Statistics of finite degree covers of torus knot complements

In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini-Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in $\mathrm{PSL}(2,\mathbb{R})\times \mathbb{R}$. As a by-product of our proofs, we obtain analogous limit theorems for high degree random covers of non-uniform Fuchsian lattices with torsion.


INTRODUCTION
A classical theorem due to Hempel [Hem87] states that the fundamental group of a tame 3-manifold is residually finite.As such, it has many finite index subgroups, or equivalently, the manifold has lots of finite degree covers.
In this paper we study the fundamental groups of torus knot complements and groups closely related to these.We ask two questions: How fast does the number of index n subgroups grow as a function of n?And what are the properties of a random index n subgroup and the corresponding degree n cover?1.1.Subgroup growth.We will study groups of the form Γ p 1 ,...,p m = x 1 , . . .
The first of our questions asks for the subgroup growth of these groups.Writing a n (Γ) for the number of index n subgroups of a group Γ, we will prove: Theorem 1.1.Let p 1 , . . ., p m ∈ N >1 such that ∑ m j=1 1 as n → ∞, where Date: May 26, 2020. 1 Here and throughout the paper, the notation f (n) ∼ g(n) as n → ∞ will indicate that lim n→∞ f (n)/g(n) → 1.
Note that all torus knot groups satisfy the condition on p 1 , . . ., p m ∈ N >1 .In general, the only groups excluded by this condition are Γ 2,2 , the fundamental group of the Klein bottle, and Z.The subgroup growth of both of these groups is well understood.
The theorem above also generalizes to free products of the form Γ p 1,1 ,...,p 1,m 1 * • • • * Γ p r,1 ...,p r,m r where ∑ j p i,m i < m i − 1 for all i = 1, . . ., r.In the case of torus knot groups, this corresponds to taking connected sums.
The analogous result is also known to hold for orientable circle bundles over surfaces [LM00].However, even if these are also central extensions of Fuchsian groups, the methods of Liskovets and Mednykh are quite different.1.2.Random subgroups and covers.In the second part of our paper, we use our results to study random finite index subgroups of Γ p 1 ,...,p m .That is, since the number of index n subgroups of Γ p 1 ,...,p m is finite, we can pick one uniformly at random and ask for its properties.Let us denote our random index n subgroup by H n .This is an example of an Invariant Random Subgroup (IRS) -i.e. a conjugation invariant Borel measure on the Chabauty space of subgroups of Γ p 1 ,...,p m (for more details see Section 2.3).
Let us also fix a classifying space X p 1 ,...,p m for Γ p 1 ,...,p m .For instance, if p, q ≥ 2 and gcd(p, q) = 1 we can take the corresponding torus knot complement.More generally, since Γ p 1 ,...,p m appears as a torsion-free lattice in PSL(2, R) × R, we may take the manifold Γ\(H 2 × R).H n gives rise to a random degree n cover of X p 1 ,...,p m .
We will study three (related) problems: • First, we will ask, given a conjugacy class K ⊂ Γ p 1 ,...,p m , how many conjugacy classes of H n the set K ∩ H n contains.We will denote this number by Z K (H n ).In topological terms, K corresponds to a free homotopy class of loops in X p 1 ,...,p m .Z K (H n ) is the number of closed lifts of that loop to the cover of X p 1 ,...,p m corresponding to H n .We note that we count these lifts as loops and not as sets.In particular, if the corresponding element in Γ p 1 ,...,p m is nonprimitive, some of these different lifts overlap.
• After this we will ask what IRS the random subgroup H n converges to as n → ∞.In topological terms, this asks for the Benjamini-Schramm limit of the corresponding random cover of X p 1 ,...,p m (see Section 2.4 for a definition of Benjamini-Schramm convergence).
• Finally, we will study the asymptotic behaviour of the real Betti numbers b k (H n ; R) of H n , or equivalently of the corresponding random cover of X p 1 ,...,p m .
For a torus knot this is the subgroup generated by the longitude.Since L p 1 ,...,p m is normal in Γ p 1 ,...,p m , it's also an IRS.We will prove Theorem 1.3.Let p 1 , . . ., p m ∈ N >1 be such that ∑ m j=1 1 in probability.
Recall that a random variable X : Ω → N is Poisson-distributed with parameter λ > 0 if and only if So (a) above gives us an explicit limit for the probability that a fixed curve lifts to any given number of curves in the cover.For example, if we denote the random degree n cover of our (p, q)-torus knot complement by X p,q (n) and γ is any free homotopy class of closed curves in X p,q (1) that is not freely homotopic to a power of the longitude we obtain: lim n→∞ P[γ lifts to exactly 3 closed curves in X p,q (n)] = 1 6e = 0.0613 . . . .(b) in particular implies that a random degree n cover of a torus knot complement does not converge to the universal cover of the given torus knot complement as n → ∞.This is different from the behaviour of random finite covers of graphs [DJPP13], surfaces [MP20] and many large volume locally symmetric spaces of higher rank [ABB + 17], that all do converge to their universal covers.
