A random cover of a compact hyperbolic surface has relative spectral gap 3 16 −ε
AA random cover of a compact hyperbolic surface has relativespectral gap − ε Michael Magee, Fr´ed´eric Naud, Doron PuderAugust 7, 2020
Abstract
Let X be a compact connected hyperbolic surface, that is, a closed connected orientablesmooth surface with a Riemannian metric of constant curvature -1. For each n ∈ N , let X n be arandom degree- n cover of X sampled uniformly from all degree- n Riemannian covering spaces of X . An eigenvalue of X or X n is an eigenvalue of the associated Laplacian operator ∆ X or ∆ X n .We say that an eigenvalue of X n is new if it occurs with greater multiplicity than in X . Weprove that for any ε >
0, with probability tending to 1 as n → ∞ , there are no new eigenvaluesof X n below − ε . We conjecture that the same result holds with replaced by . Contents ε -adapted tiled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Description of the octagons-vs-boundary algorithm . . . . . . . . . . . . . . . . . . . 284.7 Analysis of the octagons-vs-boundary algorithm . . . . . . . . . . . . . . . . . . . . . 291 a r X i v : . [ m a t h . SP ] A ug .8 Resolutions from the octagons-vs-boundary algorithm . . . . . . . . . . . . . . . . . 31 n and preliminary estimates . . . . . . . . . . . . . . . . . . . . . . . . 355.3 The zero regime of b ν , ˇ b ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 The intermediate regime of b ν , ˇ b ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5 The large regime of b ν , ˇ b ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.6 Assembly of analytic estimates for Ξ n . . . . . . . . . . . . . . . . . . . . . . . . . . 415.7 A new expression for Ξ d ν =1 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.8 Understanding | X ∗ n ( Y, J ) | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.9 Bounds on E emb n ( Y ) for ε -adapted Y . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ε -adapted surfaces . . . . . . . . . . . . . . . . . . 496.3 Part II: The contribution from ε -adapted surfaces . . . . . . . . . . . . . . . . . . . . 51 Spectral gap is a fundamental concept in mathematics and related sciences as it governs the rateat which a process converges towards its stationary state. The question that motivates this paperis whether random objects have large, or even optimal, spectral gaps. This will be made precisebelow.One of the simplest examples of spectral gap is the spectral gap of a graph.
The spectrum ofa graph G on n vertices is the collection of eigenvalues of its adjacency matrix A G . Assuming that G is d -regular, the largest eigenvalue occurs at d and is simple if and only if G is connected. Thismeans, writing λ = d ≥ λ ≥ λ ≥ · · · ≥ λ n − for the eigenvalues of A G , then there is a spectral gap between λ and λ (i.e. λ > λ ) if and only if G is connected. In fact, the Cheeger inequalities for graphs due to Alon and Milman [AM85] showthat the size of the spectral gap (i.e. λ − λ ) quantifies how difficult it is, roughly speaking, toseparate the vertices of G into two sets, each not too small, with few edges between them. This isin tension with the fact that a d -regular graph is sparse. Sparse yet highly-connected graphs arerelevant to many real-world examples .However, a result of Alon and Boppana [Nil91] puts a sharp bound on what one can achieve: fora sequence of d -regular graphs G n on n vertices, as n → ∞ , λ ( G n ) ≥ √ d − − o (1). The trivialeigenvalues of a graph occur at d , and if G has a bipartite component, at − d . A connected d -regulargraph with all its non-trivial eigenvalues in the interval [ − √ d − , (cid:112) d −
1] is called a
Ramanujangraph after Lubotzky, Phillips, and Sarnak [LPS88]. A famous conjecture of Alon [Alo86], now atheorem due to Friedman [Fri08], states that for any ε > d -regular graph on n vertices,chosen uniformly amongst such graphs, has all its non-trivial eigenvalues bounded in absolute valueby 2 √ d − ε as n → ∞ . In other words, almost all d -regular graphs have almost optimal spectral Following Barzdin and Kolmogorov [BK93], consider the network of neurons in a human brain.
In the rest of the paper, if an event depending on a parameter n holds with probability tendingto 1 as n → ∞ , then we say it holds asymptotically almost surely (a.a.s.). Friedman conjectured in [Fri03] that the following extension of Alon’s conjecture holds. Givenany finite graph G there is a notion of a degree- n cover G n of the graph. Elements of the spectrum of G n that are not elements of the spectrum of G are called new eigenvalues of G n . Friedmanconjectured that for a fixed finite graph G , for any ε > n cover of G a.a.s. has nonew eigenvalues larger than ρ ( G ) + ε , where ρ ( G ) is the spectral radius of the adjacency operator ofthe universal cover of G , acting on (cid:96) functions. Friedman’s conjecture has recently been proved ina breakthrough by Bordenave and Collins [BC19]. The focus of this paper is the extension of Alon’s and Friedman’s conjectures to compact hyper-bolic surfaces. A hyperbolic surface is a smooth surface of constant curvature −
1. In this paper, all surfaceswill be orientable. By uniformization, a connected compact hyperbolic surface can be realized asΓ \ H where Γ is a discrete subgroup of PSL ( R ) and H = { x + iy : x, y ∈ R , y > } is the hyperbolic upper half plane, upon which PSL ( R ) acts via M¨obius transformations preservingthe hyperbolic metric dx + dy y . Let X = Γ \ H be a connected compact hyperbolic surface. Topologically, X is a connected closedsurface of some genus g ≥ H on H is invariant under PSL ( R ), it descends to a differentiable operatoron C ∞ ( X ) and extends to a non-negative, unbounded, self-adjoint operator ∆ X on L ( X ). Thespectrum of ∆ X consists of real eigenvalues0 = λ ( X ) ≤ λ ( X ) ≤ · · · ≤ λ n ( X ) ≤ · · · with λ i → ∞ as i → ∞ . The same discussion also applies if we drop the condition that X isconnected . We have λ ( X ) < λ ( X ) if and only if X is connected, as for graphs. With Friedman’sconjecture in mind, we also note that the spectrum of ∆ H is absolutely continuous and supportedon the interval [ , ∞ ) [Bor16, Thm. 4.3].To state an analog of the Alon/Friedman conjecture for surfaces, we need a notion of a randomcover. Suppose X is a compact connected hyperbolic surface, and suppose ˜ X is a degree- n Rieman-nian cover of X . Fix a point x ∈ X and label the fiber above it by [ n ] def = { , . . . , n } . There is amonodromy map π ( X, x ) → Hom( π ( X, x ) , S n )that describes how the fiber of x is permuted when following lifts of a closed loop from X to ˜ X . The precise definition of a cover of a graph is not important here; only that it is analogous to a covering space ofa surface. We also take multiplicities into account in this statement. In which case X is a finite union of connected compact hyperbolic surfaces, each of which can be realized as aquotient of H . S n is the symmetric group of permutations of the set [ n ]. The cover ˜ X is uniquely determinedby the monodromy homomorphism. Let g denote the genus of X . We fix an isomorphism π ( X, x ) ∼ = Γ g def = (cid:104) a , b , a , b , . . . , a g , b g | [ a , b ] · · · [ a g , b g ] = 1 (cid:105) . (1.1)Now, given any φ ∈ X g,n def = Hom(Γ g , S n )we can construct a covering of X whose monodromy map is φ as follows. Using the fixed isomorphismof (1.1), we have a free properly discontinuous action of Γ g on H by isometries. Define a new actionof Γ g on H × [ n ] by γ ( z, i ) = ( γz, φ [ γ ]( i )) . The quotient of H × [ n ] by this action is named X φ and is a hyperbolic covering of X with mon-odromy φ . This construction establishes a one-to-one correspondence between φ ∈ X g,n and degree- n coverings with a labeled fiber X φ of X .As for graphs, any eigenvalue of ∆ X will also be an eigenvalue of ∆ X φ : every eigenfunction of∆ X can be pulled back to an eigenfunction of ∆ X φ with the same eigenvalue. We say an eigenvalueof ∆ X φ is new if it is not one of ∆ X , or more generally, appears with greater multiplicity in X φ .To pick a random cover of X , we simply use the uniform probability measure on the finite set X g,n .Recall we say an event that pertains to any n holds a.a.s. if it holds with probability tending to oneas n → ∞ . The analog of Friedman’s conjecture for surfaces is the following. Conjecture 1.1.
Let X be a compact connected hyperbolic surface. Then for any ε > , a.a.s. spec (cid:0) ∆ X φ (cid:1) ∩ (cid:20) , − ε (cid:21) = spec (∆ X ) ∩ (cid:20) , − ε (cid:21) and the multiplicities on both sides are the same.Remark . We have explained the number in terms of the spectrum of the Laplacian on thehyperbolic plane. The number also features prominently in Selberg’s eigenvalue conjecture [Sel65],that states for X = SL ( Z ) \ H , the (deterministic) family of congruence covers of X never have neweigenvalues below . Although Selberg’s conjecture is for a finite-area, non-compact hyperbolic orb-ifold, the Jacquet-Langlands correspondence [JL70] means that it also applies to certain arithmeticcompact hyperbolic surfaces.
Remark . In [Wri19, Problem 10.4], Wright asks, for random compact hyperbolic surfaces sam-pled according to the Weil-Petersson volume form on the moduli space of genus g closed hyperbolicsurfaces, whether lim inf g →∞ ( P ( λ > )) >
0. See §§ X n with genustending to ∞ such that λ ( X n ) → . Conjecture 1.1 offers a new route to resolving this problemvia the probabilistic method, since it is known by work of Jenni [Jen84] that there exists a genus2 hyperbolic surface X with λ ( X ) > and this X can be taken as the base surface in Conjecture1.1.The main theorem of the paper, described in the title, is the following. Theorem 1.4.
Let X be any closed connected hyperbolic surface. Then for any ε > , a.a.s. spec (cid:0) ∆ X φ (cid:1) ∩ (cid:20) , − ε (cid:21) = spec (∆ X ) ∩ (cid:20) , − ε (cid:21) nd the multiplicities on both sides are the same.Remark . The appearance of the number in Theorem 1.4 is essentially for the same reasonthat appears in [MN20] (note that = (1 − ), and eigenvalues of the Laplacian are naturallyparameterized as s (1 − s )). Ultimately, the appearance of can be traced back to the methodof Broder and Shamir [BS87b] who prove that a.a.s. a random 2 d -regular graph on n vertices has λ ≤ O (cid:16) d (cid:17) , using an estimate analogous to Theorem 1.10 below. Remark . More mysteriously, is also the lower bound that Selberg obtained for the smallestnew eigenvalue of a congruence cover of the modular curve SL ( Z ) \ H , in the same paper [Sel65] ashis eigenvalue conjecture. In this context, the number arises ultimately from bounds on Kloostermansums due to Weil [Wei48] that follow from Weil’s resolution of the Riemann hypothesis for curves overfinite fields. The state of the art on Selberg’s eigenvalue conjecture, after decades of intermediateresults [GJ78, Iwa89, LRS95, Iwa96, KS02], is due to Kim and Sarnak [Kim03] who produced aspectral gap of size for congruence covers of SL ( Z ) \ H .It was pointed out to us by A. Kamber that our methods also yield the following estimate onthe density of new eigenvalues of a random cover. Theorem 1.7.
Let ≤ λ i ( X φ ) ≤ λ i ( X φ ) ≤ · · · ≤ λ i k ( φ ) ( X φ ) ≤ denote the collection of new eigenvalues of ∆ X φ at most , included with multiplicity. For each ofthese, we write λ i j = s i j (1 − s i j ) with s i j = s i j ( X φ ) ∈ (cid:2) , (cid:3) . For any ε > and σ ∈ (cid:0) , (cid:1) , a.a.s. (cid:8) ≤ j ≤ k ( φ ) : λ i j < σ (1 − σ ) (cid:9) = (cid:8) ≤ j ≤ k ( φ ) : s i j > σ (cid:9) ≤ n − σ + ε . (1.2) Remark . The estimate (1.2) was established by Iwaniec [Iwa02, Thm 11.7] for congruencecovers of SL ( Z ) \ H . Although Iwaniec’s theorem has been generalized in various directions [Hux86,Sar87, Hum18], as far as we know, Iwaniec’s result has not been directly improved, so speakingabout density of eigenvalues, Theorem 1.7 establishes for random covers the best result known inthe arithmetic setting for eigenvalues above the Kim-Sarnak bound [Kim03]. Density estimatessuch as Theorem 1.7 have applications to the cutoff phenomenon on hyperbolic surfaces by work ofGolubev and Kamber [GK19].We prove Theorems 1.4 and 1.7 using Selberg’s trace formula in §
2. We use as a ‘black-box’in this method a statistical result (Theorem 1.10) about the expected number of fixed points of afixed γ ∈ Γ g under a random φ .If π ∈ S n then we write fix ( π ) for the number of fixed points of the permutation π . Given anelement γ ∈ Γ g , we let fix γ be the function fix γ : X g,n → Z , fix γ ( φ ) def = fix ( φ ( γ )) . We write E g,n [ fix γ ] for the expected value of fix γ with respect to the uniform probability measureon X g,n . In [MP20], the first and third named authors proved the following theorem. Theorem 1.9.
Let g ≥ and (cid:54) = γ ∈ Γ g . If q ∈ N is maximal such that γ = γ q for some γ ∈ Γ g ,then, as n → ∞ , E g,n [ fix γ ] = d ( q ) + O (cid:0) n − (cid:1) , here d ( q ) is the number of divisors of q . In the current paper, we need an effective version of Theorem 1.9 that controls the dependenceof the error term on γ . We need this estimate only for γ that are not a proper power. For γ ∈ Γ g ,we write (cid:96) w ( γ ) for the cyclic-word-length of γ , namely, for the length of a shortest word in thegenerators a , b , . . . , a g , b g of Γ g that represents an element in the conjugacy class of γ in Γ g . Theeffective version of Theorem 1.9 that we prove here is the following. Theorem 1.10.
For each genus g ≥ , there is a constant A = A ( g ) such that for any c > , if (cid:54) = γ ∈ Γ g is not a proper power of another element in Γ g and (cid:96) w ( γ ) ≤ c log n then E g,n [ fix γ ] = 1 + O c (cid:18) (log n ) A n (cid:19) . The implied constant in the big-O depends on c .Remark . In the rest of the paper, just to avoid complications in notation and formulas thatwould obfuscate our arguments, we give the proof of Theorem 1.10 when g = 2. The extensionto arbitrary genus is for the most part obvious: if it is not at some point, we will point out thenecessary changes.The proof of Theorem 1.10 takes up the bulk of the paper, spanning § §
6. The proof of Theorem1.10 involves delving into the proof of Theorem 1.9 and refining the estimates, as well as introducingsome completely new ideas.
The Brooks-Makover model
The first study of spectral gap for random surfaces in the liter-ature is due to Brooks and Makover [BM04] who form a model of a random compact surface asfollows. Firstly, for a parameter n , they glue together n copies of an ideal hyperbolic triangle wherethe gluing scheme is given by a random trivalent ribbon graph. Their model for this random ribbongraph is a modification of the Bollob´as bin model from [Bol88]. This yields a random finite-area,non compact hyperbolic surface. Then they perform a compactification procedure to obtain a ran-dom compact hyperbolic surface X BM ( n ). The genus of this surface is not deterministic, however.Brooks and Makover prove that for this random model, there is a non-explicit constant C > n → ∞ ) λ ( X BM ( n )) ≥ C. Theorem 1.4 concerns a different random model, but improves on the Brooks-Makover result in twoimportant ways: the bound on new eigenvalues is explicit, and this bound is independent of thecompact hyperbolic surface X with which we begin.It is also worth mentioning a recent result of Budzinski, Curien, and Petri [BCP19, Thm. 1] whoprove that the ratios diameter( X BM ( n ))log n converge to 2 in probability as n → ∞ ; they also observe that this is not the optimal value by afactor of 2. The Weil-Petersson model
Another reasonable model of random surfaces comes from the Weil-Petersson volume form on the moduli space M g of compact hyperbolic surfaces of genus g . Let6 WP ( g ) denote a random surface in M g sampled according to the (normalized) Weil-Peterssonvolume form. Mirzakhani proved in [Mir13, §§§ g → ∞ , λ ( X WP ( g )) ≥ (cid:18) log 22 π + log 2 (cid:19) ≈ . . We also note recent work of Monk [Mon20] who gives estimates on the density of eigenvalues below of the Laplacian on X WP ( g ). Prior work of the authors
In some sense, the closest result to Theorem 1.4 in the literature isdue to the first and second named authors of the paper [MN20], but it does not apply to compactsurfaces, rather to infinite area convex co-compact hyperbolic surfaces. Because these surfaces haveinfinite area, their spectral theory is more involved. We will focus on one result of [MN20] toillustrate the comparison with this paper.Suppose X is a connected non-elementary, non-compact, convex co-compact hyperbolic surface.The spectral theory of X is driven by a critical parameter δ = δ ( X ) ∈ (0 , X . If δ > then results of Patterson [Pat76] and Lax-Phillips [LP81] say that the bottom of the spectrum of X is a simple eigenvalue at δ (1 − δ ) and there are finitely many eigenvalues in the range [ δ (1 − δ ) , ).In [MN20], a model of a random degree- n cover of X was introduced that is completely analogousto the one used here; the only difference in the construction is that the fundamental group of X isa free group F r and hence one uses random φ ∈ Hom( F r , S n ) to construct the random surface X φ .The following theorem was obtained in [MN20, Thm. 1.3.]. Theorem 1.12.
