cid:10) e ˆ a ( p I ( j )) , e ˆ a ( p J ( j )) (cid:11) Sp
38n the innermost product, each matched pair appears once and is given its predetermined order.Note that for i, j ∈ [2 n ], we have (cid:104) e i , e j (cid:105) Sp = e Ti J e j = δ ˆ i,j ξ ( i ) , (A.2)where here δ is the Kronecker delta. Also notice that ∆ ( a , m ) ∈ {− , , } . We say a (cid:96) ∗ m if∆( a , m ) (cid:54) = 0. Therefore,∆ ( a , m ) = a (cid:96) ∗ m · (cid:89) I ∈I − ξ ( a ( p I (0))) ξ ( a ( p I (1))) · (cid:89) x ∈ Bj = 0 , (cid:89) ( p I ( j ) ,p J ( j ))matched by m x,j ξ (ˆ a ( p I ( j ))) . (A.3) Definition A.2.
Let m ∈ MATCH κ ≡ . Call a matching arc of m orientable if it pairs an intervalin I ± with an interval in I ∓ , and non-orientable otherwise. Let m be one of the matchings in m .In every pair (cid:0) m (2 t − , m (2 t ) (cid:1) we think of the corresponding matching arc in Σ m as directed from its origin – the interval corresponding to m (2 t − , to its terminus – the interval associated with m (2 t ) .Let D be a type- o disc of Σ m . Every interval in I that meets ∂D has an orientation coming fromthe given orientation of ∂ Σ m . We say that two intervals that meet ∂D are co-oriented (relative to D ) if their orientation induces the same orientation on ∂D , and counter-oriented otherwise. Notethat a matching arc is orientable if and only if it matches two co-oriented intervals meeting δD .In the computation of ∆ ( a , m ), we attribute every sign that appears in (A.3) to one of thetype- o discs of Σ m . Indeed, every matching arc is at the boundary of exactly one type- o disc, andevery p I ( i ) also belongs to exactly one type- o disc. Lemma A.3 (Computation of ∆( a , m )) . Assume a , a , a ∈ A and m ∈ MATCH κ ≡ , and let D be a type- o disc in Σ m .1. The number of non-orientable matching arcs along ∂D is even.2. If a (cid:96) ∗ m , the total sign contribution of D to ∆( a , m ) is the product of: ( i ) the sign of the index given by a at the origin of every matching arc with origin in I + , ( ii ) the sign of the index given by a at the terminus of every matching arc with terminus in I − , and ( iii ) ( − for every matching arc with origin in I − .3. If a (cid:96) ∗ m and p I ( k ) , p J ( k ) are matched by any m x,k then a ( p I ( k )) ≡ a ( p J ( k )) mod n , andmoreover, a ( p I ( k )) = a ( p J ( k )) if and only if m x,k corresponds to an orientable matching arc.4. For fixed m , the number of a with a (cid:96) ∗ m is (2 n ) { type- o discs of Σ m } .5. If a , a (cid:96) ∗ m , then ∆( a , m ) = ∆( a , m ) . The final statement allows us to define:
Definition A.4.
For every m ∈ MATCH κ ≡ we let ∆( m ) def = ∆( a , m ), defined by any a (cid:96) ∗ m .Proof of Lemma A.3 . Part 1. The first point is due to the fact that the boundary componentsof Σ have built-in orientation, and along the boundary of D , the orientation of intervals meeting ∂ Σ is preserved when going along an orientable matching arc, and flipped along a non-orientablematching are. But ∂D is a loop, so the number of orientation flips must be even. Here and elsewhere in this appendix, the “sign of an index” i is ξ ( i ). art 2. This follows from (A.3) by checking case by case over all possibilities.
Part 3.