(c) also has implications for the number of boundary tori in a random cover of a torus knot complement.Indeed, together with "half lives, half dies" [Hat07, Lemma 3.5], it also implies that the number of boundary components of a degree n cover is Because all the results in the theorem above are really about the group Γ p 1 ,...,p m , we can also apply them to random covers of more general spaces Y p 1 ,...,p m that have Γ p 1 ....,p m as their fundamental group (i.e.without assuming that Y p 1 ,...,p m is a classifying space for Γ p 1 ,...,p m ).In that case, the random cover Benjamini-Schramm converges to the cover of Y p 1 ,...,p m corresponding to L p 1 ,...,p m and the normalised Betti numbers converge to the 2 -Betti numbers of that cover.
Finally, we note that we prove analogous results to Theorem 1.3 for random index n subgroups of non-cocompact Fuchsian groups.
Theorem 1.4.Let Λ be a non-cocompact Fuchsian group of finite covolume.Moreover, let G n < Λ denote an index n subgroup, chosen uniformly at random.
(a) Let K 1 , . . ., K r ⊂ Λ be distinct non-trivial conjugacy classes.Then, as n → ∞, the vector of random variables converges in distribution to a vector of independent Poisson(1)-distributed random variables.(b) G n converges to the trivial group as an IRS.
Note that the analogue to Theorem 1.3(c) also holds here.However, a much stronger statement follows directly from multiplicativity of orbifold Euler characteristic.
The case of free groups in the theorem above is very similar to results on cycle counts in random regular graphs in the permutation model (see for instance [DJPP13] and also [Bol80] for a slightly different model), so our real contribution is to the case with torsion.For surface groups similar results have very recently been proved by Magee-Puder [MP20].The case of cocompact Fuchsian groups with torsion is currently open.
1.3.The structure of the proofs.Our proofs start with the count of the number of homomorphisms Γ p 1 ,...,p m → S n .Because the presentation for our groups is very explicit, we are able to write down a closed (albeit somewhat involved) formula for h n (Γ p 1 ,...,p m ) (Proposition 3.1).
The formula we find expresses h n (Γ p 1 ,...,p m ) as a sum, so the next step is to single out the largest term in this sum.The key technical results, which most of the paper rests on, are Lemmas 4.4 and 4.5, which determine the dominant term in the sum.
The idea behind the proofs of our results on random subgroups is to first prove the analogous results for random index n subgroups of C p 1 * • • • * C p m and then use the fact that most index n subgroups of Γ p 1 ,...,p m come from index n subgroups of First, we prove Poisson statistics for the number of fixed points of an element ).This uses the method of moments together with results by Volynets [Vol86] and independently Wilf [Wil86] on h n (C p ). Then we turn these into Poisson statistics for the variables Z K , where This, together with Theorem 1.2 implies the statistics in Theorem 1.3(a).In order to keep the proof a little lighter, we did not compute explicit error terms for our Poisson approximation result in (a) and used the method of moments to prove it.Error terms could be made explicit using the error terms in Müller's results [Mül96].Moreover, the Chen-Stein method (see for instance [AGG89, BHJ92, DJPP13]) would probably give sharper bounds than the method of moments.
The fact that a conjugacy class K ⊂ Γ p 1 ,...,p m typically has very few lifts to H n if it does not lie in L p 1 ,...,p m and typically has n lifts if it does (this is essentially Theorem 1.3(a)), implies that the IRS H n converges to L p 1 ,...,p m (Theorem 1.3(b)).Using results by Elek [Ele10] and Lück [Lüc94], we then also obtain that the normalised Betti numbers of H n converge to the 2 -Betti numbers of the cover of X p 1 ,...,p m /L p 1 ,...,p m .
Finally, in Section 5.5, we sketch how to complete the proof of Theorem 1.4.
1.4.Notes and references.As opposed to the case of 2-manifolds [Dix69, MP02, LS04], there are very few 3-manifolds for which the subgroup growth is well understood.For instance, to the best of our knowledge, there isn't a single hyperbolic 3-manifold group Γ for which the asymptotic behaviour of a n (Γ) is known.It does follow from largeness of these groups [Ago13] that the number grows faster than (n!) α for some α > 0, but even at the factorial scale, the growth (i.e. the optimal α) is not known.In the more general settings of lattices in PSL(2, C) it's known in one very particular case [BPR20, Section 2.5.2].One of the difficulties in determining α in general is that for a general hyperbolic 3-manifold, no proof for a factorial lower bound is known that does not rely on Agol's work.
For Seifert fibred manifolds a little more is known: the subgroup growth of orientable circle bundles over surfaces was determined by Liskovets and Mednykh [LM00] and the subgroup growth of Euclidean manifolds can be derived from general results on the subgroup growth of virtually abelian groups [dSMS99,Sul16].
One can also ask for the number of distinct isomorphism types of subgroups, in which case even less is known [FPP + 20].
Finally, results similar to our Theorems 1.1 and 1.2 are known to hold for Baumslag-Solitar groups [Kel20].
The geometry of a random cover of a graph is a classical subject in the study of random regular graphs (see for instance [AL02,Fri08,DJPP13,Pud15]).Moreover, it is known that, as n → ∞, a random 2d-regular graph sampled uniformly from the set of such graphs on n vertices as a model is contiguous to the model given by a random degree n cover of a wedge of d circles [GJKW02,Wor99].In other words, random covers are also a tool that can be used to study other models of random graphs.