Assume that δ = δ ( X ) > . Then for any σ ∈ (cid:0) δ, δ (cid:1) , a.a.s. spec (cid:0) ∆ X φ (cid:1) ∩ [ δ (1 − δ ) , σ (1 − σ )] = spec (∆ X ) ∩ [ δ (1 − δ ) , σ (1 − σ )] (1.3) and the multiplicities on both sides are the same. Although Theorem 1.12 is analogous to Theorem 1.4 (for compact X , δ ( X ) = 1), the methodsused in [MN20] have almost no overlap with the methods used here. For infinite area X , thefundamental group is free, so the replacement of Theorem 1.10 was already known by results ofBroder-Shamir [BS87b] and the third named author [Pud15]. The challenge in [MN20] was todevelop bespoke analytic machinery to access these estimates.Conversely, in the current paper, the needed analytic machinery already exists (Selberg’s traceformula) and rather, it is the establishment of Theorem 1.10 that is the main challenge here,stemming from the non-free fundamental group Γ g . First, we explain the outline of the proof of Theorem 1.4 from Theorem 1.10. Theorem 1.7 alsofollows from Theorem 1.10 using the same ideas. Both proofs are presented in full in § d -regular graph has a large spectral gap. For us, the Selberg traceformula replaces a more elementary formula for the trace of a power of the adjacency operator of agraph in terms of closed paths in the graph. 7o briefly explain this method, we let λ i ( X φ ) ≤ λ i ( X φ ) ≤ · · · denote the new eigenvalues of X φ , with multiplicities included, and for each of these write λ i k ( X φ ) = + r i k ( X φ ) . Let Γ denotethe fundamental group of X . By taking the difference of the Selberg trace formula for X φ and thatfor X we obtain a formula of the form (cid:88) new eigenvalues λ of X φ F ( λ ) = (cid:88) [ γ ] ∈ C (Γ) G ( γ ) ( fix γ ( φ ) − , (1.4)where C (Γ) is the collection of conjugacy classes in Γ, and F and G are interdependent functionsthat depend on n . The way we choose F and G together is to ensure • F ( λ ) is non-negative for any possible λ , and large if λ is an eigenvalue we want to forbid, and • G ( γ ) localizes to γ with (cid:96) w ( γ ) ≤ c log n for some c = c ( X ).By taking expectations of (1.4) we obtain E (cid:88) new eigenvalues λ of X φ F ( λ ) = (cid:88) [ γ ] ∈ C (Γ) G ( γ ) E [ fix γ ( φ ) − . (1.5)The proof will conclude by bounding the right hand side and applying Markov’s inequality toconclude that there are no new eigenvalues in the desired forbidden region. Since G is well-controlledin our proof, it remains to estimate each term E [ fix γ ( φ ) − • If γ is the identity, then G (1) is easily analyzed, and E [ fix γ ( φ ) −
1] = n − • If γ is a proper power of a non-trivial element of Γ, then we use a trivial bound E [ fix γ ( φ ) − ≤ n −
1, so we get no gain from the expectation. On the other hand, the contribution to (cid:88) [ γ ] ∈ C (Γ) G ( γ )from these elements is negligible. Intuitively, this is because the number of elements of Γ with (cid:96) w ( γ ) ≤ L and that are proper powers is (exponentially) negligible compared to the totalnumber of elements. • If γ is not a proper power and not the identity, then we use Theorem 1.10 to obtain E [ fix γ ( φ ) −
1] = O X (cid:16) (log n ) A n (cid:17) . Thus for ‘most’ summands in the right-hand side of (1.5) we obtain a significantgain from the expectation.Assembling all these estimates together gives a sufficiently upper strong bound on (1.5) to obtainTheorem 1.4 via Markov’s inequality. Proof of Theorem 1.10
To understand the proof of Theorem 1.10, we suggest that the reader first read the overview below,then § § § g = 2 in this overview andwe will forgo precision to give a bird’s-eye view of the proof.8ixing an octagonal fundamental domain for X , any X φ is tiled by octagons; this tiling comeswith some extra labelings of edges corresponding to the generators of Γ. Any labeled 2-dimensionalCW-complex that can occur as a subcomplex of some X φ is called a tiled surface. For any tiledsurface Y , we write E emb n ( Y ) for the expected number, when φ is chosen uniformly at random inHom(Γ , S n ), of embedded copies of Y in X φ .In the previous paper [MP20], we axiomatized certain collections R of tiled surfaces, dependingon γ , that have the property that E ,n [ fix γ ] = (cid:88) Y ∈R E emb n ( Y ) . (1.6)These collections are called resolutions. Here we have oversimplified the definitions to give anoverview of the main ideas.In [MP20], we chose a resolution, depending on γ , that consisted of two special types of tiledsurfaces: those that are boundary reduced or strongly boundary reduced. The motivation forthese definitions is that they make our methods for estimating E emb n ( Y ) more accurate. To give anexample, if Y is strongly boundary reduced then we prove that for Y fixed and n → ∞ , we obtain E emb n ( Y ) = n χ ( Y ) (cid:0) O Y (cid:0) n − (cid:1)(cid:1) . (1.7)However, the implied constant depends on Y , and in the current paper we have to control uniformlyall γ with (cid:96) w ( γ ) ≤ c log n . The methods of [MP20] are not good enough for this goal. To deal withthis, we introduce in Definition 4.12 a new type of tiled surface called ‘ ε -adapted’ (for some ε ≥ octagons-vs-boundary algorithm that given γ , produces a finite resolution R as in (1.6) such thatevery Y ∈ R is either • ε -adapted for some ε = ε ( g ) >
0, or • boundary reduced, with the additional condition that d ( Y ) < f ( Y ) < − χ ( Y ), where d ( Y ) isthe length of the boundary of Y and f ( Y ) is the number of octagons in Y .Any Y ∈ R has d ( Y ) ≤ c (cid:48) (log n ) and f ( Y ) ≤ c (cid:48) (log n ) given that (cid:96) w ( γ ) ≤ c log n (Corollary 4.25).The fact that we maintain control on these quantities during the algorithm is essential. However, adefect of this algorithm is that we lose control of how many ε -adapted Y ∈ R there are of a givenEuler characteristic. In contrast, in the algorithms of [MP20] we control, at least, the number ofelements in the resolution of Euler characteristic zero. We later have to work to get around this.We run the octagons-vs-boundary algorithm for a fixed ε = to obtain a resolution R . Let usexplain the benefits of this resolution we have constructed. The ε -adapted Y ∈ R contribute themain contributions to (1.6), and the merely boundary reduced Y contribute something negligible.Indeed, we prove for any boundary reduced Y ∈ R in the regime of parameters we care about,that E emb n ( Y ) (cid:28) ( A d ( Y )) A d ( Y ) n χ ( Y ) , (1.8)where A > g . This is the result of combining Corollary 3.5, Theorem 5.1,Proposition 5.11 and Lemma 4.5; the proof is by carefully effectivizing the arguments of [MP20]. Some of the notation we use in this section is detailed in Section 1.3. d ( Y ) is enough so that whencombined with d ( Y ) < f ( Y ) < − χ ( Y ) we obtain E emb n ( Y ) (cid:28) ( A d ( Y )) A f ( Y ) n − f ( Y ) (cid:28) (cid:18) c (cid:48) (log n ) n (cid:19) f ( Y ) . This is good enough that it can simply be combined with counting all possible Y with d ( Y ) ≤ c (cid:48) (log n ) and f ( Y ) ≤ c (cid:48) (log n ) to obtain that the non- ε -adapted surfaces in R contribute (cid:28) (log n ) A n to (1.6) for A = A ( g ) >
0. This is Proposition 6.1.So from now on assume Y ∈ R is ε -adapted and we explain how to control the contributions to(1.6) from these remaining Y . We first prove that there is a rational function Q Y such that E emb n ( Y ) = n χ ( Y ) (cid:18) Q Y ( n ) + O (cid:18) n (cid:19)(cid:19) (cid:18) O (cid:18) (log n ) n (cid:19)(cid:19) , (1.9)where the implied constants hold for any ε -adapted Y ∈ R as long as (cid:96) w ( γ ) ≤ c log n (Theorem5.1, Proposition 5.12 and Corollary 5.21). In fact, this expression remains approximately valid forthe same Y if n is replaced throughout by m with m ≈ (log n ) B for some B >
0; this will becomerelevant momentarily.The rational function Q Y is new to this paper; it appears through Corollary 5.15 and Lemma5.20 and results from refining the representation-theoretic arguments in [MP20]. The descriptionof Q Y is in terms of Stallings core graphs [Sta83], and related to the theory of expected number offixed points of words in the free group. One unusual thing is that our combinatorial description of Q Y does not immediately tell us the order of growth of Q Y ( n ). On the other hand, we know enoughabout Q Y (for example, the possible location of the poles, and some positivity properties) so thatwe can ‘black-box’ results from [MP20] to learn that if Y is fixed and n → ∞ , Q Y ( n ) →
1. Thisalgebraic property of Q Y , together with a priori facts about Q Y , allow us to use (1.9) to establishthe two following important inequalities: E emb n ( Y ) = n χ ( Y ) (cid:18) O c (cid:18) (log n ) n (cid:19) + O (cid:18) mn E emb m ( Y ) m χ ( Y ) (cid:19)(cid:19) (1.10) n χ ( Y ) (cid:28) mn E emb m ( Y ) , if χ ( Y ) < m ≈ (log n ) B is much smaller than n . These inequalities are provided by Proposition 5.26and Corollary 5.25.Let us now explain the purpose of (1.11) and (1.10). By black-boxing the results of [MP20]one more time, we learn that there is exactly one ε -adapted Y ∈ R with χ ( Y ) = 0, and nonewith χ ( Y ) >
0. This single Y with χ ( Y ) = 0 contributes the main term of Theorems 1.9 and 1.10through (1.10). Any other term coming from ε -adapted Y can be controlled in terms of E emb m ( Y )using (1.10) and (1.11). These errors could accumulate, but we can control them all at once byusing (1.6) in reverse with n replaced by m to obtain (cid:88) Y ∈R E emb m ( Y ) = E ,m [ fix γ ] ≤ m ≈ (log n ) B . Putting the previous arguments together proves Theorem 1.10.10 .3 Notation
The commutator of two group elements is [ a, b ] def = aba − b − . For m, n ∈ N , m ≤ n , we use thenotation [ m, n ] for the set { m, m + 1 , . . . , n } and [ n ] for the set { , . . . , n } . For q, n ∈ N with q ≤ n we use the Pochammer symbol ( n ) q def = n ( n − · · · ( n − q + 1) . For real-valued functions f, g that depend on a parameter n we write f = O ( g ) to mean there existconstants C, N > n > N , | f ( n ) | ≤ Cg ( n ). We write f (cid:28) g if there are C, N > f ( n ) ≤ Cg ( n ) for n > N . We add constants as a subscript to the big O or the (cid:28) sign tomean that the constants C and N depend on these other constants, for example, f = O ε ( g ) meansthat both C = C ( ε ) and N = N ( ε ) may depend on ε . If there are no subscripts, it means theimplied constants depend only on the genus g , which is fixed throughout most of the paper. Weuse the notation f (cid:16) g to mean f (cid:28) g and g (cid:28) f ; the use of subscripts is the same as before. Acknowledgments
We thank Amitay Kamber for the observation that our methods prove Theorem 1.7.Fr´ed´eric Naud is supported by Institut Universitaire de France. Doron Puder was supportedby the Israel Science Foundation: ISF grant 1071/16. This project has received funding fromthe European Research Council (ERC) under the European Union’s Horizon 2020 research andinnovation programme (grant agreement No 850956).
Here we describe the main tool of this §
2: Selberg’s trace formula for compact hyperbolic surfaces.Let C ∞ c ( R ) denote the infinitely differentiable real functions on R with compact support. Givenan even function ϕ ∈ C ∞ c ( R ), its Fourier transform is defined by (cid:98) ϕ ( ξ ) def = (cid:90) ∞−∞ ϕ ( x ) e − ixξ dx for any ξ ∈ C . As ϕ ∈ C ∞ c ( R ), the integral above converges for all ξ ∈ C to an entire function.Given a compact hyperbolic surface X , we write L ( X ) for the set of closed oriented geodesicsin X . A geodesic is called primitive if it is not the result of repeating another geodesic q times for q ≥
2. Let P ( X ) denote the set of closed oriented primitive geodesics on X . Every closed geodesic γ has a length (cid:96) ( γ ) according to the hyperbolic metric on X . Every closed oriented geodesic γ ∈ L ( X )determines a conjugacy class [˜ γ ] in π ( X, x ) for any basepoint x . Clearly, a closed oriented geodesicin X is primitive if and only if the elements of the corresponding conjugacy class are not properpowers in π ( X, x ). For γ ∈ L ( X ) we write Λ( γ ) = (cid:96) ( γ ) where γ is the unique primitive closedoriented geodesic such that γ = γ q for some q ≥ heorem 2.1 (Selberg’s trace formula) . Let X be a compact hyperbolic surface and let λ ( X ) ≤ λ ( X ) ≤ · · · ≤ λ n ( X ) ≤ · · · denote the spectrum of the Laplacian on X . For i ∈ N ∪ { } let r i ( X ) def = (cid:113) λ i ( X ) − if λ i ( X ) > / i (cid:113) − λ i ( X ) if λ i ( X ) ≤ / . Then for any even ϕ ∈ C ∞ c ( R ) ∞ (cid:88) i =0 (cid:98) ϕ ( r i ( X )) = area( X )4 π (cid:90) ∞−∞ r (cid:98) ϕ ( r ) tanh( πr ) dr + (cid:88) γ ∈L ( X ) Λ( γ )2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) ϕ ( (cid:96) ( γ )) . (Both sides of the formula are absolutely convergent). We will also need a bound on the number of closed oriented geodesics with length (cid:96) ( γ ) ≤ T . Infact we only need the following very soft bound from e.g. [Bus10, Lemma 9.2.7]. Lemma 2.2.
For a compact hyperbolic surface X , there is a constant C = C ( X ) such that |{ γ ∈ L ( X ) : (cid:96) ( γ ) ≤ T }| ≤ Ce T . Much sharper versions of this estimate are known, but Lemma 2.2 suffices for our purposes.Suppose that X is a connected compact hyperbolic surface. We fix a basepoint x ∈ X and anisomorphism π ( X, x ) ∼ = Γ g as in (1.1) where g ≥ X . If γ is a closed orientedgeodesic, by abuse of notation we let (cid:96) w ( γ ) denote the minimal word-length of an element in theconjugacy class in Γ g specified by γ (on page 6 we used the same notation for an element of Γ g ). Wewant to compare (cid:96) ( γ ) and (cid:96) w ( γ ). We will use the following simple consequence of the ˘Svarc-Milnorlemma [BH99, Prop. 8.19]. Lemma 2.3.
With notations as above, there exist constants K , K ≥ depending on X such that (cid:96) w ( γ ) ≤ K (cid:96) ( γ ) + K . We now fix a function ϕ ∈ C ∞ c ( R ) which has the following key properties:1. ϕ is non-negative and even.2. Supp( ϕ ) = ( − , (cid:99) ϕ satisfies (cid:99) ϕ ( ξ ) ≥ ξ ∈ R ∪ i R . Proof that such a function exists.
Let ψ be a C ∞ , even, real-valued non-negative function whosesupport is exactly ( − , ). Let ϕ = ψ (cid:63) ψ where ψ (cid:63) ψ ( x ) def = (cid:90) R ψ ( x − t ) ψ ( t ) dt. Then ϕ has the desired properties. 12e now fix a function ϕ as above and for any T > ϕ T ( x ) def = ϕ (cid:16) xT (cid:17) . Lemma 2.4.
For all ε > , there exists C ε > such that for all t ∈ R ≥ and for all T > (cid:99) ϕ T ( it ) ≥ C ε T e T (1 − ε ) t . Proof.
First observe that (cid:99) ϕ T ( it ) = T (cid:99) ϕ ( T it ) = T (cid:90) R ϕ ( x ) e T xt dx.
Using t ≥ ϕ ) = ( − ,
1) with ϕ non-negative, we have for some C ε > (cid:99) ϕ T ( it ) ≥ T (cid:90) − ε ϕ ( x ) e T xt dx ≥ T C ε e T (1 − ε ) t . Let X be a genus g compact hyperbolic surface and let X φ be the cover of X corresponding to φ ∈ Hom(Γ g , S n ) constructed in the introduction. In what follows we let T = 4 log n. For every γ ∈ L ( X ), we pick ˜ γ ∈ Γ g in the conjugacy class in Γ g corresponding to γ (so in particular (cid:96) w (˜ γ ) = (cid:96) w ( γ )). Every closed oriented geodesic γ in X φ covers, via the Riemannian covering map X φ → X, a unique closed oriented geodesic in X that we will call π ( γ ). This gives a map π : L ( X φ ) → L ( X ) . Note that (cid:96) ( γ ) = (cid:96) ( π ( γ )) . Given γ in P ( X ), | π − ( γ ) | = fix ˜ γ ( φ ), recalling that fix ˜ γ ( φ ) is the numberof fixed points of φ (˜ γ ). We have area( X φ ) = n · area( X ). Now applying Theorem 2.1 to X φ withthe function ϕ T gives ∞ (cid:88) i =0 (cid:99) ϕ T ( r i ( X φ )) = area( X φ )4 π (cid:90) ∞−∞ r (cid:99) ϕ T ( r ) tanh( πr ) dr + (cid:88) γ ∈L ( X φ ) Λ( γ )2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) ϕ T ( (cid:96) ( γ ))= n · area( X )4 π (cid:90) ∞−∞ r (cid:99) ϕ T ( r ) tanh( πr ) dr + (cid:88) γ ∈L ( X ) (cid:88) γ (cid:48) ∈ π − ( γ ) Λ( γ (cid:48) )2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) ϕ T ( (cid:96) ( γ ))= n · area( X )4 π (cid:90) ∞−∞ r (cid:99) ϕ T ( r ) tanh( πr ) dr + (cid:88) γ ∈P ( X ) fix ˜ γ ( φ ) (cid:96) ( γ )2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) ϕ T ( (cid:96) ( γ ))+ (cid:88) γ ∈L ( X ) −P ( X ) (cid:88) γ (cid:48) ∈ π − ( γ ) Λ( γ (cid:48) )2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) ϕ T ( (cid:96) ( γ )) , γ ∈ L ( X φ ), (cid:96) ( γ ) = (cid:96) ( π ( γ )), and in the thirdequality we used that if γ ∈ P ( X ), then γ (cid:48) ∈ P ( X φ ) for all γ (cid:48) ∈ π − ( γ ), so Λ( γ (cid:48) ) = Λ( γ ) = (cid:96) ( γ ).Let i , i , i , . . . be a subsequence of 1 , , , . . . such that0 ≤ λ i ( X φ ) ≤ λ i ( X φ ) ≤ · · · are the new eigenvalues of X φ . Thus λ i ( X φ ) is the smallest new eigenvalue of X φ . Taking thedifference of the above formula with the trace formula for X (with the same function ϕ T ) gives ∞ (cid:88) j =1 (cid:99) ϕ T ( r i j ( X φ )) = ( n − · area( X )4 π (cid:90) ∞−∞ r (cid:99) ϕ T ( r ) tanh( πr ) dr + (cid:88) γ ∈P ( X ) ( fix ˜ γ ( φ ) − (cid:96) ( γ )2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) ϕ T ( (cid:96) ( γ ))+ (cid:88) γ ∈L ( X ) −P ( X ) ϕ T ( (cid:96) ( γ ))2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) (cid:88) γ (cid:48) ∈ π − ( γ ) Λ( γ (cid:48) ) − Λ( γ ) . (2.1)Since ϕ T is non-negative and for any γ ∈ L ( X ), | π − ( γ ) | ≤ n , and Λ( γ (cid:48) ) ≤ (cid:96) ( γ (cid:48) ) = (cid:96) ( γ ) for all γ (cid:48) ∈ π − ( γ ), the sum on the bottom line of (2.1) is at most n (cid:88) γ ∈L ( X ) −P ( X ) ϕ T ( (cid:96) ( γ ))2 sinh (cid:16) (cid:96) ( γ )2 (cid:17) · (cid:96) ( γ ) = n (cid:88) γ ∈P ( X ) ∞ (cid:88) k =2 ϕ T ( k(cid:96) ( γ ))2 sinh (cid:16) k(cid:96) ( γ )2 (cid:17) k(cid:96) ( γ ) . (2.2)We have ∞ (cid:88) k =2 ϕ T ( k(cid:96) ( γ ))2 sinh (cid:16) k(cid:96) ( γ )2 (cid:17) k(cid:96) ( γ ) ( ∗ ) (cid:28) X (cid:96) ( γ ) ∞ (cid:88) k =2 ke − k(cid:96) ( γ )2 ( ∗∗ ) (cid:28) X (cid:96) ( γ ) e − (cid:96) ( γ ) , (2.3)where in ( ∗ ) we relied on that ϕ T is bounded, and in both ( ∗ ) and ( ∗∗ ) on that there is a positivelower bound on the lengths of closed geodesics in X . As ϕ T is supported on ( − T, T ), the left handside of (2.3) vanishes whenever (cid:96) ( γ ) ≥ T /
2. Using Lemma 2.2 we thus get n (cid:88) γ ∈P ( X ) ∞ (cid:88) k =2 ϕ T ( k(cid:96) ( γ ))2 sinh (cid:16) k(cid:96) ( γ )2 (cid:17) k(cid:96) ( γ ) (cid:28) X n (cid:88) γ ∈P ( X ): (cid:96) ( γ ) ≤ T (cid:96) ( γ ) e − (cid:96) ( γ ) ≤ n T (cid:88) m =0 (cid:88) γ ∈L ( X ) : m ≤ (cid:96) ( γ )
We write S n for the symmetric group of permutations of the set [ n ]. By convention S is thetrivial group with one element. If m ≤ n , we always let S m ≤ S n be the subgroup of permutationsfixing [ m + 1 , n ] element-wise. For k ≤ n , we will let S (cid:48) k ≤ S n be our notation for the subgroup of16ermutations fixing [ n − k ] element wise. We write C [ S n ] for the group algebra of S n with complexcoefficients. Young diagrams
A Young diagram (YD) of size n is a collection of n boxes, arranged in left-aligned rows in the plane,such that the number of boxes in each row is non-increasing from top to bottom. A Young diagramis uniquely specified by the sequence λ , λ , . . . , λ r where λ i is the number of boxes in the i th row(and there are r rows). We have λ ≥ λ ≥ · · · ≥ λ r >
0; we call such a sequence of integers a partition.
