Some Examples in the Theory of Subgroup Growth

SOME EXAMPLES IN THE THEORY OF SUBGROUP GROWTH

Thomas W. Muller¨ and Jan-Christoph Schlage-Puchta

Abstract. By estimating the subgroup numbers associated with various classes of large groups, we exhibit a number of new phenomena in the theory of subgroup growth.

Key-words: subgroup growth, large groups, one-relator groups, HNN-extensions MSC-Index: 20E07

1. Introduction

For a group Γ, denote by sn(Γ) the number of subgroups of index n in Γ. If, for instance, Γ is finitely generated or of finite subgroup rank, then sn(Γ) is finite for all n. Call a group Γ large (in the sense of Pride [8]), if Γ contains a finite index subgroup projecting onto F2, the free group of rank 2. Free products Γ of finitely many finite groups with χ(Γ) < 0 and surface groups involving three or more generators are large in this sense and their subgroup growth is by now rather well understood (at least from an asymptotic point of view); cf. [4] and [7]. However, for large groups other than the ones just mentioned, not even plausible conjectures seem to exist. The present paper arose out of an attempt to develop some feeling in this direction. In accordance with the rather informal character of our investigation, we present a number of new and interesting phenomena in a sequence of (only loosely connected) examples.

For a cyclically reduced word w = w(x1, . . . , xd) involving the generators x1, . . . , xd, denote by Γw the one-relator group

Γw = x1, . . . , xd|w(x1, . . . , xd) = 1 associated with w. In [2, §1], Lubotzky suggested that it might be fruitful to compare the subgroup growth of a d-generator one-relator group Γd with sn(Fd−1), the subgroup growth of the free group Fd−1 on d − 1 generators; in particularly he asked, for which one-relator groups Γd the limit s (Γ ) `(Γ) := lim n d n→∞ sn(Fd−1) does exist, and what spectrum of values it can attain. The subgroup growth of free d−1 groups is well known and satisfies sn(Fd) ∼ n · (n!) ; thus the existence of a finite value for `(Γ) would yield a simple asymptotic formula for sn(Γ). Our first example in particular demonstrates that the growth of a d-generator one-relator group can be arbitrarily close to the subgroup growth of Fd; in particular, we provide a host of examples where the limit `(Γ) is infinite. Moreover, we construct d-generator one- relator groups, whose asymptotic growth coincides with that of a free product of d 1 2

finite cyclic groups. In contrast to these observations, Example 2 exhibits a class of one-relator groups for which `(Γ) exists and equals 1 or 2. Example 1 might have left the reader with the impression that subgroup growth of a one-relator group with d generators significantly faster than that of Fd−1 is tied to the occurrence of large powers in the defining relation. This idea is refuted in Example 3, which shows that the group Γ = x, y|[x, y], y = 1 has subgroup growth (roughly) of the order of magnitude (n!)1/2. Section 4 is concerned with the relationship between subgroup growth and free subgroup growth. For free products Γ, the order of magnitude of sn(Γ) increases as χ(Γ) & −∞, f as does the growth of the function sn(Γ), the number of free subgroups of index n. Example 4 provides a sequence (Γp)p prime of virtually free groups satisfying sn(Γp) = p−1 p−1 2p +o(1) f p2 (n!) , while χ(Γp) % 0 and sn(Γp) is of maximal order n! . In Section 5 we consider the question how small the difference between subgroup growth functions can become without actually vanishing. We construct a sequence of pairs ¯ 1−1/k ¯ (1−1/k)/2+o(1) (∆k, ∆k)k≥2 satisfying sn(∆k) ≈ (n!) , and |sn(∆k) − sn(∆k)| < n! .

There are many instances, where the fundamental group π1(X) of some manifold X is large. From the point of view of the covering theory of such spaces, counting conjugacy classes of ﬁnite index subgroups is more natural than counting individual subgroups. From an asymptotic point of view, however, these problems turn out to be closely related. Denote by cn(Γ) the number of conjugacy classes of index n subgroups of Γ. In Section 6 we show that if Γ has subgroup growth of size (n!)µ+o(1) for some constant µ > 0, then −1 µ/2+ c(n) = n sn(Γ) + O (n!) holds for every > 0. It appears plausible that a ﬁnitely presented large group Γ should have smooth subgroup growth. In particular, we believe (and in some cases proved) that all examples in Sections 2–6 satisfy sn+1(Γ) ∼ anb. In Section 7, we show that the subgroup growth sn(Γ) of the free product Γ = Z2 ∗ Z2 of two copies of the additive group of 2-adic integers exhibits large oscillation. More precisely, we prove that

1+o(1) sn+1(Γ) log n/ log 2−3 sn(Γ) = (n!) while = Ω(n ). sn(Γ) Here we write f(n) = Ω(g(n)), if there exists some positive constant c such that the inequality |f(n)| > cg(n) has inﬁnitely many solutions, and sn(Γ) is to be understood in the topological sense; that is, it counts open subgroups of ﬁnite index.