Random covers of manifolds are much less well understood.Of course, random graph covers also give rise to random covers of punctured surfaces, so some of the graph theory results can be transported to this context.Very recently, Magee-Puder [MP20] and Magee-Naud-Puder [MNP20] studied random covers of closed hyperbolic surfaces.They proved that these covers Benjamini-Schramm converge to the hyperbolic plane and that the spectral gap of their Laplacian is eventually larger than 3 16 − ε for all ε > 0 (given that his holds for the base surface).
Invariant Random Subgroups were introduced by Abért-Glasner-Virág in [AGV14], by Bowen in [Bow14] and under a different name by Vershik in [Ver12], but had been studied in various guises before (see the references in [AGV14]).Benjamini-Schramm convergence was introduced for graphs in [BS01] and for lattices in Lie groups in [ABB + 17].The fact that Benjamini-Schramm convergence implies convergence of normalised Betti numbers was proved for sequences of simplicial complexes in [Ele10], for sequences of lattices in [ABB + 17] and for sequences of negatively curved Riemannian manifolds in [ABBG18].
Acknowledgement.We thank Jean Raimbault for useful remarks.

PRELIMINARIES
2.1.Subgroup growth.As mentioned in the introduction, our results on subgroup growth are based on the connection between finite index subgroups of a group G and transitive permutation representations of G. Indeed, an index n subgroup H < G gives rise to a transitive action of G on the finite set G/H and as such, upon labelling the elements of G/H with the numbers 1, . . ., n, a homomorphism G → S n .Here S n denotes the symmetric group on n elements.This leads to the following (see [LS03, Proposition 1.1.1]for a detailed proof): Proposition 2.1.Let G be a group and n ∈ N. Then Another result we will need is on the asymptotic number of homomorphisms C m → S n (or equivalently the number of elements of order m in S n ).The result we will use is due to Volynets [Vol86] and independently Wilf [Wil86] and fits into a large body of work, starting with classical results by Chowla-Herstein-Moore [CHM51], Moser-Wyman [MW55], Hayman [Hay56] and Harris-Schoenfeld [HS68] and culminating in a paper by Müller [Mül97] in which the asymptotic behaviour of h n (G) as n → ∞ is determined for any finite group G.It states: Here Finally, we will need two results due to Müller.The first in fact also implies the previous theorem: be a polynomial with degree m ≥ 1 and let exp(P(z)) = ∑ ∞ n=0 α n x n .Suppose further that • α n > 0 for all sufficiently large n, and Then the coefficients α n satisfy the asymptotic formula where n 0 := n/(mc m ) and The second result we will need is: In fact, Müller also provides error terms and proves the theorem for more general groups; we refer to his paper for details.

Probability theory.
For our Poisson approximation results, we will use the method of moments.Given a random variable Z : Ω → N and k ∈ N, we will write Moreover, recall that a sequence of random variables Z n : Ω n → N d is said to converge jointly in distribution to a random variable Z : Ω → N d if and only if The following theorem is classical.For a proof see for instance [Bol85].

Invariant Random Subgroups.
We will phrase our results on random subgroups in the language of Invariant Random Subgroups.For a finitely generated group Γ, Sub(Γ) will denote the Chabauty space of subgroups of Γ (see for instance [Gel18] for an introduction).
We will be interested in random index n subgroups of such a group Γ.This corresponds to studying the measure µ n on Sub(Γ), defined by where δ H denotes the Dirac mass on H ∈ Sub(Γ).µ n is an example of what is called an Invariant Random Subgroup (IRS) of Γ -i.e. a Borel probability measure on Sub(Γ) that is invariant under conjugation by Γ.We will write IRS(Γ) for the space of IRS's of Γ endowed with the weak-* topology.This space has been first studied under this name in [AGV14] and [Bow14] and under a different name in [Ver12].
We will also use a characterisation for convergence in IRS(Γ) terms of fixed points.This characterisation is probably well known, but we couldn't find the exact statement in the literature (for instance [AGV14, Lemma 16] is very similar).We will provide a proof for the sake of completeness.
Given a function f : Sub(Γ) → C, we will write µ n ( f ) for the integral of f with respect to µ n (all measures considered in our paper are finite sums of Dirac masses, so this is always well defined).
Lemma 2.6.Let Γ be a countable discrete group.Set We start with the fact that for g ∈ K, µ n ({H; g ∈ H}) = 1 n µ n (Z K ).Indeed, for any p ∈ {1, . . ., n}, the map ϕ → Stab ϕ {p} gives an (n − 1)!-to-1 correspondence between transitive homomorphisms Γ → S n and index n subgroups of Γ. Z K (ϕ) equals the number of fixed points of ϕ(g) on {1, . . ., n}.As such Now, the topology on Sub(Γ) is generated by sets of the form (see for instance [Gel18]).By the Portmanteau theorem, convergence for every open set O ⊂ Sub(Γ).This is equivalent to proving that Let us first prove that our conditions on the behaviour of µ n (Z K ) imply convergence in IRS(Γ).
We start by checking (2) for sets of the form O 1 (U).Suppose g ∈ U ∩ N. Using (1) and writing K for the conjugacy class of g, by our assumption on µ n (Z K ).Now we deal with sets of the form O 2 (V).We will write K(g) for the conjugacy class of an element g ∈ Γ.
(1) gives us by our assumptions on µ n (Z K(g) ).This proves the first direction.