We view YDs and partitions interchangeably in this paper. If (cid:80) i λ i = n we write λ (cid:96) n .Two important examples of partitions are ( n ), with all boxes of the corresponding YD in the firstrow, and (1) n def = (1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n ), with all boxes of the corresponding YD in the first column. If µ, λ areYDs, we write µ ⊂ λ if all boxes of µ are contained in λ (when both are aligned to the same top-leftborders). We say µ ⊂ k λ if µ ⊂ λ and there are k boxes of λ that are not in µ . We write ∅ forthe empty YD with no boxes. If λ is a YD, ˇ λ is the conjugate YD obtained by reflecting λ in thediagonal (switching rows and columns).A skew Young diagram (SYD) is a pair of Young diagrams µ and λ with µ ⊂ λ . This pair isdenoted λ/µ and thought of as the collection of boxes of λ that are not in µ . We identify a YD λ with the SYD λ/ ∅ so that YDs are special cases of SYDs. The size of a SYD λ/µ is the number ofboxes it contains; i.e. the number of boxes of λ that are not in µ . The size is denoted by | λ/µ | , orif λ is a YD, | λ | . Young tableaux
Let λ/µ be a SYD, with λ (cid:96) n and µ (cid:96) k . A standard Young tableau of shape λ/µ is a filling ofthe boxes of λ/µ with the numbers [ k + 1 , n ] such that each number appears in exactly one boxand the numbers in each row (resp. column) are strictly increasing from left to right (resp. top tobottom). We refer to standard Young tableaux just as tableaux in this paper. We write Tab( λ/µ )for the collection of tableaux of shape λ/µ . Given a tableau T , we denote by T | ≤ m (resp. T | >m )the tableau formed by the numbers-in-boxes of T with numbers in the set [ m ] (resp. [ m + 1 , n ]).The shape of T | ≤ m and of T | >m is a SYD in general. We let µ m ( T ) be the YD that is the shapeof T | ≤ m . If ν ⊂ µ ⊂ λ , T ∈ Tab( µ/ν ) and R ∈ Tab( λ/µ ), then we write T (cid:116) R for the tableau inTab( λ/ν ) obtained by adjoining R to T in the obvious way. Irreducible representations
The equivalence classes of irreducible unitary representations of S n are in one-to-one correspondencewith Young diagrams of size n . Given a YD λ (cid:96) n , we write V λ for the corresponding irreduciblerepresentation of S n ; each V λ is a finite dimensional Hermitian complex vector space with an actionof S n by unitary linear automorphisms. Hence V λ can also be thought of as a module for C [ S n ]. Wewrite d λ def = dim V λ . It is well-known, and also follows from the discussion of the next paragraphs,that d λ = | Tab( λ ) | . Note that d λ = d ˇ λ since reflection in the diagonal gives a bijection betweenTab( λ ) and Tab(ˇ λ ).We now give an account of the Vershik-Okounkov approach to the representation theory ofsymmetric groups from [VO04]. According to the usual ordering of [ n ] there is a filtration of17ubgroups S ≤ S ≤ S ≤ · · · ≤ S n . If W is any unitary representation of S n , m ∈ [ n ] and µ (cid:96) m , we write W µ for the span of vectorsin copies of V µ in the restriction of W to S m ; we call W µ the µ -isotypic subspace of W .It follows from the branching law for restriction of representations between S m and S m − thatfor λ (cid:96) n and T ∈ Tab( λ ) the intersection (cid:16) V λ (cid:17) µ ( T ) ∩ (cid:16) V λ (cid:17) µ ( T ) ∩ · · · ∩ (cid:16) V λ (cid:17) µ n − ( T ) is one-dimensional. Vershik-Okounkov specify a unit vector v T in this intersection. The collection { v T : T ∈ Tab( λ ) } is a orthonormal basis for V λ called a Gelfand-Tsetlin basis. Modules from SYDs If m, n ∈ N , λ (cid:96) n , µ (cid:96) m and µ ⊂ λ , then V λ/µ def = Hom S m ( V µ , V λ )is a unitary representation of S (cid:48) n − m as S (cid:48) n − m is in the centralizer of S m in S n . We write d λ/µ forthe dimension of this representation. There is also an analogous Gelfand-Tsetlin orthonormal basisof V λ/µ indexed by T ∈ Tab( λ/µ ); as such, we have d λ/µ = | Tab( λ/µ ) | . Note that when µ = λ ,Tab ( λ/µ ) = {∅} ( ∅ the empty tableau), and the representation V λ/µ is one-dimensional with basis w ∅ . One has the following consequence of Frobenius reciprocity (cf. e.g. [MP20, Lemma 2.1]). Lemma 3.1.
Let n ∈ N , m ∈ [ n ] and µ (cid:96) m . Then (cid:88) λ (cid:96) n : µ ⊂ λ d λ/µ d λ = n ! m ! d µ . Throughout the paper, we will write b λ for the number of boxes outside the first row of a YD b λ , ˇ b λ λ , and write ˇ b λ for the number of boxes outside the first column of λ . More generally, we write b λ/ν (resp. ˇ b λ/ν ) for the number of boxes outside the first row (resp. column) of the SYD λ/ν , so b λ/ν , ˇ b λ/ν b λ/ν = b λ − b ν and ˇ b λ/ν = ˇ b λ − ˇ b ν . We need the following bounds on dimensions of representations. Lemma 3.2. [MP20, Lemma 2.6] If n ∈ N , m ∈ [ n ] , λ (cid:96) n , ν (cid:96) m , ν ⊂ λ and m ≥ b λ , then ( n − b λ ) b λ b b λ λ m b ν ≤ d λ d ν ≤ b b ν ν n b λ ( m − b ν ) b ν . (3.1) Lemma 3.3.
Let λ/ν be a skew Young diagram of size n . Then d λ/ν ≤ ( n ) b λ/ν and d λ/ν ≤ ( n ) ˇ b λ/ν . roof. There are at most (cid:0) nb λ/ν (cid:1) options for the b λ/ν elements outside the first row. Given these,there are at most b λ/ν ! choices for how to place them outside the first row. The proof of the secondinequality is analogous. The Witten zeta function of the symmetric group S n is defined for a real parameter s as ζ S n ( s ) def = (cid:88) λ (cid:96) n d sλ . (3.2)This function, and various closely related functions, play a major role in this paper. One mainreason for its appearance is due to a formula going back to Hurwitz [Hur02] that states | X g,n | = | Hom(Γ g , S n ) | = | S n | g − ζ S n (2 g − . (3.3)This is also sometimes called Mednykh’s formula [Med78]. We first give the following result due toLiebeck and Shalev [LS04, Theorem 1.1] and independently, Gamburd [Gam06, Prop. 4.2]. Theorem 3.4.
For any s > , as n → ∞ ζ S n ( s ) = 2 + O (cid:0) n − s (cid:1) . This has the following corollary when combined with (3.3).
Corollary 3.5.
For any g ∈ N with g ≥ , we have | X g,n | ( n !) g − = 2 + O ( n − ) . As well as the previous results, we also need to know how well ζ S n (2 g −
2) is approximated byrestricting the summation in (3.2) to λ with a bounded number of boxes either outside the firstrow or the first column. We let Λ( n, b ) denote the collection of λ (cid:96) n such that λ ≤ n − b andˇ λ ≤ n − b . In other words, Λ( n, b ) is the collection of YDs λ (cid:96) n with both b λ ≥ b and ˇ b λ ≥ b . Aversion of the next proposition, when b is fixed and n → ∞ , is due independently to Liebeck andShalev [LS04, Prop. 2.5] and Gamburd [Gam06, Prop. 4.2]. Here, we need a version that holdsuniformly over b that is not too large compared to n . Proposition 3.6.
Fix s > . There exists a constants κ = κ ( s ) > such that when b ≤ n , (cid:88) λ ∈ Λ( n,b ) d sλ (cid:28) s (cid:18) κb s ( n − b ) s (cid:19) b . (3.4) Proof.
Here we follow Liebeck and Shalev [LS04, proof of Prop. 2.5] and make the proof uniformover b . Let Λ ( n, b ) denote the collection of λ (cid:96) n with ˇ λ ≤ λ ≤ n − b . Since d λ = d ˇ λ , (cid:88) λ ∈ Λ( n,b ) d sλ ≤ (cid:88) λ ∈ Λ ( n,b ) d sλ ,
19o it suffices to prove a bound for (cid:80) λ ∈ Λ ( n,b ) 1 d sλ . Let Λ ( n, b ) denote the elements λ of Λ ( n, b )with λ ≥ n . We write (cid:88) λ ∈ Λ ( n,b ) d sλ = Σ + Σ where Σ = (cid:88) λ ∈ Λ ( n,b ) d sλ , Σ = (cid:88) λ ∈ Λ ( n,b ) − Λ ( n,b ) d sλ . Bound for Σ . By [LS04, Lemma 2.1] if λ ∈ Λ ( n, b ) then d λ ≥ (cid:0) λ n − λ (cid:1) . Let p ( m ) denote thenumber of µ (cid:96) m . The number of λ ∈ Λ ( n, b ) with a valid fixed value of λ is p ( n − λ ). ThereforeΣ ≤ n − b (cid:88) λ = (cid:100) n (cid:101) p ( n − λ ) (cid:0) λ n − λ (cid:1) s = (cid:98) n (cid:99) (cid:88) (cid:96) = b p ( (cid:96) ) (cid:0) n − (cid:96)(cid:96) (cid:1) s . We now split the sum into two ranges to estimate Σ ≤ Σ (cid:48) + Σ (cid:48)(cid:48) whereΣ (cid:48) = b (cid:88) (cid:96) = b p ( (cid:96) ) (cid:0) n − (cid:96)(cid:96) (cid:1) s , Σ (cid:48)(cid:48) = (cid:98) n (cid:99) (cid:88) (cid:96) = b +1 p ( (cid:96) ) (cid:0) n − (cid:96)(cid:96) (cid:1) s . First we deal with Σ (cid:48) . We have p ( (cid:96) ) ≤ c √ (cid:96) for some c > (cid:96) ≤ n − (cid:96) when (cid:96) ≤ b ≤ n , (cid:18) n − (cid:96)(cid:96) (cid:19) ≥ ( n − (cid:96) ) (cid:96) (cid:96) (cid:96) . This gives Σ (cid:48) ≤ b (cid:88) (cid:96) = b c √ (cid:96) (cid:18) (cid:96)n − (cid:96) (cid:19) s(cid:96) ≤ c b b (cid:88) (cid:96) = b (cid:18) b n − b (cid:19) s(cid:96) (cid:28) s c b (cid:18) b n − b (cid:19) sb , (3.5)where the last inequality used that b ( n − b ) ≤ as we assume b ≤ n .To deal with Σ (cid:48)(cid:48) we make the following claim. Claim.
There is n > n ≥ n and (cid:96) ≤ n (cid:18) n − (cid:96)(cid:96) (cid:19) ≥ (cid:18) n (cid:19) √ (cid:96) . (3.6)20 roof of claim. Observe that when (cid:96) ≤ n (cid:18) n − (cid:96)(cid:96) (cid:19) ≥ ( n − (cid:96) ) (cid:96) (cid:96) (cid:96) = ( n − (cid:96) ) √ (cid:96) ( n − (cid:96) ) (cid:96) −√ (cid:96) (cid:96) − (cid:96) ≥ (cid:18) n (cid:19) √ (cid:96) (2 (cid:96) ) (cid:96) −√ (cid:96) (cid:96) − (cid:96) = (cid:18) n (cid:19) √ (cid:96) (cid:32) √ (cid:96) − (cid:96) (cid:33) √ (cid:96) . We have 2 √ (cid:96) − ≥ (cid:96) when (cid:96) ≥
49 which proves the claim in this case. On the other hand, it is easyto see that there is a n > n ≥ n and 1 ≤ (cid:96) < This proves theclaim.
The claim gives Σ (cid:48)(cid:48) ≤ (cid:98) n (cid:99) (cid:88) (cid:96) = b +1 (cid:16) c n s (cid:17) √ (cid:96) for some c = c ( s ) > n ≥ n . Let n = n ( s ) ≥ n be such that when n ≥ n , c n s < e − .Let q def = c n s . Then when n ≥ n , log( q ) ≤ − (cid:48)(cid:48) ≤ (cid:90) ∞ b q √ x dx = 2 q b log q (cid:18) q − b (cid:19) . We obtain Σ (cid:48)(cid:48) ≤ b + 1) q b ≤ b + 1) c b n sb . (3.7)Together with (3.5) this yields:Σ (cid:28) s c b (cid:18) b n − b (cid:19) sb + 2( b + 1) c b n sb (cid:28) s (cid:18) κb s ( n − b ) s (cid:19) b (3.8)with κ = κ ( s ) = max ( c , c ). Bound for Σ . If λ ∈ Λ ( n, b ) − Λ ( n, b ) then ˇ λ ≤ λ < n and [LS04, Prop. 2.4] gives theexistence of an absolute c > d λ ≥ c n . Thus for large enough n ,Σ ≤ (cid:88) λ ∈ Λ ( n,b ) − Λ ( n,b ) c − ns ≤ p ( n ) c − ns ≤ c √ n c − ns (cid:28) s . (3.9)Putting (3.8) and (3.9) together proves the proposition. Here we assume g = 2, and let Γ def = Γ . We write X n def = X ,n throughout the rest of the paper.Tiled surfaces were defined in [MP20, Def. 3.2]: these are subcomplexes of covering spaces of Σ ,defined in Example 4.2 below (the covering spaces inherit a CW-structure from Σ ). Here we give21n equivalent combinatorial definition. Definition 4.1 (Tiled surface) . A tiled surface X is an at most 2-dimensional CW-complex togetherwith an assignment of both a direction and a label in { a, b, c, d } to each edge (1-cell) and a cyclicordering of the half-edges incident to every vertex (0-cell). This data is subject to the followingconstraints: • At any fixed vertex v the cyclic ordering of the half-edges at v dictates a cyclic sequence oftypes of half-edges incident at v : the possible types of half-edges are a -incoming, a -outgoing, b -incoming, b -outgoing, etc. We require that the cyclic sequence of types at each vertex is acyclic subsequence of ‘ a -outgoing, b -incoming, a -incoming, b -outgoing, c -outgoing, d -incoming, c -incoming, d -outgoing’. In particular, each vertex of X has at most one incoming f -labelededge and at most one outgoing f -labeled edge, for each f ∈ { a, b, c, d } . • The cyclic ordering of half-edges at each vertex makes the 1-skeleton X (1) into a ribbon graph that yields a surface with boundary . Every oriented boundary component of this correspond-ing ribbon graph follows a cycle in X (1) and hence spells out a cyclic sequence of { a, b, c, d } and their inverses. For example, we write a if we traverse an a -labeled edge in its correctdirection, and a − if we traverse an a -labeled edge against its direction. We require that every2-cell of X is glued to X (1) along a path that spells out[ a, b ] [ c, d ] = aba − b − cdc − d − . • We require that every path that spells out [ a, b ] [ c, d ] or a cyclic shift of it is closed. Such apath is then either a boundary component of X or the boundary of a 2-cell.While technically X is not a surface (because there may be 0-cells or 1-cells not incident to any2-cell), it can always be thought of as a surface (possibly with boundary) by viewing the 2-cells asbeing glued to the ribbon graph coming from X (1) . When we want to be clear that we take thispoint of view, we refer to the surface as the thick version of X . Since each 2-cell of a tiled surface meets 8 edges, we refer to them as octagons . If X is a tiledsurface, we call X (0) the vertices of X and the connected components of X (1) \ X (0) the edges of X . We write v ( X ) for the number of vertices of X , e ( X ) for the number of edges and f ( X ) for v ( X ) , e ( X ) , f ( X ) the number of octagons. We also let ∂X denote the boundary of the thick version of X and d ( X ) ∂X, d ( X ) denote the number of directed edges along ∂X (so an edge is possibly counted twice, once for eachdirection).The combinatorial data of a tiled surface consists of the labeled directed graphs given by its1-skeleton and the incidences between its octagons and this graph. Two tiled surfaces with the samecombinatorial data are considered to be identical.A morphism from a tiled surface X to another tiled surface X is a continuous map that mapseach cell of X by a homeomorphism to a cell of X , and respects the edges’ labelings and directions.We consider two morphisms of tiled surfaces to be the same if they differ by an isotopy throughmorphisms. An embedding from X to X is an injective morphism of tiled surfaces. Example 4.2.