All our examples in some sense refute seemingly plausible ad hoc conjectures. In the final section, we formulate five groups of problems of somewhat varying degree of dif- ficulty, which appear to have a chance to be true, and whose solution would shed considerable light on the relationship between structural and asymptotic invariants of large groups. 3

We will repeatedly use the connection between subgroups and permutation representa- 1 tions. Let Sn be the symmetric group on n symbols, and write hn(Γ) = n! | Hom(Γ,Sn)|. Then we have the transformation formula1 X sk(Γ) hn−k(Γ) = nhn(Γ), n ≥ 1. (1) 1≤k≤n

For the task of computing hn(Γ) for a group Γ given by a concrete presentation Γ = hx1, . . . , xd|Ri, note that homomorphisms ϕ :Γ → Sn correspond bijectively to d- d tuples of permutations (π1, . . . , πd) ∈ Sn satisfying the relations in R; that is, counting such homomorphisms is equivalent to counting solutions of systems of equations in the symmetric group. This fact is particularly useful for one-relator groups.

2. One-relator versus free groups

Let Γ = Γd be a d-generator one-relator group. Our ﬁrst example shows that the growth of sn(Γd) can get arbitrarily close to that of sn(Fd); in particular, the limit s (Γ ) `(Γ) = lim n d n→∞ sn(Fd−1) need not exist in general. Put

X −1 α(Γ) := inf aj , {a1,...,ad} 1≤j≤d where the inﬁmum is taken over all multisets {a1, a2, . . . , ad} of positive integers such ¯ that Γ projects onto the free product Γ = Ca1 ∗ Ca2 ∗ · · · ∗ Cad . For a word w = w(x1, . . . , xd) and 1 ≤ j ≤ d denote by ej(w) the greatest common divisor of the exponents of xj in w. Clearly, the group ¯ Γ = Ce1(w) ∗ Ce2(w) ∗ · · · ∗ Ced(w) is a homomorphic image of Γw, which yields the upper bound

X −1 α(Γw) ≤ ej(w) . (2) 1≤j≤d With these preliminaries out of the way, we can now state our ﬁrst example. Example 1. (i) Let Γ be a one-relator group with d generators such that α(Γ) < d − 1. Then we have log s (Γ) lim inf n ≥ d − 1 − α(Γ); (3) n→∞ n log n in particular, if α(Γ) < 1, then sn(Γd) diverges super-exponentially fast to inﬁnity. sn(Fd−1)

1Cf. [1, Proposition 1] or [3, Proposition 1]. A far reaching generalization of this counting principle is found in [6]. 4

a1 a2 ad (ii) Let w = x1 x2 . . . xd be a word in d ≥ 2 generators with positive exponents P 1 a1, a2, . . . , ad satisfying j 1/aj ≤ 2 . Then, as n → ∞,  

 nνj /aj 1 + χ  −χ  X X  sn(Γw) ∼ sn(Ca1 ∗ · · · ∗ Cad ) ∼ K(n!) exp + log n , (4)  νj 2  1≤j≤d νj |aj  νj

P −1 exp − (2aj)   j  2|aj K := 1−χ √ ; (2π) 2 a1a2 . . . ad

log sn(Γ) P in particular, lim inf n log n = −χ, and consequently we have α(Γw) = j 1/aj.

Apart from the estimate (2) relating α(Γ) to the subgroup growth of Γ, we do not see any general way to bound α from below. This is somewhat strange, for the deﬁnition of α is purely algebraic. It would be worthwhile to determine the value of α in the above example without the use of asymptotic invariants. For a given group Γ, the set of all images which are free products of ﬁnite cyclic groups can be ordered via epimorphisms in the obvious way; however, this set may then have more than one maximal element. 6 6 8 9 For instance, if w = x1x2x1x2, then both C2 ∗C15 and C3 ∗C14 are homomorphic images of Γw.