For the other direction, suppose g ∈ N then δ N (O 1 ({g})) = 1 and hence by (2) and (1), we obtain 2.4.Benjamini-Schramm convergence.Now suppose that -as many of the groups that we study do -Γ admits a finite simplicial complex X as a classifying space.Picking a 0-cell x 0 ∈ X gives an identification Γ π 1 (X, x 0 ).Moreover, an index n subgroup H < Γ gives rise to a pointed simplicial covering space This means that the measure µ n above also gives rise to a probability measure ν n on the set for some D > 0, where two pairs (Y, y 0 ) ∼ (Y , y 0 ) if there is a simplicial isomorphism Y → Y that maps y 0 to y 0 .This set K can be metrised by setting The R-balls around y 0 and y 0 are isomorphic as pointed simplicial complexes .This allows us to speak of weak-* convergence of measures on K D .If there is a where δ [Z,z 0 ] denotes the Dirac mass on [Z, z 0 ], then we say that the random complex determined by ν n Benjamini-Schramm converges (or locally converges) to [Z, z 0 ].
We will write BS(K D ) for the space of probability measures on K D endowed with the weak-* topology.The procedure described above describes a continuous map for some D > 0, that depends on the choice of classifying space.2.5.Betti numbers.One reason for determining Benjamini-Schramm limits, is that they help determine limits of normalised Betti numbers.We will exclusively be dealing with homology with real coefficients in this paper.Given a simplicial complex X, we will write denote all the complexes in K D that can appear as an R-ball of a complex in K D .Note that this is a finite list, the length of which depends on R and D.Moreover, given a finite simplicial complex X of which all 0-cells degree at most D, we will write where V(X) denotes the set of 0-cells of X. Elek's theorem now states: Theorem 2.7 (Elek [Ele10, Lemma 6.1]).Fix D > 0 and let (X n ) n be a sequence of finite simplicial complexes in which the degree of every 0-cell is bounded by D exists for all k ∈ N.
Often, an explicit limit for these normalised Betti numbers can be determined in terms of 2 -Betti numbers.We will not go into this theory very deeply in this paper and refer the interested reader to for instance [Lüc02] or [Kam19] for more information.
If Γ is a group and X is a finite Γ-CW complex, then we will write b (2) k (X; Γ) for the k th 2 -Betti number of the pair (X, Γ).
We will rely on the Lück approximation theorem [Lüc94] (see also [Kam19, Theorem 5.26]).If Γ is a group and Theorem 2.8 (Lück approximation theorem).Let Γ be a group and X be a finite free Γ-CW complex.Moreover, let (Γ i ) i be a chain of finite index normal subgroups of Γ and set In order to prove convergence of Betti numbers we are after (Theorem 1.3(c)), we will use the approximation theorems of Elek and Lück to deduce the following lemma.Like Lemma 2.6, this lemma is probably well known but, as far as we know, not available in the literature in this form, so we will provide a proof.Lemma 2.9.Let Γ be a group that admits a finite simplicial complex X as a classifying space.Set If there exists a normal subgroup N Γ such that Γ/N is residually finite and Then for every ε > 0 and every k ∈ N, where X denotes the universal cover of X.
Proof.Recall that V(X) denotes the set of 0-cells of X and write D for the maximal degree among these 0-cells.Fix a choice of 0-cell x 0 ∈ V(X), to obtain an identification Γ π 1 (X, x 0 ) and denote the measure on K D induced by µ n by ν n ∈ BS(K D ).Finally, we will let (Z, z 0 ) → (X, x 0 ) denote the pointed cover corresponding to N.
For g ∈ K ⊂ Γ, where K is a conjugacy class, Z K (H) equals the number of lifts of x 0 at which the loop in X corresponding to g lifts to a closed loop.Now consider the set W R of all g ∈ Γ that have translation distance at most R on the universal cover X.This set consists of a finite number of conjugacy classes. If (3) then the number of lifts y in the cover of X corresponding to H, around which the R-ball B R (y) is not isometric to the R-ball B R (z 0 ) around z 0 ∈ Z is o(n) (this uses that W R consists of finitely many conjugacy classes).
Lemma 2.6 tells us that for any finite set of conjugacy classes, (3) is satisfied with asymptotic µ n -probability 1.So we obtain that for every R, ε > 0 Now, since V(X) is finite we can repeat the argument finitely many times and obtain that for each R > 0 there is a finite list B 1 , . . ., B L of finite simplicial complexes and a finite list of densities ρ 1 , . . ., ρ L > 0 such that So, by Theorem 2.7, for every ε > 0 there exists a δ > 0 such that if we fix any finite pointed complex Using the fact that Γ/N is residually finite, we can find a chain of normal subgroups H i Γ/N of finite index such that ∩ i H i = {e}.We lift this sequence of subgroups to a sequence H i Γ and obtain a sequence of pointed covers (Q i , q i ) → (X, x 0 ).Now, if we set by construction.So, for (4), we can take a (Q i , q i ) deep in the sequence we just constructed.Moreover, by Theorem 2.8 we have which finishes the proof.

A CLOSED FORMULA
Our first objective is now to derive a closed formula for h n (Γ p 1 ,...,p m ).In this section we will prove: 3.1.Counting roots.The main ingredient for the formula above is the count of the number of m th roots of a given permutation π ∈ S n -i.e. the number Note that this number only depends on the conjugacy class of π.The computation of N m (π) is a classical problem, that to the best of our knowledge has been first worked out by Pavlov [Pav82].For the sake of completeness, we will give a proof here.