We have a (cid:96) ∗ m if and only if for all ordered matched pairs p I ( k ) , p J ( k ) of any m x,k a ( p I ( k )) = (cid:40) (cid:92) a ( p J ( k )) if I and J are in the same set I ± a ( p J ( k )) if I and J are in I ± and I ∓ respectively. (A.4)This means that when a ∈ A and a (cid:96) ∗ m , there is a constraint on the values of a at every pairof points that are adjacent on the boundary of some type- o disc of Σ m . This is similar to thesituation for the orthogonal group, but the constraints are more complicated now. The constraintimplies that the values of a on the points p I ( k ) in the boundary of a fixed type- o disc D of Σ m are determined by the value at any fixed point p D on the boundary of that disc. The values of a are constant along segments of ∂D , except for segments that are matching arcs joining intervalsin the same set I ± x , across which the value of a jumps by n mod 2 n . These are the non-orientablematching arcs defined in Definition A.2. This proves Part 3.Part 4.
It now follows from Part 1 that if for each type- o disc D of Σ m , we choose a ( p ) forsome p in ∂D , then there exists a unique a ∈ A with these prescribed values and such that a (cid:96) ∗ m .Hence, for any m , there are (2 n ) { type- o discs of Σ m } elements of A with a (cid:96) ∗ m . This proves Part 4.Part 5.
Now we will prove that for m fixed, all the ∆( a , m ) have the same sign.Indeed, we collect the contribution to the sign of every o -disc D separately, and show it doesnot depend on the particular assignment of indices along ∂D . There are two options for the signs ofthese indices, where one is a complete negation of the other. Recall that the sign of ∆( a , m ) splitsup into three types of contributions according to Part 2. The contribution from ( iii ) clearly doesnot depend on a . Now consider the ( i )- and ( ii )-type contributions. • If α is an orientable matching arc, its ( i )- and ( ii )-type contributions to ∆( a i , m ) are always1 in total. This is surely the case if α is directed from I ∈ I − x to J ∈ I + x . But it is also thecase when α is directed the other way round, as the signs of both indices at its endpointsare identical. Hence the contributions of type ( i ) and type ( ii ) of orientable matching arcs toeither of ∆( a , m ) and ∆( a , m ) is equal to 1. • Note from the discussion in the proof of Part 3, that a and a are related by a sequenceof the following type of flip-moves : choose a type- o disc D of Σ m , and modify a by adding n to a ( p I ( k )) modulo 2 n , for every p I ( k ) that meets ∂D . Now for any given non-orientablematching arc α , its ( i )- and ( ii )-type contribution is the sign of one of the endpoints. Hencethe effect of a flip-move on a at a disc D is to change the type ( i ) and ( ii ) contributions to∆( a , m ) by ( − m meeting D . On the other hand, by Part 1, thetotal number of non-orientable matching arcs of m meeting D is even.This concludes the proof of Lemma A.3.We can now prove the analog of Theorem 3.4 for T r Sp w ,...,w (cid:96) ( n ). Proposition A.5.
For n ≥ N T r Sp w ,...,w (cid:96) ( n ) = (cid:88) m ∈ MATCH κ ≡ (2 n ) { type- o discs of Σ m } ∆ ( m ) (cid:89) x ∈ B Wg Sp L x ( m x, , m x, ; n ) , with ∆ ( m ) ∈ { , − } is defined as in Definition A.4.Proof. Let g ( I ) be as in § n ≥ N . We have by the same arguments as led up to (3.7),incorporating (A.1) and using Theorem 3.1 T r Sp w ,...,w (cid:96) ( n ) = (cid:88) a ∈A ( w ,...,w (cid:96) ) (cid:88) m ∈ MATCH κ ≡ ∆( a , m ) (cid:89) x ∈ B Wg Sp L x ( m x, , m x, ; n ) . his formula was the original motivation for introducing ∆( a , m ) . Now using Lemma A.3, Parts 4and 5, and interchanging the sums over a and m gives T r Sp w ,...,w (cid:96) ( n ) = (cid:88) m ∈ MATCH κ ≡ (cid:32) (cid:89) x ∈ B Wg Sp L x ( m x, , m x, ; n ) (cid:33) (cid:88) a ∈A ( w ) ∆( a , m ) = (cid:88) m ∈ MATCH κ ≡ (2 n ) { type- o discs of Σ m } ∆ ( m ) (cid:89) x ∈ B Wg Sp L x ( m x, , m x, ; n )as required.We can now prove the main result of this subsection and show that T r Sp w ,...,w (cid:96) ( n ) = ( − (cid:96) ·T r O w ,...,w (cid:96) ( − n ) for large n . Proof of Theorem 1.2.