Proof of Example 1. (i) Given ε > 0 with α(Γ) + ε < d − 1, choose numbers a1, . . . , ad ∈ N ∪ {∞} such that 1 1 1 α0 := + + ··· + ≤ α(Γ) + ε, a1 a2 ad ¯ and such that Γ projects onto Γ := Ca1 ∗ Ca2 ∗ · · · ∗ Cad . Then, for n ≥ 1, ¯ d−α0−1 d−α(Γ)−ε−1 sn(Γ) ≥ sn(Γ) ≥ (1 + o(1))(n!) ≥ (1 + o(1))(n!) , (5) ¯ where we have estimated sn(Γ) via [4, Theorem 1], using the fact that χ(Γ)¯ = α0 − d + 1 ≤ α(Γ) − d + ε + 1 < 0. Assertion (i) follows immediately from (5).

(ii) Denote by re the e-th root number function of Sn, that is, e re(g) = |{x ∈ Sn : x = g}|, g ∈ Sn. We have d a1 a2 ad | Hom(Γ,Sn)| = (g1, . . . , gd) ∈ Sn : g1 g2 . . . gd = 1 X (6) = ra1 (C1)ra2 (C2) . . . rad (Cd)N(C1,...,Cd),

C1,...,Cd where d N(C1,...,Cd) := (x1, . . . , xd) ∈ Sn : xi ∈ Ci (1 ≤ i ≤ d), x1x2 . . . xd = 1 , 5 and C1,C2,...,Cd are conjugacy classes of Sn. The contribution of the term corresponding to C1 = C2 = ··· = Cd = 1 in (6) is ¯ | Hom(Ca1 ,Sn)| · | Hom(Ca2 ,Sn)| · · · · · | Hom(Cad ,Sn)| = | Hom(Γ,Sn)|, ¯ where Γ := Ca1 ∗Ca2 ∗· · ·∗Cad , and we shall show that the sum over the remaining terms is of lesser order of magnitude. Since N(C1,...,Cd) is invariant under permutation of its arguments, we may assume that |C1| ≥ |Ci| for all i. Hence d−1 N(C1,...,Cd) = (x2, . . . , xd) ∈ Sn : xi ∈ Ci (2 ≤ i ≤ d), x2x3 . . . xd ∈ C1

≤ |C2| · |C3| · ... · |Cd|

Y 1−αj ≤ |Cj| 1≤j≤d P for any choice of non-negative real numbers α1, α2, . . . , αd satisfying j αj = 1. For a 2 P non-empty set I ⊆ [d], define SI := 1/ai. By our assumption, SI ≤ 1/2, and, by P i∈I definition, i∈I 1/(SI ai) = 1. Using the estimate given above for N(C1,...,Cd) with ¯ αj = 1/(SI aj), dividing equation (6) by | Hom(Γ,Sn)|, and interchanging the order of product and sum, we find that

1−1/(SI ai) X Y X ra (C)|C| = i . | Hom(Ca ,Sn)| I i∈I C6=1 i ∅6=I⊆[d] Consider the factor in the last expression corresponding to i ∈ I for a given set I. Grouping the conjugacy classes according to the number ` of points moved by each of its elements, this factor becomes

1−1/(SI ai) X X ra (C)|C| i = | Hom(Ca ,Sn)| 2≤`≤n C i C moves ` points

2For a positive integer n, we denote by [n] the standard set {1, 2, . . . , n} of size n. 6

Indeed, the smallest such class belongs to the partition (2`/2) if ` is even and to the `−3 1 partition (2 2 , 3 ) if ` is odd, which can be seen by splitting larger cycles and estimating |C| accordingly. Using the latter estimates, as well as the fact that X∗ rai (C)|C| ≤ `!, C and collecting terms, we ﬁnd that the contribution of terms with l ≤ n − 1 to (7) is bounded above by − 1 ( 1 −1) − 1 ( 1 −1) n ai SI 1 1 n ai SI 1 1 X − a ( 2S −1) X − ( −1) `! i I ≤ n ai SI , ` ` 2≤`≤n−1 2≤`≤n−1 1 since SI ≤ 2 . To estimate the contribution of the term l = n, we use the bound 1/a 1−1/ai n i /ai −1/2 | Hom(Cai ,Sn)| n! e n , which follows from [5, Theorem 5]. Inserting this bound into (7), we ﬁnd that the summand corresponding to l = n is bounded above by

1/a (n!)−1+1/ai e−n i /ai n1/2n!1−1/(2SI ai) n−1/SI , which is smaller than the terms corresponding to l = 2. Putting these estimates together it follows that

Our next example describes a situation where the limit `(Γ) does exist, and equals 1 or 2. We do not know of any examples where `(Γ) exists and attains a value diﬀerent from 1 or 2; however, this rather seems to indicate the limitation of our method and not an 7 actual property of one-relator groups. Indeed, expressing homomorphism numbers in terms of character values seems to yield an asymptotic formula for | Hom(Γ,Sn)| only if the contribution of the non-linear characters is negligible. By an argument similar to the one given in the proof of Example 2 below, this situation implies `(Γ) ∈ {1, 2}. From a somewhat more philosophical point of view, our approach may be viewed as a non-abelian and discrete analogue of the circle method, with the linear characters corresponding to the major arcs, and there are examples in number theory (for instance, additive questions involving smooth numbers) where the contribution of the major arcs and that of the minor arcs are of the same order of magnitude. It appears likely that such phenomena also exist in the non-abelian setting, but computing `(Γ) in such a situation might be diﬃcult.