Let us first introduce some notation.Recall that the conjugacy class of a permutation π ∈ S n is determined by its cycle type -the unordered partition of n given by the lengths of the cycles in a disjoint cycle decomposition of π.In what follows the notation 1 r 1 2 r 2 • • • n r n will denote the partition of n that has r 1 parts of size 1, r 2 parts of size 2, et cetera.K(1 r 1 2 r 2 • • • n r n ) ⊂ S n will denote the corresponding conjugacy class.In this notation, we will often omit the sizes of which there are 0 parts and write i for i 1 .
where K(m, l, r) is as in Proposition 3.1.
Note that there may be an l such that r l > 0 and K(m, l, r l ) = ∅.In this case, N m (π) = 0.
Proof.First observe that when σ ∈ K(k) ⊂ S k , then which also describes what happens to the cycles in a general permutation σ ∈ S n upon taking its m th power.
This puts restrictions on which conjugacy classes K of S n can contain m th roots of π.In order to describe these restrictions, we will split the cycles of the m th root σ according to which cycles of π they contribute.
So first assume π ∈ K(l r ) ⊂ S lr -i.e.π consists solely of l-cycles.If σ ∈ K(1 s 1 • • • (lr) s lr ) satisfies σ m = π, then the observation above tells us that all cycles of σ must have lengths that are multiples of l.Moreover, In particular, we obtain that s il = 0 for all i > m.Moreover, we have that ∑ i s il • il = rl.
We will now first completely work out the proof for π ∈ K(l r ).The expression for a more general permutation can then be obtained by multiplying the result from this special case over all cycle lengths that appear in the permutation.
So, given a conjugacy class K(1 s 1 • • • (lr) s lr ) that satisfies these conditions, we must count the number of m th roots it contains.That is, for every i such that gcd(i • l, m) = i, we must count how many i • l-cycles C we can build out of i cycles of length l from π such that C m consists exactly of these cycles of π.We claim that the number of such cycles C, given i cycles from π is (5) for these cycles from π. C will be of the form Then taking some 1 ≤ j ≤ i • l, there are i • l choices for the value of β j .Given a choice, we also know the value of β j+m , β j+2m , . . ., β j+lm , since supposing β j = α k , we obtain Hence, by assigning a value to one β, we have assignments for l β's.In this way, we have i • l ways to assign the first l values of C, i • l − l ways to assign the second l values, and so on, until we have l way to assign the last l values of C.This results in ways to place the elements of C such that C m = π.However, by rotating the first item in C through the i • l places without changing the order of elements, gives us equivalent cycles within S i•l .There are i • l of these, and so after dividing out by these, the number of possible cycles C such that C m = π is given by , where the extra factors account for the number of partitions of the cycles in π into s il sets containing i cycles and we used the fact that ∑ i s il i = r to obtain the second expression.
In order to simplify notation a little we write k i = s il .Summing over all conjugacy classes that contain m th roots of π, we get that π ∈ K(l r ) has For a general permutation π ∈ K(1 r 1 • • • n r n ) ⊂ S n , we take the product of this expression over all cycle lengths that appear in π.

The proof of Proposition 3.1.
Proof.Given a conjugacy class K ⊂ S n , we write N m (K) for the number of roots of an element π ∈ K.We have Using Proposition 3.2 and the fact that |K(1 gives the formula.

ASYMPTOTICS
The goal of this section is to prove Theorem 1.1 -the asymptotic number of index n subgroups of Γ p 1 ,...,p m as n → ∞.
First we will determine the asymptotic behaviour of h n (Γ p 1 ,...,p m ).This is done by singling out the dominant term in the expression we found for it in Proposition 3.1.After that, we show that most homomorphisms are transitive, from which the asymptotic number of index n subgroups directly follows (using Proposition 2.1) 4.1.Homomorphisms.We will prove Theorem 4.1.Let p 1 , . . ., p m ∈ N >0 such that ∑ m j=1 1 Let us write The first thing we shall need is a bound on these numbers τ p,l,r .To this end, we consider the ordinary generating function for τ p,l,r for fixed p and l, defined by Proof.By definition it holds Let the set I p,l = {i ≤ p | gcd(i • l, p) = i} =: {i 1 , . . .i m }.Then the above sum becomes This, together with Theorem 2.3 implies Corollary 4.3.
(a) Let p ∈ N. Then where Proof.Item (a) is a direct consequence of Theorem 2.3, using that K(p, 1, n) is non empty when p ≤ n -i.e. that the symmetric group contains elements of order p whenever p ≤ n -and that I p,1 consists of the divisors of p.
For (b), observe that all the coefficients in F p,l are non-negative.As such, τ p,l,r ≤ F(x 0 )/x r 0 for all x 0 ∈ (0, ∞).Setting x 0 = (r • l) 1/p and using Lemma 4.2, we get We note that any i satisfying gcd(i • l, p) = i must also satisfy i|p and hence taking the product over i|p results in a bound on taking the product over i ∈ I r,l , which proves item (b).