Start with an octagon whose edges are labeled and directed, in the given cyclicordering around the boundary, to spell out [ a, b ][ c, d ]. Gluing each pair of same labeled edgestogether, respecting their directions, yields a genus 2 surface Σ with a CW-structure coming from22he original vertices and edges in the boundary of the original octagon, together with the octagon.There is one vertex v in Σ , four edges, each of which forms a loop at v , and one octagon. The originaledge labels and directions descend to make Σ a tiled surface. Moreover, if for each f ∈ { a, b, c, d } ,we write f for the element of π (Σ , v ) corresponding to the f -labeled edge in Σ , with its givendirection, we obtain a fixed identification π (Σ , v ) = Γ = (cid:104) a, b, c, d | [ a, b ] [ c, d ] (cid:105) . Example 4.3.
The fibered product construction gives a one-to-one correspondence betweenHom(Γ , S n ) and topological degree- n covers of Σ with a labeled fiber over the basepoint v . Explic-itly, for φ ∈ Hom(Γ , S n ), we can consider the quotient X φ def = Γ \ ( U × [ n ])where U is the universal cover of Σ (an open disc) and Γ acts on U × [ n ] diagonally, by the usualaction of Γ on U on the first factor, and via φ on the second factor. The covering map X φ → Σ isinduced by the projection U × [ n ] → U .Each X φ can be given the structure of a tiled surface by pulling back the CW-complex structureof Σ and the edges’ directions and labels via the covering map. The fiber of v ∈ Σ is the collectionof vertices of X φ . We fix throughout the rest of the paper a vertex u ∈ U lying over v ∈ Σ . Thisidentifies the fiber of v in X φ with { u } × [ n ] and hence gives a fixed bijection between the verticesof X φ and the numbers in [ n ]. The map φ (cid:55)→ X φ is the desired one-to-one correspondence betweenHom(Γ , S n ) and topological degree- n covers of Σ with the fiber over v labeled bijectively by [ n ]. Example 4.4.
For any 1 ≤ γ ∈ Γ, pick a word w of minimal length in the letters a, b, c, d and theirinverses that represents an element in the conjugacy class of γ in Γ. By its definition, w is cyclicallyreduced. Now take a circle with a base point, and divide it into { a, b, c, d } -labeled and directedsegments such that following around the circle in some orientation, starting at the base point, andreading off the labels and directions spells out w . Call the resulting tiled surface C γ . Note thatgenerally C γ is not uniquely determined by γ (see [MP20, Sections 3.4 and 3.5]), and we choose oneof the options arbitrarily. We have v ( C γ ) = e ( C γ ) = (cid:96) w ( γ ).If X is a tiled surface, there are some simple relations between the quantities v ( X ), e ( X ), f ( X ), d ( X ), and χ ( X ) , the topological Euler characteristic of X . We write D ( X ) def = v ( X ) − f ( X ); this D ( X )quantity will play a major role later in the paper. We note the following relation: d ( X ) = 2 e ( X ) − f ( X ) , (4.1)which implies that 4 f ( X ) ≤ e ( X ) . (4.2)Since each vertex is incident to at most 8 half-edges we have e ( X ) ≤ v ( X ) . (4.3)The following lemma about the quantity D will be useful later. Lemma 4.5.
If the connected tiled surface X is not a single vertex, then ≤ D ( X ) ≤ d ( X ) . roof. For the first inequality we have D ( X ) = v ( X ) − f ( X ) (4.2) ≥ v ( X ) − e ( X )4 (4.3) ≥ . For the other inequality, we have8 f ( X ) = (cid:88) O an octagon of X { corners of O } = (cid:88) v a vertex of X { corners of octagons at v }≥ { vertices in X but not in ∂X } = 8 v ( X ) − { vertices in ∂X } ,so D ( X ) ≤ { vertices in ∂X } . But going through all the components of ∂X , one passes through avertex exactly d ( X ) times (a vertex may be visited more than once), so { vertices in ∂X }≤ d ( X ).The Euler characteristic χ ( X ) is also controlled by f ( X ) and d ( X ). Lemma 4.6.
Let X be a connected compact tiled surface. Then χ ( X ) ≤ − f ( X ) + d ( X )2 . Proof.
We have χ ( X ) = v ( X ) − e ( X ) + f ( X ) (4.1) = v ( X ) − f ( X ) − d ( X )2= D ( X ) − f ( X ) − d ( x )2 Lemma 4.5 ≤ − f ( X ) + d ( X )2 . Recall the definition of the tiled surface X φ from Example 4.3. Given a tiled surface Y , we define E n ( Y ) def = E φ ∈ X n [ Y → X φ ] . This is the expected number of morphisms from Y to X φ . Recall that we use the uniform probabilitymeasure on X n . We have the following result that relates this concept to Theorem 1.10. Lemma 4.7.
Given γ ∈ Γ , let C γ be as in Example 4.4. Then E n [ fix γ ] = E n ( C γ ) . (4.4) Proof.
This is not hard to check but also follows from [MP20, Lemma 3.5]. For arbitrary g ≥
2, the bound is χ ( X ) ≤ − (2 g − f ( X ) + d ( X )2 .
24e will not only need to work with E n ( Y ) for various tiled surfaces, but also the expectednumber of times that Y embeds into X φ . For a tiled surface Y , this is given by E emb n ( Y ) def = E φ ∈ X n [ Y (cid:44) → X φ ] . Here we introduce language that was used in [MP20], based on terminology of Birman and Seriesfrom [BS87a]. Let Y denote a tiled surface throughout this §§ Y , they are not necessarily directed according to the definition of Y .First of all, we augment Y by adding half-edges, which should be thought of as copies of [0 , ).Of course, every edge of Y (1) is thought of as containing two half edges, each of which inheritsa label in { a, b, c, d } and a direction from their containing edge. We add to Y { a, b, c, d } -labeledand directed half-edges to form Y + so that every vertex of Y + has exactly 8 emanating half-edges, Y + with labels and directions given by ‘ a -outgoing, b -incoming, a -incoming, b -outgoing, c -outgoing, d -incoming, c -incoming, d -outgoing’. The cyclic order we have written here induces a cyclic orderingon each of the half-edges at each vertex of Y + that we view as fixed from now on. If a half-edge of Y + does not belong to an edge of Y (hence was added to Y + ), we call it a hanging half-edge. Wewrite he ( Y ) for the number of hanging half-edges of Y . We may think of Y + as a surface too, by he ( Y )considering the thick version of Y and attaching a thin rectangle for every hanging half-edge. Wecall the resulting surface the thick version of Y + , and mark its boundary by ∂Y + . ∂Y + For two directed edges (cid:126)e and (cid:126)e of Y with the terminal vertex v of (cid:126)e equal to the source of (cid:126)e ,the half-edges between (cid:126)e and (cid:126)e are by definition the half edges of Y + at v that are strictly between (cid:126)e and (cid:126)e in the given cyclic ordering. There are m of these where 0 ≤ m ≤ cycle in Y is a cyclic sequence C =( (cid:126)e , . . . , (cid:126)e k ) of directed edges in Y (1) , such that for each1 ≤ i ≤ k the terminal vertex v i of (cid:126)e i is the initial vertex of (cid:126)e i +1 and the indices are taken modulo k . A boundary cycle of Y is a cycle corresponding to a boundary component of the thick version of Y , where the boundary component is oriented so that there are no octagons on the left of it as itis traversed.If C is a cycle in Y , a block in C is a non-empty (possibly cyclic) subsequence of successiveedges, each successive pair of edges having no half-edges between them (this means that if Y wasembedded in a tiled surface with no boundary, then C would lie at the boundary of some octagon).A half-block is a block of length 4 (in general, 2 g ) and a long block is a block of length at least 5(in general, 2 g + 1).Two blocks ( (cid:126)e i , . . . , (cid:126)e j ) and ( (cid:126)e k , . . . , (cid:126)e (cid:96) ) in a cycle C are called consecutive if ( (cid:126)e i , . . . , (cid:126)e j , (cid:126)e k , . . . , (cid:126)e (cid:96) )is a (possibly cyclic) subsequence of C and there is precisely one half-edge between (cid:126)e j and (cid:126)e k . A chain is a (possibly cyclic) sequence of consecutive blocks. A cyclic chain is a chain whose blockspave an entire cycle (with exactly one half-edge between the last block and the first blocks). A longchain is a chain consisting of consecutive blocks of lengths4 , , , . . . , , g, g − , g − , . . . , g − , g ). A half-chain is a cyclic chain consisting of consecutiveblocks of length 3 (in general, 2 g −
1) each. We also recall the following definitions from [MP20,Def. 3.7, 3.8].
Definition 4.8 (Boundary reduced) . A tiled surface Y is boundary reduced if no boundary cycle25igure 4.1: A piece P of ∂Y + is shown in black line. The broken black line marks parts of ∂Y + adjacent to but not part of P and the yellow stripe marks the side of the internal side of Y . Thispiece consists of 9 full directed edges and 9 hanging half-edges, so Defect ( P ) = − Y contains a long block or a long chain. Definition 4.9 (Strongly boundary reduced) . A tiled surface Y is strongly boundary reduced if noboundary cycle of Y contains a half-block or a half-chain.The following concepts of a piece and its defect play a crucial role in the paper. Definition 4.10. A piece P of ∂Y is a (possibly cyclic) path along ∂Y + , consisting of whole directededges and/or whole hanging half-edges. We write e ( P ) for the number of full directed edges in P , he ( P ) for the number of hanging half-edges in P , and | P | def = e ( P ) + he ( P ). We letDefect( P ) def = e ( P ) − he ( P ) . (In general, Defect( P ) def = e ( P ) − (2 g − he ( P ) . ) See Figure 4.1 for an illustration of a piece. The boundary reduction algorithm was defined in [MP20, §§ Input.
An embedding of tiled surfaces
Y (cid:44) → Z where Y is compact and Z has no boundary. Output.
A boundary reduced compact tiled surface Y (cid:48) embedded in Z , with a given embedding Y (cid:44) → Y (cid:48) , such that the embedding Y (cid:44) → Z factors through Y (cid:44) → Y (cid:48) (cid:44) → Z . In other words, Y (cid:48) extends Y within Z . Algorithm.
Let Y (cid:48) = Y . (a) If Y (cid:48) is boundary reduced, terminate the algorithm and return Y (cid:48) . (b) If some boundary cycle of Y (cid:48) contains a long block, then in Z , this long block follows theboundary of some octagon that is not in Y (cid:48) . Add this octagon, and all its incident vertices andedges to Y (cid:48) to form a new sub-complex of Z . This also includes the case that the boundary cycleis the boundary of an octagon in Z , in which case we add only the octagon itself. Note that thisstep decreases d ( Y (cid:48) ) by at least 2. Call the new subcomplex Y (cid:48) and return to (a) . (c) If some boundary cycle of Y (cid:48) contains a long chain, each block of the chain follows the boundaryof an octagon in Z that is not in Y (cid:48) . Octagons corresponding to successive blocks of the chainmeet along a hanging half-edge of Y (cid:48) + , which is part of a full edge of Z . Add all these octagons,and all their incident vertices and edges to Y (cid:48) . Again, this decreases d ( Y (cid:48) ) by at least 2. Call thenew subcomplex Y (cid:48) and return to (a) . 26e compile some properties of the boundary reduction algorithm into the following proposition.Except for one, all either follow immediately from the description of the algorithm, or were provedin [MP20]. Proposition 4.11.
Let
Y (cid:44) → Z be an embedding of a compact tiled surface Y into a tiled surface Z without boundary.1. The boundary reduction algorithm applied to Y (cid:44) → Z always terminates in finitely manyiterations. (Each iteration decreases d ( Y (cid:48) ) .)2. [MP20, Prop. 3.9] The result of the boundary reduction algorithm does not depend on anychoices made during the algorithm and hence we can define BR ( Y (cid:44) → Z ) to be the boundary BR ( Y (cid:44) → Z ) reduced tiled sub-surface of Z obtained from applying the boundary reduction algorithm to Y (cid:44) → Z .3. (Obvious) If Y (cid:48) = BR ( Y (cid:44) → Z ) , and Y (cid:48) (cid:54) = Y , then d ( Y (cid:48) ) < d ( Y ) .4. [MP20, Lemma 3.10] If Y (cid:48) = BR ( Y (cid:44) → Z ) , then e ( Y (cid:48) ) ≤ e ( Y ) + d ( Y ) .
5. If Y (cid:48) = BR ( Y (cid:44) → Z ) , then f ( Y (cid:48) ) ≤ f ( Y ) + d ( Y ) . Proof of item 5.
In each of steps (b) or (c) , d ( Y (cid:48) ) decreases by at least two, so there are at most d ( Y )2 steps where octagons are added. In step (b) exactly one octagon is added (which is at most d ( Y )3 because otherwise Y is boundary reduced). In step (c) , if the long chain consists of (cid:96) blocks,it is of length 3 (cid:96) + 2 ≤ d ( Y (cid:48) ), and at most (cid:96) ≤ d ( Y (cid:48) ) − < d ( Y )3 new octagons are added. In total, atmost d ( Y )2 · d ( Y )3 = d ( Y ) new octagons are added throughout the boundary reduction algorithm. ε -adapted tiled surfaces In this §§ Y (cid:48) is either • very well adapted to our methods, so that we can give an estimate for E emb n ( Y (cid:48) ) with aneffective error term, or alternatively, • the number of octagons of Y (cid:48) is larger than the length of the boundary of Y (cid:48) .The first type of output needs to be quantified by a constant ε ≥
0. To explain this, we make thefollowing definition. Recall Definition 4.10 of a piece P of ∂Y and the quantities Defect( P ) and | P | . Definition 4.12.
Let ε ≥ Y be a tiled surface. A piece P of ∂Y is ε -adapted if it satisfies Defect( P ) ≤ χ ( P ) − ε | P | . (4.5) For larger values of the genus g , we could get a tighter bound, but the stated bound holds and is good enough. In general, if Defect( P ) ≤ g · χ ( P ) − ε | P | .
27e have χ ( P ) = 0 if P is a whole boundary component and χ ( P ) = 1 otherwise. We say that apiece P is ε -bad if (4.5) does not hold, i.e., if Defect( P ) > χ ( P ) − ε | P | . We say that Y is ε -adaptedif every piece of Y is ε -adapted. The following lemma shows that this notion quantifies the notionof strongly boundary reduced tiled surfaces. Lemma 4.13.
Let Y be a tiled surface.1. Y is boundary reduced if and only if it is -adapted.2. Y is strongly boundary reduced if and only if every piece of ∂Y is ε -adapted for some ε > .If Y is compact, this is equivalent to that Y is ε -adapted for some ε > .Proof. It is immediate that if Y is 0-adapted (resp. ε -adapted for some ε >
0) then it is boundaryreduced (resp. strongly boundary reduced). The converse implication is not hard and can be foundin [MP20, proof of Lemma 4.20].
Lemma 4.14. If P is an ε -bad piece of Y , then | P | < d ( Y )3 − ε . (4.6) Proof. If P is ε -bad, then by definition e ( P ) − · he ( P ) > χ ( P ) − ε | P | . So(3 − ε ) | P | < e ( P ) + he ( P )) + ( e ( P ) − · he ( P ) − χ ( P )) ≤ · e ( P ) ≤ d ( Y ) . The algorithm depends on a positive constant ε >
0; we shall see below that fixing ε = works finefor our needs (for arbitrary g ≥ ε = g .) To force the algorithm to be deterministic,we a priori make some choices: Notation . For every compact tiled surface Y which is boundary reduced but not ε -adapted,we pick an ε -bad piece P ( Y ) of ∂Y .With the ambient parameter ε fixed as well as the choices of ε -bad pieces, the octagons-vs-boundary (OvB) algorithm is as follows. Input.
An embedding of tiled surfaces
Y (cid:44) → Z where Y is compact and Z has no boundary. Output.
A compact tiled surface Y (cid:48) and a factorization of the input embedding Y (cid:44) → Z by Y (cid:44) → Y (cid:48) (cid:44) → Z where both maps are embeddings. Algorithm.
Let Y (cid:48) = Y . (a) Let Y (cid:48) = BR ( Y (cid:48) (cid:44) → Z ). If θ ( Y (cid:48) ) def = f ( Y (cid:48) ) − d ( Y (cid:48) ) > Y (cid:48) . (b) If Y (cid:48) is not ε -adapted, add all the octagons of Z meeting a P ( Y (cid:48) ) to Y (cid:48) , and go to (a) .Return Y (cid:48) . a To be sure, an octagon O in Z is said to meet P ( Y (cid:48) ) if some directed edge or hanging-half-edge of P ( Y (cid:48) ) liesat ∂O . For arbitrary g ≥
2, the bound is | P | < g · d ( Y )2 g − − ε . emark . The reason why it is useful to have the inequality f ( Y (cid:48) ) > d ( Y (cid:48) ) as a terminatingcondition of the algorithm is that it forces the Euler characteristic of Y (cid:48) to be linearly comparableto the number of octagons in Y (cid:48) by Lemma 4.6. Of course, we would like to know when/if this algorithm terminates.In step (a) , if the boundary reduction algorithm affects Y (cid:48) then d ( Y (cid:48) ) decreases by at least two,and f ( Y (cid:48) ) increases by at least one. So θ ( Y (cid:48) ) increases by at least three.In step (b) , if Y (cid:48) changes, the following lemma shows that θ ( Y (cid:48) ) increases by at least oneprovided that ε ≤ . Lemma 4.17.