Example 2. Let w = w(x1, . . . , xd) be a word of one of the forms ( v(x1, . . . , xd−4)[xd−3, xd−2][xd−1, xd] w = 2 2 2 v(x1, . . . , xd−3)xd−2xd−1xd with d ≥ 4 or d ≥ 3, respectively, and an arbitrary word v. Then `(Γw) exists and equals 1 or 2, and `(Γw) = 2 if and only if every generator occurring in v has even exponent sum.

Proof. For a word w = w(x1, . . . , xd) and g ∈ Sn denote by Nw(g) the number of solutions of the equation w(x1, . . . , xd) = g in Sn. We focus on the second case, the argument for w of the ﬁrst kind being completely analogous. Deﬁne `∗(v) to be 2 or 1, according to whether all exponent sums in v are even, or not. We have

X −1 Nw(1) = Nv(g)Nx2y2z2 (g )

g∈An

2 X X −1 2 = 2(n!) Nv(g) + Nv(g) Nx2y2z2 (g ) − 2(n!)

g∈An g∈An

∗ d−1 d−3 2 = ` (v)(n!) + O (n!) max Nx2y2z2 (g) − 2(n!) . (8) g∈An Only the computation of P N (g) needs justiﬁcation. Denoting byx ¯ the image of g∈An v ∼ d−3 x ∈ Sn in Sn/An = C2, we have v(x1, . . . xd−3) ∈ An for a tuple (x1, . . . , xd−3) ∈ Sn if and only if v(¯x1,..., x¯d−3) = 1 in C2. Hence, X n!d−3 n o N (g) = (y , . . . , y ) ∈ Cd−3 : v(y , . . . , y ) = 1 . v 2 1 d−3 2 1 d−3 g∈An The second factor on the right-hand side is the number of solutions of the linear equation v(y1, . . . , yd−3) = 0 in the vector space of dimension d − 3 over GF(2). This number of solutions equals 2d−4 or 2d−3, depending on whether `∗(v) = 1 or 2. These observations yield (8). By [7, Proposition 1],

2 2 X χ(g) 2 X −1 N 2 2 2 (g) − 2(n!) = (n!) ≤ (n!) (χ(1)) . x y z (χ(1)))2 χ χ χ(1)>1 χ(1)>1 8

The last expression is On−1(n!)2 by [7, Corollary 2]. Putting this estimate back into (8), we ﬁnd that ∗ −1 d−1 Nw(1) = ` (v) + O(n ) (n!) . Our claim follows now from an argument similar to the one given in the proof of Example 1 (ii).

3. Large growth without powers

The examples of Section 2 might have left the reader with the impression that subgroup growth of a one-relator group faster than that of the corresponding free group is tied to the occurrence of large powers in its deﬁning relation. This idea is refuted by the following result. Denote by [x, y] = xyx−1y−1 the commutator of x and y. Example 3. Let Γ = x, y|[x, y], y = 1 .

Then √ 1/2 −n/2+O(n/ log n) sn(Γ) = (n!) 2 . (9)

Proof. We have [x, y, y] = (xyx−1y−1)y(yxy−1x−1)y−1 = (xyx−1)y(xy−1x−1)y−1 = [yx, y]

Fix a conjugacy class c, an element σ ∈ c, and let µi be the number of i-cycles of σ. ∼ Q Then CSn (σ) = i Ci o Sµi . For π ∈ c ∩ CSn (σ), deﬁne µij to be the number of j-cycles of π, whose points are in distinct i-cycles of σ, and let λj be the number of j-cycles of π, which are powers of a j-cycle of σ. Counting the number of j-cycles of π, we obtain the equations X µij + λj = µj, (11) i while counting the number of i-cycles of σ, we get X j µ + λ = µ . (12) i ij i i j

Finally, by deﬁnition of the µi, we have X iµi = n (13) i 9

Given parameters µij and λj, the number of elements π realizing these parameters equals