Note that our proof for (a) does not work for τ p,l,n with l = 1 and p ≥ 2, since it does not hold that τ p,l,n = 0 for all large n.To see this, let n be prime.Then the only i ∈ N satisfying gcd(i • l, p) = i is when i = p.Hence, the only vectors k ∈ K(p, l, n) have to be of the form k = (0, . . ., 0, n p ).However, if n > p is prime then n p will never be an integer.
We start with the terms in which r 1 is "small", this is the longest part of the proof.
Proof.This will follow from Corollary 4.3.Let us write In the product above, we have r l l ≤ n.Using this and the fact that (r l l) j/p i /l ≤ r l to bound the exponential factors, we obtain Now we use that ∑ l≥2 r l ≤ n−r 1 2 and get using the fact that the number of partitions of n is bounded by exp(π √ 2n/3) (see for instance [Apo76, Theorem 14.5]).Using Robbins's [Rob55] version of Stirling's approximation, one can write r 1 !≤ C • √ r 1 (r 1 /e) r 1 for some universal constant C > 0, whenever r 1 > 0.Moreover, the term corresponding to r 1 = 0 in the sum above is smaller than that corresponding to r 1 = n − δ, if we increase the constant C a little (depending on p 1 , . . ., p m ), we may write On the other hand, Corollary 4.3(a), together with Stirling's approximation, implies that So, there is a constant C > 0, depending on p 1 , . . ., p m only, such that for some D > 0, depending on p 1 , . . ., p m only.This tends to 0 as n → ∞, using our assumption on δ.
For the remaining terms in the sum, we have: . Then for any δ > 0, it holds that ∑ r 1 ,...,r n ≥0 s.t.∑ l r l •l=n and n−δ≤r which is a uniformly bounded number in the sum we consider.As such, there exists some constant C > 0, depending on p 1 , . . ., p m only such that ∑ r 1 ,...,r n ≥0 s.t.∑ l r l •l=n and n−δ≤r Because this is a finite sum, we may apply Corollary 4.3(a), which implies that for two constants D, D > 0. Filling this in, we see that there exists a constant C > 0 such that ∑ r 1 ,...,r n ≥0 s.t.∑ l r l •l=n and n−δ≤r The latter tends 0 as n → ∞, using our assumption that m − 1 > ∑ i We are now ready to prove the asymptotic equivalent for h n (Γ p 1 ,...,p m ).

Recall that
Φ p 1 ,...,p m : is the surjection that sends the generator x i ∈ Γ p 1 ,...,p m to a generator of the i th factor on the right.The lemmas above also prove: Proof.This can be done indirectly by comparing Theorem 4.1 to the asymptotic equivalent for h n (C p 1 * • • • * C p m ) due to Volynets [Vol86] and independently Wilf [Wil86].The fact that these two sequences are asymptotic to each other implies the result.It can also be seen directly from Lemmas 4.4 and 4.5.Indeed, they imply that h n (Γ p 1 ,...,p m ) is asymptotic to the term corresponding to (r 1 , r 2 , . . ., r n ) = (n, 0, . . ., 0) in (6).In the proof of this formula, these vectors (r 1 , r 2 , . . ., r n ) that are summed over correspond to the conjugacy classes that roots are counted of.The term that determines the asymptotic are the roots of unity in S n , i.e. maps is the identity element in S n .These are exactly the maps that factor through Φ p 1 ,...,p m .4.2.Subgroups.We are now ready to prove our main theorem -the asymptotic behaviour of the number of index n subgroups of Γ p 1 ,...,p m ).We shall do this by showing that h n (Γ p 1 ,...,p m ) ∼ t n (Γ p 1 ,...,p m ) as n → ∞, that is for large n most of the homomorphisms from Γ p 1 ,...,p m to S n are transitive.After that, Proposition 2.1, together with Theorem 4.1 gives the asymptote.
Proof.The quickest way to prove that most homomorphisms are transitive, is to use the fact that asymptotically almost all homomorphisms Γ p 1 ,...,p m factor through the homomorphism Müller (Theorem 2.4) proved that asymptotically almost all homomorphisms C p 1 * • • • * C p m → S n are transitive, which, together with Proposition 2.1 and Stirling's approximation, gives the result.For a more direct proof (that essentially goes along the same lines as that of Müller), we can use that the number of transitive homomorphisms G → S n can be recursively computed from the sequence (h n (G)) n .That is, we have (for a proof see [LS03, Lemma 1.1.3]): Combining this with the bounds from Theorem 4.1, a further computation and Proposition 2.1 also gives the result.

RANDOM SUBGROUPS AND COVERS
In this section we will study the properties of random index n subgroups of Γ p 1 ,...,p m and random degree n covers of torus knot complements.
The basic idea is to prove that a random index n subgroup of )) converges to the trivial subgroup.This, together with Theorem 1.2 will then imply that a random index n subgroup of Γ p 1 ,...,p m converges to L p 1 ,...,p m .Both of these results will be quantitative in the sense that we have control over the number of conjugacy classes a given conjugacy class of either C p 1 * • • • * C p m or Γ p 1 ,...,p m lifts to in a random index n subgroup of the corresponding subgroup (Theorem 1.3(a)).This then immediately implies the fact that a random degree n cover of X p 1 ,...,p m Benjamini-Schramm converges to X Φ p 1 ,...,p m .Combined with Lemma 2.9, this convergence implies our result on Betti numbers.5.1.Set-up.Given a group Γ and n ∈ N, we will write is a conjugacy class then we will write for the random variable that measures the number of conjugacy classes that K splits into, i.e.