It follows from Proposition A.5 and Lemma 3.2 that T r Sp w ,...,w (cid:96) ( n ) = (cid:88) m ∈ MATCH κ ≡ (2 n ) { type- o discs of Σ m } ∆ ( m ) (cid:89) x ∈ B Wg Sp L x ( m x, , m x, ; n )= (cid:88) m ∈ MATCH κ ≡ (2 n ) { type- o discs of Σ m } (cid:89) x ∈ B ( − L x · sign (cid:16) σ − m x, · σ m x, (cid:17) · Wg O L x ( m x, , m x, ; − n ) · ∆ ( m )= (cid:88) m ∈ MATCH κ ≡ ( − n ) { type- o discs of Σ m } (cid:89) x ∈ B Wg O L x ( m x, , m x, ; − n ) · Ξ ( w , . . . , w (cid:96) ; m ) , (A.5)where Ξ ( w , . . . , w (cid:96) ; m ) def = ( − { type- o discs of Σ m } · ( − L · ∆ ( m ) · (cid:89) x ∈ B sign (cid:16) σ − m x, σ m x, (cid:17) . We now show that Ξ ( w , . . . , w (cid:96) ; m ) is independent of m and equal to ( − (cid:96) . This will completethe proof by combining (A.5) with Theorem 3.4.Our strategy for proving that Ξ ( w , . . . , w (cid:96) ; m ) ≡ ( − (cid:96) consists of three parts:1. The fact there are r = | B | different types of letters in w , . . . , w (cid:96) can be ignored, and all lettersmay be considered as identical.2. If I = I + , there is one particular set of matchings m for which Ξ ( w , . . . , w (cid:96) ; m ) = ( − (cid:96) .3. The value of Ξ ( w , . . . , w (cid:96) ; m ) does not change if we make small local changes: ( a ) flipping thedirection of one matching arc, ( b ) exchanging the termini of two matching arcs, or ( c ) flippingthe orientation of one of the letters in the word from positive to negative or vice versa.During this proof we consider the matchings m x,i of m as matchings of the letters of the words w , . . . , w (cid:96) , this is possible since the letters are in one to one correspondence with the intervals I . Part I: Consider all letters as identical
First, recall the definition from § σ m ∈ S k associated with the matching m belonging to m , and note that the order of the pairs in m does not affect the sign of σ m , nor ∆ ( m ),so we ignore it here. (In contrast, the order within each pair does affect these quantities.) As aresult, we can treat all matchings { m x, } x ∈ B as a single matching m ∈ M L of the whole collection41igure A.1: On the left hand side, there is one word of even length (6 in this example) with allletters positive, and m = m match I → I , I → I and I → I . An easy computation givesthat Ξ ( w ; m , m ) = − m = m match I → I , I → I , I → I , I (cid:48) → I (cid:48) , I (cid:48) → I (cid:48) and I → I (cid:48) . An easy analysis gives that Ξ ( w , w ; m , m ) = 1 in thiscase.of intervals I , where we keep track of the order within each pair, namely, of which endpoint is theorigin and which the terminus of every matching arc. Similarly, we replace { m x, } x ∈ B with a singlematching m ∈ M L of I . The corresponding permutations σ m and σ m lie in S L . From every pairof matchings m , m ∈ M L , we can construct a corresponding surface Σ m ,m as in § m , m ) accordingly. It is thus enough to show that for every m , m ∈ M L ,Ξ ( w , . . . , w (cid:96) ; m , m ) def = ( − { type- o discs of Σ m ,m } · ( − L · ∆ ( m , m ) · sign (cid:0) σ − m σ m (cid:1) = ( − (cid:96) . (A.6) Part II: Particular matchings m , m Next, we show that the equality (A.6) holds for a particular pair m , m ∈ M L , when all let-ters in w , . . . , w (cid:96) are positive, namely, when I = I + . The pair will satisfy m = m , and sosign (cid:0) σ − m σ m (cid:1) = 1. We partition the words w , . . . , w (cid:96) into singletons of even-length words andpairs of odd-length words. The matchings m and m will only pair letters of words in the sameblock of this partition. It is enough to prove (A.6) for every connected component of Σ m ,m separately.First consider the case of a single, even-length word w (which, by abuse of notation, has length2 L ). Let each of m , m pair the first interval to the second, the third to the fourth, and so forth.It is easy to check that in this case, there is exactly one type- o disc, with 2 L non-orientable match-ing arcs at its boundary, all directed, say, clockwise. In every compatible assignment of indices a (cid:96) ∗ ( m , m ), the sign ξ flips along every matching arc, and as all letters are positive, exactly halfof the matching arcs contribute ( −