λi Q (j−1)µij /i Y µi!ϕ(i) j i , (14) λ ! Q (µ /i)!jµij /i i i j ij where ϕ is Euler’s function. To see this, we can restrict attention to a single factor of the product, which corresponds to a direct factor of CSn (σ) . First we choose the λi i-cycles which are powers of i-cycles of σ. This can be done in µi ϕ(i)λi ways. The λj image π of π under the projection onto Sµi−λi contains µij/i cycles of length j, hence π can be chosen in (µi − λi)! Q µij /i j(µij!j ) j−1 many ways. Finally, each j-cycle of π can be realized by i tuples of cycles in CSn (σ). Formula (14) follows from these observations. Summing over all possible values of the parameters, we ﬁnd that

λi Q (j−1)µij /i X∗ Y µi!ϕ(i) j i |c ∩ CSn (σ)| = , λ ! Q (µ /i)!jµij /i i i j ij P∗ where the sum extends over all tuples of non-negative integral µij’s and λj’s satisfying (11) and (12) for all j respectively i. Next we bound the number of such tuples. We have λ ≤ µ , thus the λ can be chosen in at most Q (µ +1) ways. Since a partition of j √ j j √ i i n has O( n) parts of distinct size, all but O( n) factors in the last product are equal to 1; hence √ Y c n log n (µi + 1) ≤ e i with some absolute constant c > 0. Next, observe that µij = 0, unless µi ≥ j and µj ≥ i. The number k of pairs (i, j) satisfying these conditions is √ [ n] X 3/4 k = (i, j): µi ≥ j, µj ≥ i ≤ 2 (i, j): µi ≥ j n , j=1 since, given j ∈ [n], a partition of n has at most O(pn/j) parts of distinct sizes, which P P are repeated at least j times. Furthermore, i,j µij ≤ j µj ≤ n, hence the number of possible choices for the µij is bounded above by the number of non-negative integral solutions of the inequality x1 + ... + xk ≤ n, which equals n + k 3/4 < (n + k)k = eO(n log n). k Hence, we have

µ !ϕ(i)λi Q i(j−1)µij /i O(n3/4 log n) Y i j |c ∩ CSn (σ)| = e max λ ! Q (µ /i)!jµij /i i i j ij

λi µi 3/4 Y µi!(ϕ(i)/i) i = eO(n log n) max , λ ! Q (µ /i)!(ij)µij /i i i j ij 10 where the maximum is taken over all tuples satisfying (11) and (12). The substitution λi → λi − i, µii → µii + i changes the value of the term to be maximized by a factor i (i/ϕ(i)) λi(λi − 1) ··· (λi − i + 1) 2 , i (µii/i + 1) 1/i which is > 1, provided that λi > (in) + i. On the other hand, for each positive constant A, X X n λ ≤ µ ≤ , i i A i>A i>A hence a maximizing tuple satisﬁes X X 1/i X 2/3 λi ≤ (n + i) + µi ≤ 2n . i 2≤i≤n1/3 i>n1/3

Thus, removing all λi both from the conditions (11) and (12) and from the term to be maximized changes the expression by a factor eO(n2/3 log n), and we obtain

µi 3/4 Y µi!i |c ∩ C (σ)| = eO(n log n) max Sn Q µ /i µij (µij/i)!(ij) ij P i j ∀i: j jµij /i=µi P ∀j: i µij =µj As a candidate for the maximum consider the class c = [σ], where σ consists of an bn/2c-cycle and dn/2e ﬁxed points. We have dn/2e! |c ∩ C (σ)| = ϕ(bn/2c) + ≥ (bn/2c + 1)!. Sn bn/2c √ Q µi −c n Hence, in the sum (10) we can neglect all classes c with i µi!i < bn/2c!e . On n! the other hand, we have |c ∩ CSn (σ)| ≤ |c| = Q µi ; hence, classes with i µi!i √ Y µi n/2 c n µi!i > bn/2c!2 e i are negligible as well. Moreover, only classes c contribute signiﬁcantly to | Hom(Γ,Sn)|, for which the parameters µij can be chosen in an admissible way such that

Y µij /i n (µij/i)!(ij) < 2 . i,j Suppose that X n n iµi ≥ + √ . √ 2 log n i≤ log n Then, for π, σ ∈ c, there are at least √2n points of [n], which lie in cycles of lengths √ log n ≤ log n of both π and σ. It follows that, for each admissible tuple of µij, we have X 2n jµ ≥ √ , ij log n i,j from which we deduce that log n 2n/ log n Y 2n 2n n (µij/i)! ≥ 2 ! ≥ 2 ≥ 2 ; √ log n e log n i,j≤ log n 11 thus, we may also assume that X n n iµi < + √ . √ 2 log n i≤ log n Since X n n µ < − √ i 2 log n i would imply √ Y µi −c n µi!i < bn/2c!e , i we may restrict attention to conjugacy classes satisfying n 5n n n − √ ≤ µ ≤ + √ , 2 log n 1 2 log n and X 3n iµi ≤ √ . √ log n i≤ log n

P √4n Moreover, we have i≥2 µi ≤ log n .