Z K (H) = |(K ∩ H)/H|
where H acts on K ∩ H by conjugation.Note that if we fix any g ∈ K and ϕ : Γ → S n is a transitive homomorphism corresponding to H (cf. Proposition 2.1), then Our goal now is to show that these random variables are asymptotically Poissondistributed.

Poisson statistics for random elements of
Our first step is to enlarge our probability space and prove our results there.Concretely, the expression for Z K in terms of fixed points is well-defined for any homomorphism, not just for transitive ones.As such, we can interpret Z K as a random variable Before we prove this theorem, we observe that this immediately implies that on Hom(C p 1 * • • • * C p m , S n ), the random variables Z K are asymptotically Poisson-distributed and independent.
Proof of Theorem 5.1.We will write Λ = C p 1 * • • • * C p m and H n (Λ) = Hom(Λ, S n ).Let us once and for all fix g i ∈ K i for i = 1, . . ., r and write these elements as words in the generators x 1 , . . ., x m , i.e. we write where x j i,t and x j i,t+1 are distinct for all t = 1, . . ., l i − 1.By potentially changing the conjugate, we may also assume that x j i,l i = x j i,1 .Moreover, we will choose the unique representative such that 0 < s i,t < p j i,t for all t = 1, . . ., l i .We will write |g i | for the word length of g i .So Now, if we want v ∈ {1, . . ., n} to be a fixed point of ϕ(g i ) for some ϕ ∈ H n (Λ), then there need to be sequences (w t,0 w t,1 . . .w t,s i,t ), for t = 1, . . ., l i , such that (7) ϕ(x j i,t )(w t,q ) = w t,q−1 , q = 1, . . ., s i,t w 1,1 = w l i ,s i,l i = v.
In other words, if we want v to be a fixed point of g i , then certain sequences (for which there are many choices) need to appear in the disjoint cycle decompositions of the images of the generators x 1 , . . ., x m .Figure 1 gives an example of the situation.
We will call such a sequence of sequences corresponding to v being a fixed point for g i a g i -cycle based at v. The sequences (w t,0 w t,1 . . .w t,s i,t ) appearing in the cycle will be called the words in the cycle.The elements from {1, . . ., n} appearing in the words will be called the labels in them.If ϕ satisfies (7) for a given g i -cycle ω, we will say that ϕ satisfies ω.
Observe that the random variable H n (Γ) → N counts r-tuples (F 1 , F 2 , . . ., F r ) where F i is a sequence of k i distinct fixed points of ϕ(g i ).As such, we may write where α i a k i -tuple of g i -cycles based at different elements of{1, . . ..n} and 1 α : H n (Λ) → {0, 1} satisfies 1 α (ϕ) = 1 if and only if ϕ satisfies all the g i -cycles contained in α for all i = 1, . . ., r.Note that many of these indicators are constant 0 functions, because the combination of labels involved leads to a contradiction about the properties of ϕ(x j ) for some j ∈ {1, . . ., m} We will write where A 1 (n) = {α ∈ A; every label appears at most once in α} and The remainder of the proof now consists of proving two facts, namely We start with estimating Observe that ; ϕ satisfies all the g i -cycles contained in α for all i = 1, . . ., r h n (Λ) .
In order to count the numerator on the right hand side, we need to count the number of ways to complete the information given in α to a homomorphism Λ → S n .We do this as follows.The words from the g i -cycle must appear as parts of cycles in a disjoint cycle decomposition of the x j 's.So, a choice needs to be made for the lengths of these cycles, which words appear together in a cycle, and which other labels appear in these cycles.Once these cycles have been completed, this determines m homomorphisms C p j → S D j , where D j depends on the chosen cycle lengths.To complete this into a homomorphism C p j → S n , we have the choice out of h n−D j (C p j , S n ) homomorphisms.This, as n → ∞, gives a total of ∼ m ∏ j=1 ∑ {S 1 ,...,S t }|=W j (α) ∑ d 1 ,...,d t |p j d q ≥∑ w∈S q (w) C(S, d) • n ∑ q d q −∑ w∈W j (α) (w) h n−∑ q d q (C p j ) ways to complete the information in α to a homomorphism, where • W j (α) is the set of words that appear in α and pose a condition on ϕ(x j ), • the notation {S 1 , . . ., S t } |= W j (α) means that {S 1 , . . ., S t } forms a set partition of W j (α) (these are the groups of words that are going to appear together in cycles in ϕ(x j )), • the numbers d 1 , . . ., d t are going to be the lengths of the cycles containing the words in the sets {S 1 , . . ., S t }, • (w) is the number of labels in a word w, • C(S, d) is a combinatorial constant that counts the number of ways to distribute the words over cycles in according to {S 1 , . . ., S t } and d 1 , . . ., d t .Moreover, if the set partition {S 1 , . . ., S t } consists of singletons and d 1 = d 2 = . . .= d t = p j then C(S, d) = 1 • and we have already made one simplification: the powers of n should in reality take the form of a falling factorial.However, since we are only interested in asymptotics and all the products involved are of fixed bounded length, we replaced them by powers of n, whence the "∼".Now we notice that all the sums and products involved are finite, we may apply Theorem 2.2 to single out the largest term.This implies that, as n → ∞, Another important thing to observe is that E Hom n [1 α ] is constant on A 1 (n): it does not depend on the labels involved.This implies that (8) it is the number of ways to the label the g i -cycles with distinct elements from {1, . . ., n}.Together with (8), this proves our claim that In order to prove that the other term tends to zero, we argue in a similar fashion.Indeed, we will think of the g i -cycles as labelled graphs: the vertices are the labels and the edges are determined by the conditions in (7).In this language the graphs in A 1 (n) are exactly those that consist of disjoint circuits.The graphs in A 2 (n) come in finitely many isomorphism types and all have more edges than vertices.