1) (see Lemma A.3, Part 2), so ∆ ( m , m ) = ( − L in this case.Hence the left hand side of (A.6) is ( − · ( − L · ( − L · − (cid:96) = 1. See the left hand side of Figure A.1. Second, consider the case of a pair of odd-length words w , w , of total length 2 L . Let each of m , m pair the first interval of each word with the secondone, the third with the fourth and so on, and pair the last interval of C ( w ) with the last intervalof C ( w ). Again, it is easy to verify there is a single type- o disc in Σ m ,m , with 2 L non-orientablematching arcs. At the boundary of the type- o disc there are | w | successive o -points of C ( w ), andthen | w | o-points of C ( w ), where the matching arcs separating these two sequences are the twomatchings arcs connecting the last interval of w with the last interval of w . In every compatible42igure A.2: The left part depicts a type- o disc D with two co-directed matching arcs α and α atits boundary. Switching their termini results in splitting D into two separate type- o discs: D and D , as in the right hand side. This move flips both sign (cid:0) σ − m σ m (cid:1) and ( − { type- o discs of Σ m ,m } ,but leaves ∆ ( m , m ), and therefore also Ξ ( w , . . . , w (cid:96) ; m , m ), unchanged.assignment a (cid:96) ∗ ( m , m ), the signs ξ alternate, and so every pair of matching arcs connectingthe same two intervals of the same word contributes ( −
1) to ∆ ( m , m ). However, both matchingarcs connecting the last intervals have the same sign at their origins, and so their contributionis 1. This shows that ∆ ( m , m ) = ( − L − in this case. Hence the left hand side of (A.6) is( − · ( − L · ( − L − · (cid:96) = 2. See the right hand side ofFigure A.1. Part III: Ξ is invariant under local modifications Finally, we show that the three local modifications we specified above do not alter the value ofΞ ( w , . . . , w (cid:96) ; m , m ). As applying suitable steps of all three types leads from the instance de-scribed in part II of this proof to any given pair of matchings m , m and to any orientation ofthe 2 L letters (positive/negative), this will complete the proof. Note that none of these changesaffect the total number of letters, L , so we ought to show that they do not alter the product( − { type- o discs of Σ m ,m } · ∆ ( m , m ) · sign (cid:0) σ − m σ m (cid:1) .We begin with flipping the direction of one matching arc. Obviously, this does not change thenumber of type- o discs. It does change the sign of one of σ m or σ m , and therefore the sign of σ − m σ m , but it also changes the contribution of this matching arc to ∆ ( m , m ): this follows froma simple case-by-case analysis of whether the origin of the matching arc is in I + or in I − , andlikewise the terminus of the arc. The analysis is based on Lemma A.3 and (A.4).Next, consider a switch between the termini of two matching arcs α and α of, say, m . Thisswitch changes the sign of σ m and therefore of σ − m σ m . We distinguish between three cases andshow that in each one of them, there is one more sign change that cancels with the change insign( σ − m σ m ): • Assume that α and α both belong to the same type- o disc D and are directed along thesame orientation of ∂D . Then switching the termini splits D into two discs, and so the signof ( − { type- o discs of Σ m ,m } flips. Any compatible assignment a before the switch remainscompatible after it, and the combined sign contribution of the two arcs (as in Lemma A.3,Part 2) remains unchanged. See Figure A.2 • Assume that α and α both belong to the same type- o disc D and are directed along differentorientations of ∂D . Of the two components of ∂D \ ( α ∪ α ), one, denoted C o , has the originsof α and α as endpoints, and the other, denoted C t , has the two termini as endpoints.Switching the termini corresponds to reflecting C t – see Figure A.3. By the definition of43igure A.3: The left part depicts a type- o disc D with two counter-directed matching arcs α and