Now let σ ∈ c and π ∈ c ∩ CSn (σ). Let π1 ∈ S|π1| be the permutation obtained from π by restricting to points which are moved by π and ﬁxed by σ; similarly, let π2 ∈ S|π2| be the permutation obtained from π by restricting to points which are moved by π and

σ. The class√ of π2 is determined by a partition of some integer ≤ n, hence there are c n at most e possible conjugacy classes of π2’s. The set A of points which π2 moves√ is determined by some set of cycles of σ with length ≥ 2, hence there are at most 24n/ log n choices for A. Given A, there are at most

|CSym(A)(σ|A)| choices for π2. Since σ acts without ﬁxed points on A, we have |A| |π2| |C (σ| )| ≤ !2|A|/2 = !2|π2|/2. Sym(A) A 2 2

µ1 The points of π1 can be chosen in ways, while the action of π1 on its points can |π1| be chosen in at most |π1|! ways. Hence, √ µ |π | 4n/ log n X X 1 2 |π2|/2 |c ∩ CSn (σ)| ≤ 2 !2 |π1| 2 0≤ν≤min(µ1,n−µ1) π1∈Sν

√ X µ1! n − µ1 − ν = !2(n−µ1−ν)/2eO(n/ log n). (µ1 − ν)! 2 0≤ν≤min(µ1,n−µ1) If ν is increased by 2, then every summand is changed by a factor 4(µ − ν)(µ − ν − 1) 1 1 , n − µ1 − ν 12 √ which is > 2, if µ − ν > n. For µ ≤ n , we can restrict the summation to the range √ 1 1 2 (µ1 − n, µ1], and we obtain ln m √ |c ∩ C (σ)| ≤ µ ! − µ !2n/2−µ1 eO(n/ log n) Sn 1 2 1 lnm √ ≤ ! eO(n/ log n). 2 n If µ1 > 2 , similar reasoning gives √ µ1! O(n/ log n) |c ∩ CSn (σ)| ≤ e , (2µ1 − n)! n √ the right-hand side being decreasing for µ1 > 2 + n, whence lnm √ |c ∩ C (σ)| ≤ ! eO(n/ log n) Sn 2 n for µ1 > 2 as well. Summarizing, we have shown that lnm √ | Hom(Γ,S )| = n! ! eO(n/ log n). n 2

Next, we compare hn(Γ) with hn+1(Γ). Note ﬁrst that for any class c of Sn, there 0 is a class c of Sn+1, obtained from c by enlarging one cycle by one point, such that 0 |c | ≥ n|c|. Indeed, if we choose an index i with µi+1 < µi, and turn one of the i-cycles into an (i + 1)-cycle, then we obtain a class c0 with µ i + 1 |c0| = i n|c| ≥ n|c|. µi+1 + 1 i √ In the situation discussed above, assume ﬁrst that µ11 ≤ n. Then, increasing µ1√by 1, n increases the possible choices of the set B of points moved by π1 by a factor > 3 . If √ 0 0 n 0 µ11 > n, then we can pass from π1 to some π1 with |[π1]| ≥ 3 |[π1]|; since |π1| = |π1|+1, the number of choices for the set B is decreased by a factor µ − µ + 1 √ 1 11 < n, µ11 √ 0 n hence the contribution of c compared to that of c increases by a factor ≥ 3 . Thus, √ h (Γ) n n+1 ≥ . hn(Γ) 4

In order to deduce an estimate for sn(Γ), we start from the equation n−1 X sn(Γ) = nhn(Γ) − sν(Γ)hn−ν(Γ). ν=1

We want to show that the sum is of smaller order of magnitude than the term nhn(Γ). We have n−1 n−1 1 X 1 X s (Γ)h (Γ) ≤ νh (Γ)h (Γ) nh (Γ) ν n−ν nh (Γ) ν n−ν n ν=1 n ν=1

X hν(Γ)hn−ν(Γ) ≤ . hn(Γ) 1≤ν≤n/2 13

Our estimate for | Hom(Γ,Sn)| yields h (Γ)h (Γ) bn/2c−1 √ ν n−ν = eO(n/ log n), hn(Γ) bν/2c