With notation as above, if Y (cid:48) is modified in step (b) , then1. d ( Y (cid:48) ) increases by less than ε | P ( Y (cid:48) ) | .2. θ ( Y (cid:48) ) increases by more than (cid:0) − ε (cid:1) | P ( Y (cid:48) ) | , so the increase is positive when ε ≤ . Note that θ ( Y (cid:48) ) is an integer, so any positive increase is an increase by at least one. Proof.
Suppose that in step (b) Y (cid:48) is modified. Let Y (cid:48)(cid:48) denote the result of this modification andlet P = P ( Y (cid:48) ). Let k denote the number of new octagons added. First assume that P is a path,so χ ( P ) = 1. We have k ≤ he ( P ) + 1 because every hanging half-edge along P marks the passingfrom one new octagon to the next one. Every new octagon borders 8 edges in Z . For most newoctagons, two of these edges contain hanging half-edges of P and are internal edges in Y (cid:48)(cid:48) , so if j ofthe edges belong to P , the net contribution of the octagon to d ( Y (cid:48)(cid:48) ) − d ( Y (cid:48) ) is at most 6 − j . Theexceptions are the two extreme octagons which possibly meet only one hanging half-edge of P , andcontribute a net of at most 7 − j . The sum of the parameter j over all new octagons is exactly e ( P ). In total, we obtain: d (cid:0) Y (cid:48)(cid:48) (cid:1) − d (cid:0) Y (cid:48) (cid:1) ≤ k + 2 − · e ( P ) ≤ he ( P ) + 1) + 2 − · e ( P )= 2 (3 · he ( P ) − e ( P )) + 8 < ε | P | − χ ( P )) + 8 = 2 · ε | P | , where the last inequality comes from the definition of an ε -bad piece. If P is a whole boundarycycle of Y (cid:48) + , we have k ≤ he ( P ) and all octagons contribute at most 6 − j to d ( Y (cid:48)(cid:48) ) − d ( Y (cid:48) ), so d (cid:0) Y (cid:48)(cid:48) (cid:1) − d (cid:0) Y (cid:48) (cid:1) ≤ k − · e ( P ) ≤ · he ( P ) − · e ( P ) < ε | P | − χ ( P )) = 2 ε | P | . This proves Part 1.There is a total of 8 k directed edges at the boundaries of the new octagons. Of these, e ( P )are edges of P . Each of the remaining 8 k − e ( P ) can ‘host’ two hanging half-edges of P , and eachhanging half-edge appears in exactly 2 directed edges of new octagons. This gives2 he ( P ) ≤ k − e ( P )) , For arbitrary g ≥ θ ( Y (cid:48) ) increases by more than (cid:16) g − ε (cid:17) | P ( Y (cid:48) ) | , so we need ε ≤ g .
29o 8 k ≥ he ( P ) + e ( P ) = | P | . Hence θ (cid:0) Y (cid:48)(cid:48) (cid:1) − θ (cid:0) Y (cid:48) (cid:1) = k − (cid:0) d (cid:0) Y (cid:48)(cid:48) (cid:1) − d (cid:0) Y (cid:48) (cid:1)(cid:1) > | P | − ε | P | = (cid:18) − ε (cid:19) | P | . The upshot of the previous observations and Lemma 4.17 is that, provided ε ≤ , every timestep (a) of the algorithm is reached, except for the first time, Y (cid:48) has changed in step (b) , so θ ( Y (cid:48) )has increased by at least one. Since θ ( Y ) = f ( Y ) − d ( Y ) ≥ − d ( Y ) , and it may increase only up to 1, this implies the following lemma: Lemma 4.18. If ε ≤ , then during the octagons-vs-boundary algorithm, step (a) is reached atmost d ( Y ) + 2 times. In particular, the algorithm always terminates. Now that we know the algorithm always terminates (assuming ε ≤ ), and it clearly hasdeterministic output due to our a priori choices and the fact that the boundary reduction algorithmhas deterministic output (Proposition 4.11 Part 2), if Y (cid:44) → Z is an embedding of a compact tiledsurface Y into a tiled surface Z without boundary we write OvB ε ( Y (cid:44) → Z ) for the output of the OvB ε ( Y (cid:44) → Z ) OvB algorithm with parameter ε applied to Y (cid:44) → Z . Thus OvB ε ( Y (cid:44) → Z ) is a tiled surface Y (cid:48) withan attached embedding Y (cid:44) → Y (cid:48) . We can now make the following easy observation. Lemma 4.19.
Let ε ≤ , let Y (cid:44) → Z be an embedding of a compact tiled surface Y into a tiledsurface Z without boundary, and let Y (cid:48) = OvB ε ( Y (cid:44) → Z ) . Then at least one of the following holds: • Y (cid:48) is ε -adapted. • Y (cid:48) is boundary reduced and f ( Y (cid:48) ) > d ( Y (cid:48) ) . We also want an upper bound on how d ( Y (cid:48) ) and f ( Y (cid:48) ) increase during the OvB algorithm. Lemma 4.20.
Assume ε ≤ . Let Y be a compact tiled surface, Z be a boundary-less tiledsurface and Y denote the output of the OvB algorithm applied to an embedding Y (cid:44) → Z . Then d ( Y ) ≤ d ( Y ) , (4.8) f ( Y ) ≤ f ( Y ) + 4 d ( Y ) + d ( Y ) . (4.9) Proof.
If step (a) is only reached once, then the result of the algorithm, Y , is equal to BR ( Y (cid:44) → Z ).In this case we have d (cid:0) Y (cid:1) ≤ d ( Y ) and f (cid:0) Y (cid:1) ≤ f ( Y ) + d ( Y ) by Proposition 4.11 part 5, so thestatement of the lemma holds. So from now on suppose step (a) is reached more than once.Let Y = Y (cid:48) at the penultimate time that step (a) is completed. Between the penultimate timethat step (a) is completed and the algorithm terminates, step (b) takes place to form Y = Y (cid:48) , andthen step (a) takes place one more time to form Y = Y which is the output of the algorithm.First we prove the bound on d ( Y ). We have θ ( Y ) ≤
0, so θ ( Y ) − θ ( Y ) ≤ − ( f ( Y ) − d ( Y )) ≤ d ( Y ) . For arbitrary g ≥
2, we pick ε ≤ g . The statement of the lemma holds as is.
30e claim that in every step of the OvB algorithm, the increase in θ is larger then the increase in d .Indeed, this is obviously true in step (a) , where θ does not decrease and d does not increase. It isalso true in step (b) by Lemma 4.17 and our assumption that ε ≤ . Therefore, d ( Y ) − d ( Y ) ≤ θ ( Y ) − θ ( Y ) ≤ d ( Y ) , and we conclude that d ( Y ) ≤ d ( Y ).Let P = P ( Y ). By Lemma 4.17, d ( Y ) ≤ d ( Y ) + 2 ε | P | (4.6) ≤ d ( Y ) + 2 ε · d ( Y )3 − ε = d ( Y ) (cid:20) ε − ε (cid:21) ≤ . · d ( Y ) ≤ . · d ( Y ) , where the penultimate inequality is based on that ε ≤ . Finally, d ( Y ) ≤ d ( Y ), so (4.8) is proven.For the number of octagons, note first that f ( Y ) = θ ( Y ) + d ( Y ) ≤ d ( Y ) ≤ d ( Y ) . Let k denote the number of new octagons added in step (b) to form Y from Y . As noted in theproof of Lemma 4.17, k ≤ he ( P ) + 1. As P = P ( Y ) is ε -bad, we have he ( P ) ≤
13 ( e ( P ) + ε | P | ) (4.6) ≤ d ( Y ) (cid:18) ε − ε (cid:19) < d ( Y ) ≤ d ( Y ) , the penultimate inequality is based again on that ε ≤ . Thus f ( Y ) − f ( Y ) ≤ he ( P ) + 1 ≤ d ( Y ).Finally, by Proposition 4.11 part 5, f ( Y ) − f ( Y ) ≤ d ( Y ) ≤ d ( Y ) , and we conclude f ( Y ) = f ( Y ) + [ f ( Y ) − f ( Y )] + [ f ( Y ) − f ( Y )] ≤ d ( Y ) + 2 d ( Y ) + d ( Y ) = 4 d ( Y ) + d ( Y ) , which proves (4.9) in this case as well. We fix g = 2 and ε = in this section and throughout the rest of the paper. We recall the followingdefinition from [MP20, Def. 3.23]. Definition 4.21 (Resolutions) . A resolution R of a tiled surface Y is a collection of morphisms oftiled surfaces R = { f : Y → W f } , such that every morphism h : Y → Z of Y into a tiled surface Z with no boundary decomposesuniquely as Y f → W f h (cid:44) → Z , where f ∈ R and h is an embedding.The point of this definition is the following lemma that was also recorded in [MP20, Lemma3.24]. 31 emma 4.22. If Y is a compact tiled surface and R is a finite resolution of Y , then E n ( Y ) = (cid:88) f ∈R E emb n ( W f ) . (4.10)The type of resolution we wish to use in this paper is the following. Definition 4.23.
For a compact tiled surface Y , let R ε ( Y ) denote the collection of all morphisms Y f −→ W f obtained as follows: • F : Y → Z is a morphism of Y into a boundary-less tiled surface Z . • U F is the image of F in Z . Hence there is a given embedding ι F : U F (cid:44) → Z . • W f is given by W f = OvB ε ( U f (cid:44) → Z ) and f = ι F ◦ F : Y → W f . Theorem 4.24.
Given a compact tiled surface Y , the collection R ε ( Y ) defined in Definition 4.23is a finite resolution of Y .Proof. To see that R ε ( Y ) is finite, note that there are finitely many options for U F (this is a quotientof the compact complex Y ). For any such U F we have f ( U F ) ≤ f ( Y ) and d ( U F ) ≤ d ( Y ), and henceby Lemma 4.20 there is a bound on f ( W f ) depending only on Y . As we add a bounded number ofoctagons to obtain W f , there is a bound also on v ( W f ) and on e ( W f ). This means that W f is oneof only finitely many tiled surfaces, and there are finitely many morphisms of Y to one of these.Now we explain why R ε ( Y ) is a resolution – this is essentially the same as [MP20, proof of Thm.3.29]. Let F : Y → Z be a morphism with ∂Z = ∅ . By the definition of R ε ( Y ), it is clear that F decomposes as Y f → W f (cid:44) → Z for the f ∈ R that originate in F . To show uniqueness, assume that F decomposes in an additional way Y f (cid:48) → W f (cid:48) (cid:44) → Z where W f (cid:48) is the result of the OvB algorithm for some F (cid:48) : Y → Z (cid:48) with ∂Z (cid:48) = ∅ . We claim thatboth decomposition are precisely the same decompositions of h (namely W f (cid:48) = W f and f (cid:48) = f ).Indeed, the OvB algorithm with input F (cid:48) ( Y ) (cid:44) → Z (cid:48) takes place entirely inside W f (cid:48) , and does notdepend on the structure of Z (cid:48) \ W f (cid:48) : the choices are made depending only on the structure of theboundary of Y (cid:48) in step (b) of the OvB algorithm, as well as in every step of the boundary reductionalgorithm invoked in (a) . Moreover, the result of these steps depends only on the octagons of Z immediately adjacent to the boundary of Y (cid:48) . But W f (cid:48) is embedded in Z , and so it must be identicalto W f and f (cid:48) identical to f .It is the following corollary of the previous results, applied to a tiled surface C γ as in Example4.4, that will be used in the rest of the paper. Corollary 4.25.
Let (cid:54) = γ ∈ Γ . For any f : C γ → W f in R ε ( C γ ) , either1. W f is boundary reduced, and χ ( W f ) < − f ( W f ) < − d ( W f ) , or2. W f is ε -adapted.Moreover, in either case, d ( W f ) ≤ (cid:96) w ( γ ) (4.11) f ( W f ) ≤ (cid:96) w ( γ ) + 4 ( (cid:96) w ( γ )) . (4.12)32 roof. The inequalities (4.11) and (4.12) are from Lemma 4.20 and the fact that d ( C γ ) = 2 (cid:96) w ( γ )and f ( C γ ) = 0. It follows from the construction of R ε ( C γ ) using the OvB algorithm that if f ∈ R ε ( Y )with f : Y → W f , and W f is not ε -adapted, then W f is boundary reduced and d ( W f ) < f ( W f ).Combined with Lemma 4.6 this gives χ ( W f ) ≤ − f ( W f ) + 12 d ( W f ) < − f ( W f ) + 12 f ( W f ) ≤ − f ( W f ) . In the current section §§ § § g = 2. Throughout this entire § Y is a fixedcompact tiled surface. We let v = v ( Y ), e = e ( Y ), f = f ( Y ) denote the number of vertices, edges,and octagons of Y , respectively. We fix a bijective map J : Y (0) → [ v ], and as in [MP20, §
4] foreach n ∈ N we modify J by letting J n : Y (0) → [ n − v + 1 , n ] , J n ( v ) def = J ( v ) + n − v . (5.1)We use the map J n to identify the vertex set of Y with [ n − v + 1 , n ]. Throughout this section, f will stand for one of the letters a, b, c, d . For each letter f ∈ { a, b, c, d } , let e f denote the number of f -labeled edges of Y . Let V − f = V − f ( Y ) ⊂ [ n − v + 1 , n ] be the subset of vertices of Y with outgoing f -labeled edges, and V + f ⊂ [ n − v + 1 , n ] those vertices of Y with incoming f -labeled edges. Notethat e f = |V − f | = |V + f | . For each f ∈ { a, b, c, d } we fix g f ∈ S (cid:48) v such that for every pair of vertices i, j of Y in [ n − v + 1 , n ] with a directed f -labeled edge from i to j , we have g f ( i ) = j .In [MP20, §§ σ + f , σ − f , τ + f , τ − f ∈ S (cid:48) v ⊂ S n for each f ∈ { a, b, c, d } satisfying the following four properties (as well as a fifth one that is notrelevant here): P1 For all f ∈ { a, b, c, d } , σ ± f ( V ± f ) = τ ± f ( V ± f ) = [ n − e f + 1 , n ]. P2 For all f ∈ { a, b, c, d } , ( σ + f ) − σ − f = ( τ + f ) − τ − f = g f . P3 For all f ∈ { a, b, c, d } , σ ± f | [ n ] \V ± f = τ ± f | [ n ] \V ± f . P4 Each of the permutations σ − b (cid:0) σ + a (cid:1) − , τ + a (cid:0) σ + b (cid:1) − , τ + b (cid:0) τ − a (cid:1) − , σ − c (cid:0) τ − b (cid:1) − , σ − d (cid:0) σ + c (cid:1) − , τ + c (cid:0) σ + d (cid:1) − , τ + d (cid:0) τ − c (cid:1) − , σ − a (cid:0) τ − d (cid:1) − fixes every element of [ n − f + 1 , n ].We hence forth view these as fixed, given Y . In the prequel paper [MP20, Thm. 4.12] the followingtheorem was proved. 33 heorem 5.1. For n ≥ v we have E emb n ( Y ) = ( n !) | X n | · ( n ) v ( n ) f (cid:81) f ( n ) e f · Ξ n ( Y ) (5.2) where Ξ n ( Y ) def = (cid:88) ν ⊂ v − f λ (cid:96) n − f d λ d ν (cid:88) ν ⊂ µ f ⊂ e f − f λ d µ a d µ b d µ c d µ d Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:17) , (5.3)Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:17) def = (cid:88) r + f , r − f ∈ Tab ( µ f /ν ) s f , t f ∈ Tab ( λ/µ f ) M (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) (5.4) and M ( { σ ± f , τ ± f , r ± f , s f , t f } ) is the following product of matrix coefficients: M (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) def = (cid:68) σ − b (cid:0) σ + a (cid:1) − w r + a (cid:116) s a , w r − b (cid:116) s b (cid:69) (cid:68) τ + a (cid:0) σ + b (cid:1) − w r + b (cid:116) s b , w r + a (cid:116) t a (cid:69) · (cid:68) τ + b (cid:0) τ − a (cid:1) − w r − a (cid:116) t a , w r + b (cid:116) t b (cid:69) (cid:68) σ − c (cid:0) τ − b (cid:1) − w r − b (cid:116) t b , w r − c (cid:116) s c (cid:69) · (cid:68) σ − d (cid:0) σ + c (cid:1) − w r + c (cid:116) s c , w r − d (cid:116) s d (cid:69) (cid:68) τ + c (cid:0) σ + d (cid:1) − w r + d (cid:116) s d , w r + c (cid:116) t c (cid:69) · (cid:68) τ + d (cid:0) τ − c (cid:1) − w r − c (cid:116) t c , w r + d (cid:116) t d (cid:69) (cid:68) σ − a (cid:0) τ − d (cid:1) − w r − d (cid:116) t d , w r − a (cid:116) s a (cid:69) . (5.5)In light of Theorem 5.1, we will repeatedly discuss ν, { µ f } , λ satisfying ν ⊂ v − e f µ f ⊂ e f − f λ (cid:96) n − f ∀ f ∈ { a, b, c, d } (5.6)and { r ± f , s f , t f } satisfying r + f , r − f ∈ Tab( µ f /ν ) , s f , t f ∈ Tab( λ/µ f ) ∀ f ∈ { a, b, c, d } . (5.7)To give good estimates for Ξ n ( Y ), we need an effective bound for the quantities M ( { σ ± f , τ ± f , r ± f s f , t f } )that was obtained in [MP20]. Before giving this bound, we recall some notation. For T ∈ Tab( λ/ν ),we write top( T ) for the set of elements in the top row of T (the row of length λ − µ which may beempty) and left( T ) for the set of elements in the left-most column of T (same here). For any twosets A, B in [ n ], we define d ( A, B ) = | A \ B | . Given { r ± f , s f , t f } as in (5.7), we define D top (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) def = (5.8) d (cid:16) σ − b (cid:0) σ + a (cid:1) − top( r + a (cid:116) s a ) , top( r − b (cid:116) s b ) (cid:17) + d (cid:16) τ + a (cid:0) σ + b (cid:1) − top( r + b (cid:116) s b ) , top( r + a (cid:116) t a ) (cid:17) + d (cid:16) τ + b (cid:0) τ − a (cid:1) − top( r − a (cid:116) t a ) , top( r + b (cid:116) t b ) (cid:17) + d (cid:16) σ − c (cid:0) τ − b (cid:1) − top( r − b (cid:116) t b ) , top( r − c (cid:116) s c ) (cid:17) + d (cid:16) σ − d (cid:0) σ + c (cid:1) − top( r + c (cid:116) s c ) , top( r − d (cid:116) s d ) (cid:17) + d (cid:16) τ + c (cid:0) σ + d (cid:1) − top( r + d (cid:116) s d ) , top( r + c (cid:116) t c ) (cid:17) + d (cid:16) τ + d (cid:0) τ − c (cid:1) − top( r − c (cid:116) t c ) , top( r + d (cid:116) t d ) (cid:17) + d (cid:16) σ − a (cid:0) τ − d (cid:1) − top( r − d (cid:116) t d ) , top( r − a (cid:116) s a ) (cid:17) . We also define D left ( { σ ± f , τ ± f , r ± f , s f , t f } ) analogously to D top with left in place of top. Lemma 5.2. [MP20, Lemma 4.16] Let ν, { µ f } , λ be as in (5.6) and { r ± f , s f , t f } be as in (5.7). . If λ + ν > n − f + ( v − f ) , then (cid:12)(cid:12)(cid:12) M (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:18) ( v − f ) λ + ν − ( n − f ) (cid:19) D top ( { σ ± f ,τ ± f ,r ± f ,s f ,t f } ) .