α at its boundary. The two connected components of D \ ( α ∪ α ) are denoted C o and C t .Switching the termini of α and α results in reflecting C t , as in the right hand side. This moveflips both sign (cid:0) σ − m σ m (cid:1) and ∆ ( m , m ), but leaves ( − { type- o discs of Σ m ,m } , and therefore alsoΞ ( w , . . . , w (cid:96) ; m , m ), unchanged.compatible assignments, every piece of ∂D ∩ ∂ Σ is assigned a well-defined index in [2 n ], andby Lemma A.3, Part 3, two different pieces of ∂D ∩ ∂ Σ are assigned the same index if andonly if the corresponding orientations induced by ∂ Σ induce, in turn, the same orientationon ∂D . This means that if we preserve the assignment along C o , the signs along C t must beflipped. The number of type- o discs is preserved. In the terminology of Lemma A.3, type-( iii )contributions to ∆ ( m , m ) do not change. The sign contributions of the matching arcs along C o do not change. Also, the contribution of orientable arcs along C t does not change, nor doesthe type-( i ) contribution of α and α . However, ∆ ( m , m ) does flip. To see this, denoteby ∂ , ∂ the two connected components of ∂D ∩ ∂ Σ which contain (as endpoints) the twotermini t ( α ) and t ( α ), respectively ( ∂ and ∂ may be equal). Let i and i be the indicescorresponding to ∂ and ∂ in some compatible assignment (before the flip of α and α ).If the orientation of ∂ Σ along ∂ and ∂ induces the same orientation on ∂D , then i = i andof the two intervals at the termini, one is in I + and the other in I − . Thus the total type-( ii )contribution of α and α flips. As C t contains an even number of non-orientable matchingarcs in this case, the total sign contribution of the non-orientable arcs along C t is preserved(as in the proof of Lemma A.3, Part 5).If the orientation of ∂ Σ along ∂ and ∂ induces different orientations on ∂D , then i = (cid:98) i and the two letters at the termini are both positive or both negative. In this case, the totaltype-( ii ) contribution of α and α is unchanged, but the contribution of non-orientable arcsalong C t is flipped because the number of these arcs is odd. • The third and last case is the one where α and α belong to different type- o discs. Switchingtheir termini then leads to merging the two discs into one. In the united type- o disc, thetwo arcs are “co-oriented”, so this case is the reverse of the first one, and ∆ ( m , m ) remainsunchanged.The final small change we consider is that of flipping some letter from being positive to negative,namely, of flipping an interval in some C ( w k ) from I ± to I ∓ . Here, sign (cid:0) σ − m σ m (cid:1) is unchanged.By the first local modification in this part of the proof, we may assume without loss of generalitythat this letter is at the termini of two matching arcs, α and α . A similar argument as in theprevious paragraph would show that: 44 Assume that α and α belong to the same type- o disc D with the same orientation. The flipof the letter then splits D into two type- o discs. Denote by ∂ and ∂ the pieces of ∂D ∩ ∂ Σ atthe termini. They must be counter-oriented. We may preserve the same assignment of indicesas before the flip of the letter, but then the type-( ii ) contribution of both arcs flips when theletter is flipped. No other change in sign contributions occurs. • Assume that α and α belong to the same type- o disc D with opposite orientations. Theflip of the letter preserves the number of type- o discs and corresponds to reflecting C t . Here ∂ and ∂ are co-oriented and the signs along C t must be flipped. There is no change to∆ ( m , m ): the total type-( ii ) contribution of α and α is 1 before and after the flip, andthe number of non-orientable arcs along C t is even. • If α and α belong to different type- o disc, the flip is the reverse of the first case.This completes the proof of Theorem 1.2.We now have the analog of Corollary 3.5 for G = Sp: Corollary A.6.
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Doron Puder,School of Mathematical Sciences,Tel Aviv University,Tel Aviv, 6997801, Israel [email protected]@gmail.com