1 √ n n √ n which is ≤ n2 for log n < ν ≤ 2 . For 1 ≤ ν ≤ log n , we have !ν ν hn−ν(Γ) 4 4 4 4 5 ≤ √ · √ · ... · √ ≤ p √ ≤ √ . hn(Γ) n − 1 n − 2 n − ν n − n/ log n n Hence, for such ν, h (Γ)h (Γ) jν k √ 13ν ν/2 √ Aν ν/2 ν n−ν ≤ !ecν/ log ν5νn−ν/2 ≤ ecν/ log ν < , hn(Γ) 2 n n for some constant A. Furthermore, we have ν/2 r ν/2 X Aν A A log2 n X A 1 ≤ + + √ √ ; √ n n n log n n 1≤ν≤n/ log n ν≥log n it follows that √ 1/2 −n/2 O(n/ log n) sn(Γ) ∼ nhn(Γ) = (n!) 2 e , and the proof of (9) is complete.

4. Subgroup growth versus free subgroup growth

Consider the sequence of groups p Γp = x, y| [x, y], y = y = 1 , p prime.

For each p,Γp is a homomorphic image of the group Γ considered in the previous section, and it is isomorphic to the HNN-extension a, b, t|ap = bp = [a, b] = 1, at = b Example 4. Let p be a prime. Then ((n!)1/4eO(n3/4 log2 n), p = 2 sn(Γp) = . (15) p−1 − p−1 n+O(n2/3 log2 n) (n!) 2p 2 2p , p ≥ 3

1/2 Observe that as p grows, the subgroup growth of Γp approaches (n!) from below. Moreover, while the subgroup growth of Γp increases with p, this is not the case for the free subgroup growth, that is, the growth of the function f sn(Γp) = number of free subgroups of index n in Γp. In fact, we infer from [3, Theorem 5] that ( p−1 p pn p2 2 f (1 + o(1)) 2π (n!) , p |n sn(Γp) = 0, p2 - n, that is, the free subgroup growth of Γp decreases with p. 14

Proof of (15). As in the proof of Example 3, we have X∗ | Hom(Γp,Sn)| = n! |c ∩ CSn (σ)|, c where this time the summation is restricted to classes c with σp = 1. Deﬁning the parameters µi, µij, λi as in the last section, and writing λ for λp, we infer that

λ (p−2)µpp/p X X µ1!µp!(p − 1) p | Hom(Γp,Sn)| = n! µ λ!µ11!µ1p!(µp1/p)!(µpp/p)!p 1p µ1,µp λ,µ11,µ1p,µp1,µpp µ1+pµp=n µ11+µp1=µ1 λ+µ1p+µpp=µp µ11+pµ1p=µ1 1 λ+ p µp1+µpp=µp p|µpp

λ (p−2)µpp/p X X µ1!µp!(p − 1) p = n! 2 µ . λ!µ11!(µ1p!) (µpp/p)!p 1p µ1,µp λ,µ11,µ1p,µpp µ1+pµp=n λ+µ1p+µpp=µp µ11+pµ1p=µ1 p|µpp The sum ranges over O(n3) tuples of indices, thus we may neglect all terms which are smaller than the largest contribution by a factor of n4. Under the substitution λ → λ + p, µpp → µpp − p, the summand is changed by a factor (p − 1)pµ /p pp , (λ + 1)(λ + 2) ··· (λ + p)pp−2 hence, for µp, µ1p ﬁxed, the summand is maximal for some λ satisfying

1/p −1+1/p 1/p −1+1/p µpp p (p − 1) − p ≤ λ ≤ µpp p (p − 1).

1/p Moreover, all terms with λ ≥ n are negligible. Next consider the substitution µ11 → 2 µ11 +p , µ1p → µ1p −p, µpp → µpp +p. This procedure changes the value of the summand by a factor 2(p−1) 2 2 2 p µ1p(µ1p − 1) ··· (µ1p − p + 1) 2 . (µ11 + 1)(µ11 + 2) ··· (µ11 + p )(µpp/p + 1) 1 2/p 2/p −1/p2 This expression is < 4 , provided that µ11 > p µ1p (µpp/p + 1) , thus, terms with 2/p µ11 > 3n are again negligible.