2. If ˇ λ + ˇ ν > n − f + ( v − f ) , then (cid:12)(cid:12)(cid:12) M (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:18) ( v − f ) ˇ λ + ˇ ν − ( n − f ) (cid:19) D left ( { σ ± f ,τ ± f ,r ± f ,s f ,t f } ) . Recall from §§ b ν is the number of boxes of a Young diagram ν outside the first row,and ˇ b ν is the number of boxes outside the first column. We have the following trivial upper boundsfor D top ( { σ ± f , τ ± f , r ± f , s f , t f } ) and D left ( { σ ± f , τ ± f , r ± f , s f , t f } ): D top (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) ≤ b λ − b ν ) (5.9) D left (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) ≤ b λ − ˇ b ν ) (5.10)We recall the following estimate obtained in [MP20, Prop. 4.24]. Proposition 5.3.
Let ε ≥ . Suppose that ν, { µ f } , λ are as in (5.6) and { r ± f , s f , t f } are as in (5.7).If Y is ε -adapted then D top (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) ≥ b λ + 3 b ν − b µ a − b µ b − b µ c − b µ d + εb λ/ν . (5.11) The same result holds replacing D top with D left and b • by ˇ b • . Ξ n and preliminary estimates In this §§ . Y is ε -adapted leads to bounds on Ξ n . We continueto view Y as fixed and hence suppress dependence of quantities on Y . As on page 23, we write D def = v − f . We will use the notation Ξ P ( ν ) n where P is a proposition concerning ν to meanΞ P ( ν ) n def = (cid:88) ν ⊂ v − f λ (cid:96) n − f P ( ν ) holds true d λ d ν (cid:88) ν ⊂ µ f ⊂ e f − f λ d µ a d µ b d µ c d µ d Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:17) . We will continue to use this notation, for various propositions P , throughout the rest of the paper.We want to give bounds for various Ξ P ( ν ) n under the condition that Y is either boundary reduced(namely, 0-adapted) or, moreover, ε -adapted for some ε >
0. We will always assume v ≤ n and soalso D = v − f ≤ n . Then for n (cid:29) n = Ξ ν =(1 n − v ) n + Ξ ν =( n − v ) n + Ξ n + Ξ < ˇ b ν ≤ D ; b ν > n + Ξ b ν , ˇ b ν > D n . (5.12)This is according to three regimes for b ν and ˇ b ν : • The zero regime : when b ν or ˇ b ν equal 0. The contribution from here is Ξ ν =(1 n − v ) n + Ξ ν =( n − v ) n . • The intermediate regime: when b ν , ˇ b ν > D . The contribution35rom this regime is Ξ n + Ξ < ˇ b ν ≤ D ; b ν > n . (Note that b ν ≤ D and ˇ b ν ≤ D cannot holdsimultaneously as D ≤ v ≤ n and ν (cid:96) n − v .) • The large regime: when both b ν , ˇ b ν > D . The contribution from this regime is Ξ b ν , ˇ b ν > D n .The strategy for bounding these different contributions is to further partition the tuples ( ν, { µ f } , λ )according to the data b λ , { b µ f } , b ν , ˇ b λ , { ˇ b µ f } , ˇ b ν . Definition 5.4.
For B = (cid:0) B λ , { B µ f } , B ν , ˇ B λ , { ˇ B µ f } , ˇ B ν (cid:1) we write ( ν, { µ f } ,λ ) (cid:96) B ( ν, { µ f } , λ ) (cid:96) B if (5.6) holds, and ν , { µ f } and λ have the prescribed number of blocks outside the first row andoutside the first column, namely, b λ = B λ , ˇ b λ = ˇ B λ , b ν = B ν , ˇ b ν = ˇ B ν and ∀ f ∈ { a, b, c, d } b µ f = B µ f , ˇ b µ f = ˇ B µ f . We denote by B n ( Y ) the collection of tuples B which admit at least one tuple of YDs ( ν, { µ f } , λ ).Finally, we let Ξ Bn = Ξ Bn ( Y ) def = (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d λ d ν d µ a d µ b d µ c d µ d Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:17) . (5.13)Note that Ξ n ( Y ) = (cid:80) B ∈B n ( Y ) Ξ Bn . Also, note that B ∈ B ( Y ) imposes restrictions on thepossible values of B λ , { B µ f } , B ν , ˇ B λ , { ˇ B µ f } , ˇ B ν . For example, for every f ∈ { a, b, c, d } , 0 ≤ B µ f − B ν ≤ v − e f and 0 ≤ B λ − B µ f ≤ e f − f , and likewise for the ˇ B ’s. In addition, B ν + ˇ B ν + 1 ≥ n − ν ,and so on.We first give a general estimate for the quotient of dimensions in the summands in (5.13). Lemma 5.5.
Suppose that v ≤ n and that ( ν, { µ f } , λ ) satisfy (5.6). If b ν ≤ D then d λ d ν d µ a d µ b d µ c d µ d (cid:28) d ν b b λ λ n (cid:16) b λ +3 b ν − (cid:80) f b µf (cid:17) . (5.14) If ˇ b ν ≤ D , an analogous inequality holds with all b ’s replaced with ˇ b ’s, and D top replaced with D left .Proof. The two cases in the statement of the lemma are dual, so we will only prove the first. ByLemma 3.2, d ν d µ f ≤ b b µf µ f ( n − v ) b ν ( n − e f − b µ f ) b µf ≤ b b λ λ n b ν (cid:0) n − n / (cid:1) b µf , where the second inequality is based on that e f + b µ f ≤ e f +( b ν + v − e f ) = b ν + v ≤ n / . Similarly, d λ d ν ≤ b bνν ( n − f ) bλ ( n − v − b ν ) bν ≤ b bλλ n bλ ( n − n / ) bν . Altogether, d λ d ν d µ a d µ b d µ c d µ d ≤ b b λ λ n ( b λ +4 b ν ) (cid:0) n − n / (cid:1) b ν + (cid:80) f b µf = b b λ λ n (cid:16) b λ +3 b ν − (cid:80) f b µf (cid:17) (cid:18) − n − / (cid:19) b ν + (cid:80) f b µf ≤ b b λ λ n (cid:16) b λ +3 b ν − (cid:80) f b µf (cid:17) · (cid:18) − n − / (cid:19) n / . (cid:16) − n − / (cid:17) n / n →∞ →
1, the right hand side of the last inequality is at most 2 b b λ λ n (cid:16) b λ +3 b ν − (cid:80) f b µf (cid:17) for large enough n .We next give bounds for the individual Ξ Bn . Lemma 5.6.
There is κ > such that if Y is ε -adapted for ε ≥ , v ≤ n and B ν ≤ D , then (cid:12)(cid:12)(cid:12) Ξ Bn (cid:12)(cid:12)(cid:12) (cid:28) B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:18) κ D ( n − v − D ) (cid:19) B ν . A dual statement holds if all B ’s in the assumption and inequality are replaced with ˇ B ’s.Proof. The two cases of the lemma are dual, so we prove only the first one. By assumption, B ν ≤ D ≤ v ≤ n . So for every ( ν, { µ f } , λ ) (cid:96) B , λ + ν − ( n − f ) = ( n − f − B λ ) + ( n − v − B ν ) − ( n − f ) ≥ n − v − B ν − ( B ν + D ) ≥ n − v , and Lemma 5.2 Part 1 gives that whenever (cid:110) r ± f , s f , t f (cid:111) satisfy (5.7), (cid:12)(cid:12)(cid:12) M (cid:16)(cid:110) σ ± f , τ ± f , r ± f s f , t f (cid:111)(cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:18) D n − v (cid:19) D top ( { σ ± f ,τ ± f ,r ± f ,s f ,t f } ) . (5.15)Proposition 5.3 gives B λ + 3 B ν − (cid:88) f B µ f ≤ D top (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) − ε ( B λ − B ν ) , so by Lemma 5.5 d λ d ν d µ a d µ b d µ c d µ d (cid:12)(cid:12)(cid:12) M (cid:16)(cid:110) σ ± f , τ ± f , r ± f s f , t f (cid:111)(cid:17)(cid:12)(cid:12)(cid:12) (cid:28) B B λ λ n − ε ( B λ − B ν ) (cid:18) n D n − v (cid:19) D top ( { σ ± f ,τ ± f ,r ± f ,s f ,t f } ) . Now using the trivial upper bound D top (cid:16)(cid:110) σ ± f , τ ± f , r ± f , s f , t f (cid:111)(cid:17) ≤ B λ − B ν ) in (5.9) and B λ − B ν ≤ v − f ≤ v ≤ n , we obtain that for large enough n , (cid:18) n D n − v (cid:19) D top ( { σ ± f ,τ ± f ,r ± f ,s f ,t f } ) ≤ D B λ − B ν ) (cid:18) − n − / (cid:19) n / ≤ D B λ − B ν ) . Therefore, d λ d ν d µ a d µ b d µ c d µ d (cid:12)(cid:12)(cid:12) M (cid:16)(cid:110) σ ± f , τ ± f , r ± f s f , t f (cid:111)(cid:17)(cid:12)(cid:12)(cid:12) (cid:28) B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν . (cid:12)(cid:12)(cid:12) Ξ Bn (cid:12)(cid:12)(cid:12) (cid:28) B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d ν (cid:88) r + f , r − f ∈ Tab ( µ f /ν ) s f , t f ∈ Tab ( λ/µ f ) 1 ≤ B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d ν since there are at most ( D ) ( B λ − B ν ) ≤ D ( B λ − B ν ) choices of r + f (cid:116) s f or of r − f (cid:116) t f for all f , by Lemma3.3. For fixed ν above, there are at most B B λ λ choices of { µ f } and λ such that ( ν, { µ f } , λ ) (cid:96) B .For example, the boxes outside the first row of λ uniquely determine λ and form a YD of size B λ ;there are at most B λ ! ≤ B B λ λ of these. Hence (cid:12)(cid:12)(cid:12) Ξ Bn (cid:12)(cid:12)(cid:12) (cid:28) B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:88) ν (cid:96) n − v : b ν = B ν d ν . We can finally apply Proposition 3.6 to obtain for the same κ = κ (2) > (cid:12)(cid:12)(cid:12) Ξ Bn (cid:12)(cid:12)(cid:12) (cid:28) B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:18) κB ν ( n − v − B ν ) (cid:19) B ν ≤ B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:18) κ D ( n − v − D ) (cid:19) B ν . Since Lemma 5.6 is only useful for B ν or ˇ B ν small compared to n we have to supplement it withthe following weaker bound. Lemma 5.7. If Y is any tiled surface and B ∈ B n ( Y ) then (cid:12)(cid:12)(cid:12) Ξ Bn (cid:12)(cid:12)(cid:12) ≤ ( D !) (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d λ d ν . Proof.
Since M ( { σ ± f , τ ± f , r ± f s f , t f } ) is a product of matrix coefficients of unit vectors in unitaryrepresentations, we obtain |M ( { σ ± f , τ ± f , r ± f s f , t f } ) | ≤
1. Therefore, with assumptions as in thelemma, and arguing similarly as in the proof of Lemma 5.6, we obtain (cid:12)(cid:12)(cid:12) Ξ Bn (cid:12)(cid:12)(cid:12) ≤ (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d λ d ν d µ a d µ b d µ c d µ d (cid:88) r + f , r − f ∈ Tab ( µ f /ν ) s f , t f ∈ Tab ( λ/µ f ) 1 ( ∗ ) ≤ ( D !) (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d λ d ν d µ a d µ b d µ c d µ d ≤ ( D !) (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d λ d ν , ∗ ) we used the fact there are at most | λ/ν | ! = ( v − f )! choices of r + f (cid:116) s f and of r − f (cid:116) t f . b ν , ˇ b ν We only need analytic estimates for Ξ ν =(1 n − v ) n and Ξ ν =( n − v ) n when Y is boundary reduced (so 0-adapted); when Y is ε -adapted for ε > ν =(1 n − v ) n and Ξ ν =( n − v ) n in §§ Lemma 5.8. If Y is boundary reduced and v ≤ n then (cid:12)(cid:12)(cid:12) Ξ ν =(1 n − v ) n (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) Ξ ν =( n − v ) n (cid:12)(cid:12)(cid:12) (cid:28) D D . Proof.
The proofs for Ξ ν =(1 n − v ) n and Ξ ν =( n − v ) n are dual so we only give the bound for the latter. If ν = ( n − v ) then B ν = 0. Inserting the bounds from Lemma 5.6 with ε = 0 (since Y is boundaryreduced, see Lemma 4.13) and B ν = 0 gives (cid:12)(cid:12)(cid:12) Ξ ν =( n − v ) n (cid:12)(cid:12)(cid:12) (cid:28) (cid:88) B ∈B n ( Y ): B ν =0 B B λ λ D B λ . Because B ∈ B n ( Y ), we have B λ ≤ v − f = D . In B n ( Y ), the set of B (cid:48) s with B ν = 0 and a fixed valueof B λ is of size at most D : indeed, ˇ B ν = n − v −
1, there are at most min( B λ , λ − ν ) ≤ D optionsfor B µ f for each f , there are at most B λ ≤ D possible values of ˇ B λ and at most B µ f ≤ B λ ≤ D choices for ˇ B µ f . Hence (cid:12)(cid:12)(cid:12) Ξ ν =( n − v ) n (cid:12)(cid:12)(cid:12) (cid:28) D D (cid:88) B λ =0 (cid:0) B λ D (cid:1) B λ ≤ D D (cid:88) B λ =0 ( D ) B λ (cid:28) D D . b ν , ˇ b ν Lemma 5.9.
Assume that v ≤ n / .
1. If Y is boundary reduced with D ≤ n then (cid:12)(cid:12)(cid:12) Ξ n (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) Ξ < ˇ b ν ≤ D ; b ν > n (cid:12)(cid:12)(cid:12) (cid:28) (cid:0) D (cid:1) D +1 ( n − v − D ) . (5.16)
2. For any ε ∈ (0 , , there is η = η ( ε ) ∈ (0 , ) such that if Y is ε -adapted, with D ≤ n η then (cid:12)(cid:12)(cid:12) Ξ n (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) Ξ < ˇ b ν ≤ D ; b ν > n (cid:12)(cid:12)(cid:12) (cid:28) ε n . (5.17) Proof.
When D = 0, the inequalities 0 < b ν ≤ D cannot hold, and so | Ξ n | = | Ξ < ˇ b ν ≤ D ; b ν > n | =0 by definition, and both statements hold. So assume D ≥
1. The beginning of the proofs of bothstatements is identical, and in both we may assume that D ≤ n / . The bounds for Ξ n and Ξ < ˇ b ν ≤ D ; b ν > n are similar, using the two dual statements of Lemma 5.6, so we just show how tobound Ξ n . 39or any ε ≥
0, the bounds from Lemma 5.6 give (cid:12)(cid:12)(cid:12) Ξ n (cid:12)(cid:12)(cid:12) (cid:28) (cid:88) B ∈B n ( Y ):0 B B λ λ (cid:0) D n − ε (cid:1) B λ − B ν (cid:18) κ D ( n − v − D ) (cid:19) B ν . Arguing similarly as in the proof of Lemma 5.8, the number of B ’s in the sum above with a fixedvalue of B ν and B λ is bounded by D . We obtain (cid:12)(cid:12)(cid:12) Ξ n (cid:12)(cid:12)(cid:12) (cid:28) D (cid:88) n (cid:12)(cid:12)(cid:12) (cid:28) D · κB λ D ( n − v − D ) D (cid:88) t =0 (cid:0) D n − ε (cid:1) t (cid:28) D ( n − v − D ) D (cid:88) t =0 (cid:0) D n − ε (cid:1) t . (5.18)If Y is boundary reduced, it is 0-adapted (Lemma 4.13), and D n − ≥ , so (5.18) yields (cid:12)(cid:12)(cid:12) Ξ n (cid:12)(cid:12)(cid:12) (cid:28) D ( n − v − D ) · (cid:0) D (cid:1) D ≤ (cid:0) D (cid:1) D +1 ( n − v − D ) proving the first statement.For the second statement, given ε >
0, let η = ε and assume 1 ≤ D ≤ n η . The choice of η implies that for n (cid:29) ε D n − ε ≤ , so (5.18) gives (cid:12)(cid:12)(cid:12) Ξ n (cid:12)(cid:12)(cid:12) (cid:28) ε D ( n − v − D ) (cid:28) ε n . b ν , ˇ b ν In the large regime of b ν and ˇ b ν we use the same estimate for any type of tiled surface. Lemma 5.10. If v ≤ n / and D ≤ n / then (cid:12)(cid:12)(cid:12) Ξ b ν , ˇ b ν > D n (cid:12)(cid:12)(cid:12) (cid:28) D ( n − v − D ) . roof. Using the bound from Lemma 5.7 gives (cid:12)(cid:12)(cid:12) Ξ b ν , ˇ b ν > D n (cid:12)(cid:12)(cid:12) ≤ (cid:88) B ∈B n ( Y ): B ν , ˇ B ν > D ( D !) (cid:88) ( ν, { µ f } ,λ ) (cid:96) B d λ d ν ≤ ( D !) (cid:88) ν (cid:96) n − v ,b ν > D d − ν (cid:88) ν ⊂ v − f λ d λ (cid:88) ν ⊂ µ f ⊂ e f − f λ ≤ ( D !) (cid:88) ν (cid:96) n − v ,b ν > D d − ν (cid:88) ν ⊂ v − f λ d λ ≤ D D ( n − f )!( n − v )! (cid:88) ν (cid:96) n − v ,b ν > D d − ν (cid:28) D D n D κ ( D + 1) (cid:16) n − v − ( D + 1) (cid:17) D +1 = κn D ( D + 1) (cid:16) n − v − ( D + 1) (cid:17) D κ ( D + 1) (cid:16) n − v − ( D + 1) (cid:17) . The second-last inequality used Lemma 3.1 and the final inequality used Proposition 3.6. Since D +1 D is bounded and we assume D ≤ n / and v ≤ n / we obtain the stated result. Ξ n Now we combine the estimates obtained in §§ Y is boundary reduced. Proposition 5.11.