We write where the sum is over isomorphism types types G of graphs appearing in A 2 (n) and A G (n) consists of all α ∈ A 2 (n) whose graph has isomorphism type G.
Suppose G is such an isomorphism type with v(G) vertices and e(G) edges.Again E Hom n [1 α ] is the same for all α ∈ A G (n).Moreover, with exactly the same arguments as above we have Because v(G) < e(G) for all G appearing in the sum, the sum indeed tends to zero, which finishes the proof.
be distinct conjugacy classes.Then, as n → ∞, the vector of random variables Proof.We will again write Λ = C p 1 * • • • * C p m .Using the (n − 1)!-to-1 correspondence between transitive permutation representations Γ → S n and index n subgroups of Γ (i.e.Proposition 2.1), what we need to prove is that for all We have , by Corollary 5.2 and Theorem 2.4 (note that this uses that ∑ m i=1 1
The distribution of Z K i is the same on T n (Γ) as it is on A n (Γ).By Theorem 1.2, as n → ∞ a typical element of T n (Γ) factors through Φ p 1 ,...,p m .So the limiting distribution of the Z K i is the same as that on T n (Γ) Φ := {ϕ ∈ T n (Γ); ϕ factors through Φ p 1 ,...,p m }.
Now if K i ⊂ L p 1 ,...,p m = ker(Φ p 1 ,...,p m ) then Z K i is constant and equal to n on T n (Γ) Φ .If K i ⊂ L p 1 ,...,p m , then the limiting distribution of Z K i on T n (Γ) Φ is given by Theorem 5.3.Finally, Theorem 5.3 gives us the asymptotic independence among the Z K i for K i ⊂ L p 1 ,...,p m and the independence of the whole set follows from the fact that constant random variables are independent of any other random variable.
Since both orbifold Euler characteristic and 2 -Betti numbers are multiplicative with respect to finite index subgroups (see for instance [Lüc02, Theorem 1.35(9)] or [Kam19, Theorem 3.18(iv)] for the latter), the lemma follows.
Proof of Theorem 1.3(c).This is now direct from Theorem 1.3(b) and Lemmas 5.5 and 2.9.
5.5.Random index n subgroups of Fuchsian groups.In this last section we discuss applications of our results to random subgroups of Fuchsian groups.We have: Proof sketch.First of all note that non-cocompact Fuchsian group of finite covolume are exactly groups of the form F r * C p 1 * • • • * C p m , with −r + m − 1 − ∑ m i=1 1/p i < 0, where F r denotes the free groups on r generators.
If r = 0, (a) and (b) are the content of Theorem 5.3 and Corollary 5.4 respectively.If r > 0, the proof of Theorem 5.3 needs to be adapted slightly: r of the generators are now allowed to have any permutation of their image and not just permutations of a fixed order.With exactly the same strategy (and slightly easier computations, which we leave to the reader) the analogue of Theorem 5.3 can now be proved (if m = 0, much better bounds are in fact available [DJPP13]).In order to prove the analogue of Corollary 5.4, the only new ingredient that is needed is that t n (Γ)/h n (Γ) → 1.When m = 0, this is a direct consequence of Dixon's theorem [Dix69].For the remaining cases, the proof has not been written down, but a similar strategy does the trick.Indeed, the results by by Volynets-Wilf (Theorem 2.2) together with Stirling's approximation that for p > 1,

Lemma 4. 4 .
Let p 1 , . . ., p m ∈ N >0 such that m − 1 > ∑ m j=1 1 p j. Then for any δ as well, where we equip Hom(C p 1 * • • • * C p m , S n ) with the uniform measure P Hom n .We will denote the expected value with respect to this measure by E Hom n .We have: Theorem 5.1.Let p 1 , . . ., p m ∈ N and let K 1 , . . ., K r ⊂ C p 1 * • • • * C p m be distinct conjugacy classes.Then for any k 1 , . . ., k r ∈ N we have lim n→∞ E Hom n

5. 3 .
Poisson statistics for random subgroups of C p 1 * • • • * C p m and Benjamini-Schramm convergence.From the above we also obtain that Z K are asymptotically independent Poisson-distributed variables when seen as random variables on the set of index n subgroups of C p 1 * • • • * C p m .Theorem 5.3.Let p 1 , . . ., p m ∈ N such that ∑ m i=1 1 by Corollary 5.2 and Theorem 2.4, which proves the result.Our next goal is to use this to prove convergence of a random index n subgroup of C p 1 * • • • * C p m : Corollary 5.4.Let p 1 , . . ., p m be such that ∑ m i=1 1
d<p i → ∞, where B is a constant depending on (r, p 1 , . .., p m ).(n − k) d/p i + k d/p i − n d/p i d as n → ∞, which settles the remaining cases.