There is A > such that if Y is boundary reduced, v ≤ n / , and D ≤ n / ,then | Ξ n | (cid:28) ( A D ) A D . Proof.
With assumptions as in the proposition, splitting Ξ n as in (5.12) and using Lemmas 5.8,5.9(1), and 5.10 gives | Ξ n | (cid:28) D D + (cid:0) D (cid:1) D +1 ( n − v − D ) + D ( n − v − D ) . If D = 0 this gives | Ξ n | (cid:28) ≤ D ≤ n / we obtain | Ξ n | (cid:28) ( A D ) A D as required.Next we show that if Y is ε -adapted, then D can be as large as a fractional power of n while Ξ n is still very well approximated by Ξ ν =( n − v ) n + Ξ ν =(1 n − v ) n . Proposition 5.12.
For any ε ∈ (0 , , there is η = η ( ε ) ∈ (0 , ) such that if Y is ε -adapted with D ≤ n η and v ≤ n / , then (cid:12)(cid:12)(cid:12) Ξ n − Ξ ν =( n − v ) n − Ξ ν =(1 n − v ) n (cid:12)(cid:12)(cid:12) (cid:28) ε n . Proof.
Lemmas 5.9(2) and 5.10 yield that given ε ∈ (0 , η = η ( ε ) ∈ (0 , ), such that if D ≤ n η , v ≤ n / and Y is ε -adapted, then (cid:12)(cid:12)(cid:12) Ξ n − Ξ ν =( n − v ) n − Ξ ν =(1 n − v ) n (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Ξ < ˇ b ν ≤ D ; b ν > n + Ξ n + Ξ b ν , ˇ b ν > D n (cid:12)(cid:12)(cid:12) (cid:28) ε n + D ( n − v − D ) (cid:28) n . .7 A new expression for Ξ d ν =1 n We continue to fix a compact tiled surface Y . LetΞ d ν =1 n def = Ξ ν =(1 n − v ) n + Ξ ν =( n − v ) n . The goal of this section is to give a formula for each of Ξ ν =(1 n − v ) n and Ξ ν =( n − v ) n that is more precisethan is possible to obtain with the methods of the previous section. This will be done by refiningthe methods of [MP20, § ν =(1 n − v ) n = Ξ ν =( n − v ) n .The first step is to relate these quantities to a counting problem for homomorphisms from afree group to S n . Let F be the free group on generators a, b, c, d . Recall the definition of the sets V ± f from §§ f ∈ { a, b, c, d } let G ± f be the subgroup of S n fixing pointwise V ± f . Let G ± def = G ± a × G ± b × G ± c × G ± d ≤ S n . Recall from §§ g f ∈ S (cid:48) v such that for every pairof vertices i, j of Y in [ n − v + 1 , n ] with a directed f -labeled edge from i to j , we have g f ( i ) = j .We let g = ( g a , g b , g c , g d ) ∈ S n . We see that g f ( V − f ) = V + f and G + = g G − ( g ) − . Our formulas for Ξ ν =(1 n − v ) n and Ξ ν =( n − v ) n will involve the size of the set X ∗ n ( Y, J ) def = (cid:8) ( α a , α b , α c , α d ) ∈ G + g G − (cid:12)(cid:12) W ( α a , α b , α c , α d ) ∈ S n − v (cid:9) (5.19)where W ( g a , g b , g c , g d ) def = g − d g − c g d g c g − b g − a g b g a . Note that a similar set, denoted X n ( Y, J ) in[MP20, Section 4.1], is the set in which the condition is that W ( α a , α b , α c , α d ) = 1 rather than theidentity only when restricted to [ n − v + 1 , n ], as in (5.19). This smaller set X n ( Y, J ) counts thenumber of coverings φ ∈ Hom (Γ , S n ) in which ( Y, J ) embeds.The main result of this §§ Proposition 5.13.
With notations as above, Ξ ν =(1 n − v ) n = Ξ ν =( n − v ) n = ( n ) v | X ∗ n ( Y, J ) | (cid:81) f ∈ a,b,c,d ( n − e f )!( n ) f . Recall that ( n ) q is the Pochhammer symbol as defined in §§ In the rest of the paper, wheneverwe write an integral over a group, it is performed with respect to the uniform measure on the relevantgroup.
Let I = (cid:90) h ± f ∈ G ± f (cid:90) π ∈ S n − v (cid:8) W (cid:0) h + a g a h − a , h + b g b h − b , h + c g c h − c , h + d g d h − d (cid:1) π = 1 (cid:9) ,I = (cid:90) h ± f ∈ G ± f (cid:90) π ∈ S n − v (cid:8) W (cid:0) h + a g a h − a , h + b g b h − b , h + c g c h − c , h + d g d h − d (cid:1) π = 1 (cid:9) · sign( π ) . Lemma 5.14.
We have I = I and | X ∗ n ( Y, J ) | = | S n − v | (cid:12)(cid:12) G + g G − (cid:12)(cid:12) I .Proof. First, every permutation in the image of W is a product of commutators of permutationsand hence even. This shows that I = I . The quantity | S n − v | · | G + | · | G − | · I is the number ofelements ( h + , h − ) ∈ G + × G − such that h + g h − = ( g a , g b , g c , g d ) with W ( g a , g b , g c , g d ) ∈ S n − v . But The reason we use this word instead of the relator [ g a , g b ][ g c , g d ] of Γ is the same as in [MP20]: the one-to-one correspondence between X n and degree- n covers of a genus 2 surface uses the version of the symmetric groupwhere permutations are multiplied as functions acting from the right , whereas in this section we want to multiplypermutations as functions on [ n ] acting from the left . g a , g b , g c , g d arises from exactly | G + |·| G − || G + g G − | pairs ( h + , h − ). Hence | X ∗ n ( Y, J ) | = | S n − v | · | G + | · | G − | · | G + g G − | · I | G + | · | G − | = | S n − v | · | G + g G − | · I . For a Young diagram λ of size m , we write χ λ for the trace of the irreducible representation of S m on V λ . Corollary 5.15.
For ν = ( n − v ) or (1 n − v ) we have | X ∗ n ( Y, J ) | = (cid:81) f ∈ a,b,c,d ( n − e f )!( n ) v (cid:88) λ (cid:96) n d λ Θ νλ ( Y, J n ) where Θ νλ ( Y, J n ) def = (cid:90) h ± f ∈ G ± f (cid:90) π ∈ S n − v χ λ (cid:0) W (cid:0) h + a g a h − a , h + b g b h − b , h + c g c h − c , h + d g d h − d (cid:1) π (cid:1) χ ν ( π ) . (5.20) Proof.
Using Schur orthogonality, write { g = 1 } = 1 n ! (cid:88) λ (cid:96) n d λ χ λ ( g ) , hence I and I are equal to 1 n ! (cid:88) λ (cid:96) n d λ Θ νλ ( Y, J n )with ν = ( n − v ) and (1 n − v ), respectively. We have | G + g G − | = (cid:81) f ∈{ a,b,c,d } ( n − e f )! (cf. [MP20,Lemma 4.4]). Hence by Corollary 5.14, for ν = ( n − v ) or ν = (1 n − v ), | X ∗ n ( Y, J ) | = ( n − v )! (cid:89) f ∈{ a,b,c,d } ( n − e f )! · n ! (cid:88) λ (cid:96) n d λ Θ νλ ( Y, J n )= (cid:81) f ∈ a,b,c,d ( n − e f )!( n ) v (cid:88) λ (cid:96) n d λ Θ νλ ( Y, J n ) . Consider the vector space W λ def = V λ ⊗ ˇ V λ ⊗ V λ ⊗ ˇ V λ ⊗ V λ ⊗ ˇ V λ ⊗ V λ ⊗ ˇ V λ as a unitary representation of S n . This is a departure from [MP20, §
4] where W λ was thoughtof as a representation of S n ; we take a more flexible setup here. The reader may find it useful tosee [MP20, §§ V λ givesan isomorphism V λ ∼ = ˇ V λ , v (cid:55)→ ˇ v . Let B λ ∈ End( W λ ) be defined as in [MP20, eq. (4.4)] by the43ormula (cid:104) B λ ( v ⊗ ˇ v ⊗ v ⊗ ˇ v ⊗ v ⊗ ˇ v ⊗ v ⊗ ˇ v ) , w ⊗ ˇ w ⊗ w ⊗ ˇ w ⊗ w ⊗ ˇ w ⊗ w ⊗ ˇ w (cid:105) def = (cid:104) v , w (cid:105)(cid:104) v , v (cid:105)(cid:104) w , v (cid:105)(cid:104) w , w (cid:105)(cid:104) v , w (cid:105)(cid:104) v , v (cid:105)(cid:104) w , v (cid:105)(cid:104) w , w (cid:105) . (5.21)We note the following, extending [MP20, Lemma 4.7]. Lemma 5.16.
For any ( g , g , g , g , g , g , g , g ) ∈ S n , we have tr W λ ( B λ ◦ ( g , g , g , g , g , g , g , g )) = χ λ ( g − g − g g g − g − g g ) . Proof.
The proof is a direct calculation directly generalizing [MP20, Lemma 4.7].Let Q ± be the orthogonal projection in W λ onto the vectors that are invariant by G ± acting on W λ by the map ( g a , g b , g c , g d ) ∈ G ± (cid:55)→ ( g a , g a , g b , g b , g c , g c , g d , g d ) ∈ S n . These projections appeared also in [MP20, §§ Lemma 5.17.
For ν = (1 n − v ) or ( n − v ) we have Θ νλ ( Y, J n ) = tr W λ ( p ν B λ Q + g Q − ) where p ν denotes the operator p ν def = (cid:90) π ∈ S n − v χ ν ( π ) ( π, , , , , , , ∈ End (cid:16) W λ (cid:17) . Remark . Note that p ν is the projection in End( W λ ) onto the ν -isotypic subspace for the actionof S n − v on the first factor of W λ (while being the identity on the remaining seven factors). This isa self-adjoint operator. Proof.
Recall the definition of Θ νλ ( Y, J n ) in (5.20). Using Lemma 5.16, for every set of fixed valuesof the h ± f and π , we have χ λ (cid:16) W (cid:16) h + a g a h − a , h + b g b h − b , h + c g c h − c , h + f g d h − d (cid:17) π (cid:17) =tr W λ (cid:16) B λ ◦ (cid:16) h + a g a h − a π, h + a g a h − a , h + b g b h − b , h + b g b h − b , h + c g c h − c , h + c g c h − c , h + f g d h − d , h + f g d h − d (cid:17)(cid:17) Therefore, Θ νλ ( Y, J n ) = tr W λ ( B λ Q + g Q − p ν ) = tr W λ ( p ν B λ Q + g Q − ) . Using Lemma 5.17, we now find a new expression for Θ νλ ( Y, J n ) by calculating tr W λ ( p ν B λ Q + g Q − ). Proposition 5.19.
For ν = ( n − v ) or (1 n − v ) we have Θ νλ ( Y, J n ) = (cid:88) ν ⊂ µ f ⊂ e f − f λ (cid:48) ⊂ f λ d λ/λ (cid:48) d µ a d µ b d µ c d µ d Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:48) (cid:17) . (5.22) Proof.
This calculation is very similar to the proof of [MP20, Prop. 4.11] where tr W λ ( B λ Q + g Q − )was calculated. The only difference here is the presence of the additional operator p ν . Therefore we44ill not give all the details. The proof follows [MP20, proof of Prop. 4.11] using properties P1 - P4 of σ ± f , τ ± f . One also uses that p ν is a self-adjoint projection. The role that p ν plays in the proof isthat instead of obtaining a summation over all ν ⊂ v λ , the projection p ν forces only the relevant ν to appear. Proof of Proposition 5.13.
Fix ν = ( n − v ) or ν = (1 n − v ) and write Ξ v n for Ξ v =( n − v ) n or Ξ v =(1 n − v ) n ,respectively. Combining Corollary 5.15 and Proposition 5.19 we obtain, | X ∗ n ( Y, J ) | = (cid:81) f ∈{ a,b,c,d } ( n − e f )!( n ) v (cid:88) λ (cid:96) n d λ (cid:88) ν ⊂ µ f ⊂ e f − f λ (cid:48) ⊂ f λ d λ/λ (cid:48) d µ a d µ b d µ c d µ d Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:48) (cid:17) = (cid:81) f ∈ a,b,c,d ( n − e f )!( n ) f ( n ) v (cid:88) ν ⊂ µ f ⊂ e f − f λ (cid:48) (cid:97) n − f d λ (cid:48) d µ a d µ b d µ c d µ d Υ n (cid:16)(cid:110) σ ± f , τ ± f (cid:111) , ν, { µ f } , λ (cid:48) (cid:17) = (cid:81) f ∈ a,b,c,d ( n − e f )!( n ) f ( n ) v Ξ νn , where the second equality used Lemma 3.1 and the third used d ν = 1. This gives the result. | X ∗ n ( Y, J ) | Recall the definition of X ∗ n ( Y, J ) in (5.19). Because these 4-tuples of permutations generally donot correspond to coverings of the surface Σ , they are better analyzed as n -degree coverings of thebouquet of four loops, namely, as graphs on n vertices labeled by [ n ] with directed edges labeledby a, b, c, d , and exactly one incoming f -edge and one outgoing f -edge in every vertex and every f ∈ { a, b, c, d } . Equivalently, these graphs are the Schreier graphs depicting the action of S n on [ n ]with respect to the four permutations α a , α b , α c , α d .Such a Schreier graph G corresponds to some 4-tuple ( α a , α b , α c , α d ) ∈ X ∗ n ( Y, J ) if and only if thefollowing two conditions are satisfied. The assumption that ( α a , α b , α c , α d ) ∈ G + g G − means that Y (1) , the 1-skeleton of Y , is embedded in G , in an embedding that extends J n on the vertices. Thecondition that W ( α a , α b , α c , α d ) ∈ S n − v , means that at every vertex of G with label in [ n − v + 1 , n ],there is a closed path of length 8 that spells out the word [ a, b ][ c, d ].In Lemma 5.20 below we show that the number of such graphs (equal to | X ∗ ( Y, J ) | ) is rationalin n . To this end, we apply techniques based on Stallings core graphs, in a similar fashion to thetechniques applied in [Pud14, PP15].Construct a finite graph ˆ Y as follows. Start with Y (1) , the 1-skeleton of Y . At every vertexattach a closed cycle of length 8 spelling out [ a, b ] [ c, d ]. Then fold the resulting graph, in the senseof Stallings , to obtain ˆ Y . In other words, at each vertex v of Y (1) , if there is a closed path at v spelling [ a, b ] [ c, d ], do nothing. Otherwise, find the largest prefix of [ a, b ] [ c, d ] that can be read ona path p starting at v and the largest suffix of [ a, b ] [ c, d ] that can be read on a path s terminatingat v . Because Y is a tiled surface, | p | + | s | <
8. Attach a path of length 8 − | p | − | s | between theendpoint of p and the beginning of s which spells out the missing part of the word [ a, b ] [ c, d ]. Inthis description, no folding is required. Note, in particular, that Y (1) is embedded in ˆ Y .By the discussion above, the Schreier graphs G corresponding to X ∗ ( Y, J ) are the graphs inwhich there is an embedding of Y (1) which extends to a morphism of directed edge-labeled graphs of Folding a graph with directed and labeled edges means that as long as there is a vertex with two incoming edgeswith the same label, or two outgoing edges with the same label, these two edges are merged, and so are their otherendpoints. It is well known that this process has a unique outcome. Y . We group these G according to the image of ˆ Y . So denote by Q ( Y ) the possible images of ˆ Y in the graphs G : these are the quotients of ˆ Y which restrict to a bijection on Y (1) . As ˆ Y is a finitegraph, the set Q ( Y ) is finite. Lemma 5.20.
For every n ≥ v ( Y ) , | X ∗ n ( Y, J ) | ( n !) = 1( n ) v ( Y ) (cid:88) H ∈Q ( Y ) ( n ) v ( H ) (cid:81) f ∈{ a,b,c,d } ( n ) e f ( H ) . (5.23) Proof.
By the discussion above it is enough to show that for every H ∈ Q ( Y ) and n ≥ v ( Y ),the number of Schreier graphs G on n vertices where the image of ˆ Y is H , is precisely n ) v ( Y ) · ( n ) v ( H ) (cid:81) f ∈{ a,b,c,d } ( n ) e f ( H ) . First, note that v ( H ) ≤ v (cid:16) ˆ Y (cid:17) ≤ v ( Y ), so under the assumption that n ≥ v ( Y ), H can indeed be embedded in Schreier graphs on n vertices. The number of possiblelabeling of the vertices of H , which must extend the labeling of the vertices of Y (1) , is( n − v ( Y )) ( n − v ( Y ) − · · · ( n − v ( H ) + 1) = ( n ) v ( H ) ( n ) v ( Y ) . There are exactly e a constraints on the permutation α a for it to agree with the data in the vertex-labeled H , so a random permutation satisfies these constraints with probability ( n − e a )! n ! = n ) e a . Thesame logic applied to the other letters gives the required result.Combining Lemma 5.20 with Proposition 5.13 gives the following corollary. Corollary 5.21.
For n ≥ v ( Y ) we have Ξ d ν =1 n ( Y ) = 2 (cid:81) f ∈{ a,b,c,d } ( n ) e f ( Y ) ( n ) f ( Y ) (cid:88) H ∈Q ( Y ) ( n ) v ( H ) (cid:81) f ∈{ a,b,c,d } ( n ) e f ( H ) . In particular, if Y is fixed and n → ∞ , we have Ξ d ν =1 n ( Y ) = 2 (cid:88) H ∈Q ( Y ) n e ( Y ) − f ( Y )+ χ ( H ) (cid:18) O (cid:18) n (cid:19)(cid:19) . (5.24)Note that in the construction of ˆ Y from Y (1) , we add a “handle” (a sequence of edges) to thegraph for every vertex of Y that does not admit a closed cycle spelling [ a, b ] [ c, d ]. Hence the Eulercharacteristic of ˆ Y is equal to that of Y (1) minus the number of such vertices in Y . If Y has anoctagon attached along every closed cycle spelling [ a, b ] [ c, d ], there are v ( Y ) − f ( Y ) such vertices,so χ (cid:16) ˆ Y (cid:17) = χ (cid:16) Y (1) (cid:17) − ( v ( Y ) − f ( Y )) = f ( Y ) − e ( Y ) . (5.25)In particular, this is the case when Y is (strongly) boundary reduced. This is important because ofthe role of χ ( H ) in (5.24) for H ∈ Q ( Y ). It turns out that when Y is strongly boundary reduced,ˆ Y has Euler characteristic strictly larger than all other graphs in Q ( Y ): Lemma 5.22. If Y is strongly boundary reduced, then for every H ∈ Q ( Y ) \ { ˆ Y } , χ ( H ) < χ (cid:16) ˆ Y (cid:17) . roof. We use [MP20, Prop. 4.28] that states that if Y is strongly boundary reduced, then as n → ∞ , Ξ n ( Y ) = 2 + O (cid:0) n − (cid:1) . (5.26)When Y is fixed and n → ∞ , it follows from Lemmas 5.9(1) and 5.10 thatΞ n ( Y ) = Ξ d ν =1 n ( Y ) + O (cid:0) n − (cid:1) . (5.27)Combining (5.26) and (5.27) gives Ξ d ν =1 n ( Y ) = 2 + O (cid:0) n − (cid:1) . (5.28)Comparing (5.28) with (5.24) shows that there is exactly one H ∈ Q ( Y ) with χ ( H ) = f − e , andall remaining graphs in Q ( Y ) have strictly smaller Euler characteristic. Finally, (5.25) shows this H must be ˆ Y itself. E emb n ( Y ) for ε -adapted Y In this section we give the final implications of the previous sections for E emb n ( Y ) for ε -adapted Y .Recall the definition of Q ( Y ) from §§ Lemma 5.23.
Let n ∈ N and q ∈ N ∪ { } with q ≤ n . Then n q (cid:18) − q n (cid:19) ≤ n q exp (cid:18) − q n (cid:19) ≤ ( n ) q ≤ n q . Proof.
The first inequality is based on 1 − x ≤ e − x . The second one is based on writing ( n ) q = n q (cid:0) − n (cid:1) · · · (cid:16) − q − n (cid:17) and using e − x ≤ − x which holds for x ∈ (cid:2) , (cid:3) . The third inequality isobvious. Proposition 5.24.
Let ε ∈ (0 , and η = η ( ε ) ∈ (0 , ) be the parameter provided by Proposition5.12 for this ε . Let n ∈ N and M = M ( n ) ≥ . Let Y be ε -adapted, and suppose that D ( Y ) ≤ n η and v ( Y ) , e ( Y ) , f ( Y ) ≤ M ≤ n . Then as n → ∞ , E emb n ( Y ) n χ ( Y ) = (cid:18) O ε (cid:18) M n (cid:19)(cid:19) (cid:88) H ∈Q ( Y ) \{ ˆ Y } n χ ( H )+ e ( Y ) − f ( Y ) . (5.29) Proof.
Assume all parameters are as in the statement of the proposition. By Theorem 5.1 andProposition 5.12 we have E emb n ( Y ) n χ ( Y ) = ( n !) | X n | · ( n ) v ( Y ) ( n ) f ( Y ) (cid:81) f ( n ) e f ( Y ) n χ ( Y ) (cid:20) Ξ d ν =1 n ( Y ) + O ε (cid:18) n (cid:19)(cid:21) . By Lemma 5.23, ( n ) v ( Y ) ( n ) f ( Y ) (cid:81) f ( n ) e f ( Y ) n χ ( Y ) = 1+ O (cid:16) M n (cid:17) . By Corollary 3.5, ( n !) | X n | = + O (cid:0) n (cid:1) . With Corollary47.21, this gives E emb n ( Y ) n χ ( Y ) = (cid:20)
12 + O (cid:18) M n (cid:19)(cid:21) (cid:81) f ( n ) e f ( Y ) ( n ) f ( Y ) (cid:88) H ∈Q ( Y ) ( n ) v ( H ) (cid:81) f ( n ) e f ( H ) + O ε (cid:18) n (cid:19) Lem. 5.23 = (cid:20) O (cid:18) M n (cid:19)(cid:21) (cid:88) H ∈Q ( Y ) n e ( Y ) − f ( Y )+ χ ( H ) + O ε (cid:18) n (cid:19) , (5.30)where the use of Lemma 5.23 is justified since for every H ∈ Q ( Y ), v ( H ) ≤ v ( ˆ Y ) ≤ v ( Y ) ≤ M ,and e ( H ) ≤ e ( ˆ Y ) ≤ e ( Y ) + 8 v ( Y ) ≤ M . In the summation in (5.30), the top power of n is realizedby ˆ Y and is equal to zero (by (5.25) and Lemma 5.22), so we obtain E emb n ( Y ) n χ ( Y ) = (cid:20) O (cid:18) M n (cid:19)(cid:21) (cid:88) H ∈Q ( Y ) \{ ˆ Y } n χ ( H )+ e ( Y ) − f ( Y ) + O ε (cid:18) n (cid:19) , which yields (5.29).The drawback of Proposition 5.24 is that we do not know how to directly estimate the sum over H ∈ Q ( Y ) \{ ˆ Y } that appears therein. Instead, we take a more indirect approach. It turns out,as explained in the remaining results of this section, that for ε -adapted Y we can control E emb n ( Y )using E emb m ( Y ) with m much smaller than n . Corollary 5.25.
Let ε ∈ (0 , , and η = η ( ε ) ∈ (0 , ) be the parameter provided by Proposition5.12 for this ε . Let m ∈ N . Let Y be ε -adapted and suppose that D ( Y ) ≤ m η and v ( Y ) , e ( Y ) , f ( Y ) ≤ m . Then as m → ∞ , E emb m ( Y ) m χ ( Y ) (cid:29) ε (cid:88) H ∈Q ( Y ) \{ ˆ Y } m χ ( H )+ e ( Y ) − f ( Y ) . Proposition 5.26.
Let ε ∈ (0 , , η as in Proposition 5.12 and K > . Let n ∈ N and m = m ( n ) ∈ N with m < n and m n →∞ → ∞ . Let Y be ε -adapted and suppose that v ( Y ) , e ( Y ) , f ( Y ) ≤ ( K log n ) ≤ m / and that D ( Y ) ≤ K log n ≤ m η . Then as n → ∞ , E emb n ( Y ) n χ ( Y ) = 1 + O ε,K (cid:18) (log n ) n (cid:19) + O ε,K (cid:18) mn E emb m ( Y ) m χ ( Y ) (cid:19) . (5.31) Proof.
With assumptions as in the proposition, Proposition 5.24 gives E emb n ( Y ) n χ ( Y ) = (cid:18) O ε,K (cid:18) (log n ) n (cid:19)(cid:19) (cid:88) H ∈Q ( Y ) \{ ˆ Y } n χ ( H )+ e ( Y ) − f ( Y ) = 1 + O ε,K (cid:18) (log n ) n (cid:19) + O ε,K (cid:88) H ∈Q ( Y ) \{ ˆ Y } n χ ( H )+ e ( Y ) − f ( Y ) . H ∈ Q ( Y ) \ { ˆ Y } we have χ ( H ) + e ( Y ) − f ( Y ) ≤ − m < n , (cid:88) H ∈Q ( Y ) \{ ˆ Y } n χ ( H )+ e ( Y ) − f ( Y ) = (cid:88) H ∈Q ( Y ) \{ ˆ Y } (cid:16) nm (cid:17) χ ( H )+ e ( Y ) − f ( Y ) m χ ( H )+ e ( Y ) − f ( Y ) ≤ mn (cid:88) H ∈Q ( Y ) \{ ˆ Y } m χ ( H )+ e ( Y ) − f ( Y ) Cor. 5.25 (cid:28) ε mn E emb m ( Y ) m χ ( Y ) , concluding the proof of the proposition. We are given c >
0, and an element γ ∈ Γ of cyclic word length (cid:96) w ( γ ) ≤ c log n . We assume that γ is not a proper power of another element of Γ. Recall that C γ is an annular tiled surface associatedto γ as in Example 4.4. By Lemma 4.7 E n [ fix γ ] = E n ( C γ ) . As above, fix ε = and let R ε ( C γ ) be the resolution of C γ provided by Definition 4.23 and Theorem4.24. By Lemma 4.22 we have for any n ≥ E n [ fix γ ] = (cid:88) h ∈R ε ( C γ ) E emb n ( W h ) . (6.1)By Corollary 4.25, there is a constant K = K ( c ) >
0, such that for each h ∈ R ε ( C γ ), and for n ≥ d ( W h ) ≤ K log n, f ( W h ) ≤ K (log n ) . By Lemma 4.5 we have D ( W h ) ≤ d ( W h ) so D ( W h ) ≤ K log n, and v ( W h ) ≤ d ( W h ) + f ( W h ). We also have e ( W h ) ≤ v ( W h ) by (4.3). Hence by increasing K ifnecessary we can also ensure v ( W h ) , e ( W h ) ≤ K (log n ) . ε -adapted surfaces Our first goal is to control the contribution to E n [ fix γ ] in (6.1) from non- ε -adapted surfaces. Let R ε (non- ε -ad) ( C γ ) denote the set of morphisms h : C γ → W h in R ε ( C γ ) such that W h is not ε -adapted.In particular, such W h is boundary reduced and f ( W h ) > d ( W h ). Proposition 6.1.
There is a constant
A > (depending on g in general) such that for any c > , f (cid:96) w ( γ ) ≤ c log n , then (cid:88) h ∈R ε ( non- ε -ad ) ( C γ ) E emb n ( W h ) (cid:28) c (log n ) A n . Proof.
We first do some counting. Let us count h ∈ R ε (non- ε -ad) ( C γ ) by their value of D ( W h ) and f ( W h ). By Corollary 4.25 every h ∈ R ε (non- ε -ad) ( C γ ) has χ ( W h ) < − f ( W h ) < − d ( W h ) . (6.2)Combining (6.2) with Lemma 4.5 yields0 ≤ D ( W h ) ≤ d ( W h ) < f ( W h ) . (6.3)First we bound the number of possible W h with D ( W h ) = D and f ( W h ) = f for fixed D < f .Note that in this case v ( W h ) = v = D + f . We may over-count the number of W h with v verticesby counting the number of W h together with a labeling of their vertices by [ v ]. We first constructthe one-skeleton of such a tiled surface: there are at most v v choices for the a -labeled edges, andalso for the b -labeled edges etc. Because W h are all boundary reduced, there is an octagon attachedto any closed [ a, b ] [ c, d ] path, so the one-skeleton completely determines the entire tiled surface.Hence there are at most v v choices for W h with v ( W h ) = v .We also have to estimate how many ways there are to map C γ into such a W h . Fixing arbitrarilya vertex v of C γ , any morphism C γ → W h is uniquely determined by where v goes; hence there areat most v morphisms and so in total there are at most v v +10 ≤ v v = ( D + f ) D + f ) ≤ (2 f ) f elements h ∈ R ε (non- ε -ad) ( C γ ) with D ( W h ) = D and f ( W h ) = f . Hence there are at most K log n · (2 f ) f elements h ∈ R ε (non- ε -ad) ( C γ ) with f ( W h ) = f .By Proposition 5.11 there is A > h ∈ R ε (non- ε -ad) ( C γ ) | Ξ n ( W h ) | (cid:28) K ( A D ( W h )) A D ( W h ) ≤ ( A f ( W h )) A f ( W h ) , (6.4)so by Theorem 5.1, Corollary 3.5, and Lemma 5.23 we get E emb n ( W h ) Thm 5 . = n ! | X n | ( n ) v ( W h ) ( n ) f ( W h ) (cid:81) f ( n ) e f ( W h ) Ξ n ( W h ) Cor . . (cid:28) ( n ) v ( W h ) ( n ) f ( W h ) (cid:81) f ( n ) e f ( W h ) Ξ n ( W h ) Lemma 5 . (cid:28) K n χ ( W h ) Ξ n ( W h ) (6.4) (cid:28) K n χ ( W h ) ( A f ( W h )) A f ( W h ) . Therefore, for every 1 ≤ f ≤ K (log n ) , (cid:88) h ∈R ε (non- ε -ad) ( C γ ) f ( W h )= f E emb n ( W h ) (cid:28) K ( A f ) A f (cid:88) h ∈R ε (non- ε -ad) ( C γ ) f ( W h )= f n χ ( W h ) (6.2) ≤ ( A f ) A f (cid:88) h ∈R ε (non- ε -ad) ( C γ ) f ( W h )= f n − f ≤ K log n (cid:32) ( A f ) A (2 f ) n (cid:33) f ≤ K log n · (cid:32) A A (cid:0) K (log n ) (cid:1) A +10 n (cid:33) f . (cid:88) h ∈R ε (non- ε -ad) ( C γ ) E emb n ( W h ) = K (log n ) (cid:88) f =1 (cid:88) h ∈R ε (non- ε -ad) ( C γ ) f ( W h )= f E emb n ( W h ) (cid:28) k K log n · K (log n ) (cid:88) f =1 (cid:32) A A (cid:0) K (log n ) (cid:1) A +10 n (cid:33) f (cid:28) K (log n ) A +21 n , where the last inequality is based on that A A ( K (log n ) ) A n ≤ for n (cid:29) K ε -adapted surfaces Write R ε ( ε -ad) ( C γ ) ⊂ R ε ( C γ ) for the collection of morphisms h : C γ → W h in R ε ( C γ ) such that W h is ε -adapted. In light of Proposition 6.1 it remains to deal with the contributions to E n [ fix γ ] from R ε ( ε -ad) ( C γ ). Indeed we have by Proposition 6.1 and (6.1) E n [ fix γ ] = (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb n ( W h ) + O c (cid:18) (log n ) A n (cid:19) . (6.5)Recall that if W h is ε -adapted, it is, in particular, strongly boundary reduced, and so by [MP20,Page 8], E emb n ( W h ) = n χ ( W h ) (cid:2) O (cid:0) n − (cid:1)(cid:3) as n → ∞ . By Theorem 1.9, E n [ fix γ ] = 1 + O (cid:0) n − (cid:1) .Comparing this with (6.5), we conclude that there is exactly one h ∈ R ε ( C γ ) with χ ( W h ) = 0.This h also satisfies that W h is ε -adapted .Still, we are missing some information about R ε ( ε -ad) ( C γ ) that we will need: for example, theability to count how many h : C γ → W h there are in R ε ( ε -ad) ( C γ ) with different orders of contributions(smaller than 1) to (6.5). We are going to use a trick to get around this missing information.Let η ∈ (0 , ) be the parameter provided by Proposition 5.12 for the current ε = . Let m be an auxiliary parameter given by m = (cid:108) ( K log n ) /η (cid:109) so that when n (cid:29) c
1, for all h ∈ R ε ( ε -ad) ( C γ ), D ( W h ) ≤ K log n ≤ m η and v ( W h ) , e ( W h ) , f ( W h ) ≤ K (log n ) ≤ m . Moreover, (log n ) (cid:28) c m (cid:28) c (log n ) η . To exploit the fact that each E emb n ( W h )is controlled by E emb m ( W h ) (Corollary 5.25 and Proposition 5.26), we will at two points use theinequality m ≥ E m [ fix γ ] (6.1) = (cid:88) h ∈R ε ( C γ ) E emb m ( W h ) ≥ (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb m ( W h ) . (6.6) It can be shown that h is the result of the OvB algorithm when applied to the embedding C γ (cid:44) → (cid:104) γ (cid:105) \ U with U the universal cover of Σ – see [MP20, Section 3].
51e begin with (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb n ( W h ) Prop. 5.26 = (cid:88) h ∈R ε ( ε -ad) ( C γ ) n χ ( W h ) (cid:20) O c (cid:18) (log n ) n (cid:19) + O (cid:18) mn E emb m ( W h ) m χ ( W h ) (cid:19)(cid:21) = (cid:88) h ∈R ε ( ε -ad) ( C γ ) n χ ( W h ) (cid:20) O c (cid:18) (log n ) n (cid:19)(cid:21) + O mn (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb m ( W h ) (6.6) = (cid:88) h ∈R ε ( ε -ad) ( C γ ) n χ ( W h ) (cid:18) O c (cid:18) (log n ) n (cid:19)(cid:19) + O (cid:18) m n (cid:19) . (6.7)The middle estimate above used that χ ( W h ) ≤ h ∈ R ε ( ε -ad) ( C γ ), and so (cid:0) nm (cid:1) χ ( W h ) ≤
1. Thecontribution to (6.7) from h is 1 + O c (cid:16) (log n ) n (cid:17) . So we obtain (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb n ( W h ) = 1 + O c (cid:18) (log n ) n (cid:19) + O (cid:18) m n (cid:19) + O (cid:88) h ∈R ε ( ε -ad) ( C γ ) χ ( W h ) < n χ ( W h ) . (6.8)To deal with the last error term, we relate it to the expectations at level m . Indeed, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) h ∈R ε ( ε -ad) ( C γ ) χ ( W h ) < n χ ( W h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) h ∈R ε ( ε -ad) ( C γ ) χ ( W h ) < (cid:16) nm (cid:17) χ ( W h ) m χ ( W h ) ≤ mn (cid:88) h ∈R ε ( ε -ad) ( C γ ) m χ ( W h )Cor. 5.25 (cid:28) mn (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb m ( W h ) (6.6) ≤ m n . Incorporating this estimate into (6.8) gives (cid:88) h ∈R ε ( ε -ad) ( C γ ) E emb n ( W h ) = 1 + O c (cid:18) (log n ) n (cid:19) + O (cid:18) m n (cid:19) = 1 + O c (cid:18) (log n ) A n (cid:19) , where A = η . Combining this with (6.5) and increasing A if necessary we obtain E n [ fix γ ] = 1 + O c (cid:18) (log n ) A n (cid:19) as required. This concludes the proof of Theorem 1.10. References [Alo86] N. Alon. Eigenvalues and expanders.
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