Suppose from now on that p ≥ 3. If (µ1, µ2, µ11µ1p, µpp, λ) is a tuple of indices with n n 1/3 2/3 2p ≤ µp ≤ p , which has not been discarded so far, then λ < n , µ11 < 3n . By Stirling’s formula

2 λ (p−2)µpp/p (p−2)(pµp−µ1)/p µ !µ !(p − 1) p µ !µ !p 2/3 2 1 p = 1 p · eO(n log n) (16) 2 µ1p µ λ!µ11!(µ1p!) (µpp/p!)p 2 p µ1 µ1/p ((µ1/p)!) p − p2 !p 2 1− 2 µ µ −(pµp−µ1)/p µ ( p ) 1 µ p pµ − µ = 1 p p 1 e e ep2

2 2/3 2 ×p(p−2)(µp/p+µ1/p ) · eO(n log n). 15

9n n For µp ≥ 10p , we have µ1 ≤ 10 , and the last expression is bounded above by

1− 2 −1 l n m p lnm l 4n m n 1 − 1 − 1 ! ! ! p ≤ n! 2 2p 18 , 10 p 5p2 hence, these terms can be neglected against the term coming from the tuple with 2 n 2 n 2 2 n 2 µ1 = n − p b n 2p c, µ1p = µp = pb n 2p c, µ11 = 2p b n 2p c, µpp = λ = 0, which is of size 1 1 2 − 2p +o(1) n 9n 2 (n!) . For 2p ≤ µp ≤ 10p , consider the substitution µp → µp − p, µ1 → µ1 + p . The right-hand side of (16) is changed by a factor

2 p −2p µp µ1 2 µ1 ( p − p2 ) p 3p−4 µpp

µp µ1 which is ≥ 2, provided that p ≥ 5 and n is suﬃciently large, or p = 3 and ( p − p2 ) ≥ 4 n 2/3 2 · 3 . In any case, we may assume that µp ≤ 2p + O(n log n), and we ﬁnd that the n total contribution of terms with µp ≥ 2p equals

dn/2e! O(n2/3 log2 n) p−1 − p−1 n O(n2/3 log2 n) · e = (n!) 2p 2 2p · e . dn/(2p)e!pn/(2p) n If µ1 > 2 , the summation conditions imply that µ11 ≥ µ1 − pµp, and we use the 2 substitutions µ11 → µ11 + p , µ1p → µ1p − p, µpp → µpp + p to bound µpp from above via 2p 2p −p2 µpp < p (µ1/2) (µ1 − pµp) + 5p log n. 3/5 3/5 In particular, we have µpp < n , provided that µ1 − pµp > p · n . In the latter case, we have n−µ1 λ (p−2)µpp/p µ ! ! µ1!µp!(p − 1) p 1 p O(n3/5 log n) = n−µ e 2 µ1p 1 λ!µ11!(µ1p!) (µpp/p)!p n−µ1 2 p (2µ1 − n)( p !) p

If µ1 is increased by p, then the last expression is changed by a factor

p n−µ1 µ1( p )p 2p , (2µ1 − n) 1 2/3 n 2/3 which is < 2 , provided that 2µ1 − n > n . Hence, terms with µ1 ≥ 2 + 5n log n are n n 2/3 negligible. Finally, for 2 < µ1 < 2 + 5n log n, we have

λ (p−2)µpp/p µ1!µp!(p − 1) p µ1!dn/2pe! 2 µ 2 µ p λ!µ11!(µ1p!) (µpp/p)!p 1p (µ1p!) (µ1 − pµ1p)!p 1

dn/2e!dn/2pe! 3/5 2 = eO(n log n) (dn/(2p)e!)2pn/(2p)

p−1 − p−1 n O(n3/5 log2 n) = (n!) 2p 2 2p e . Summarizing, we have shown that

p−1 − p−1 n O(n2/3 log2 n) hn(Γp) = (n!) 2p 2 2p e .

Next, we want to obtain lower bounds for hn+1(Γp)/hn(Γp) and hn+2p(Γp)/hn(Γp). We n 2/3 2 2/p have already seen that summands with µ1 < 2 − n log n or µ11 > 2n make 16 negligible contributions to hn(Γp). For all other summands, simultaneously increasing 2 µ1+1 1 1− n, µ1, and µ11 by 1, results in a change by a factor ≥ n p , hence µ11+1 5

hn+1(Γp) 1 1− 2 ≥ n p . hn(Γp) 6

In a similar vein, the substitutions n → n + 2p, µ1 → µ1 + p, µp → µp + 1, and µ1p → µ1p + 1, when applied to a relevant term, result in an increase by a factor

p−1 (µ1 + 1)(µ1 + 2) ... (µ1 + p)(µp + 1) n 2 µ +1 = (1 + o(1)) , (µ1p + 1) p 1p 2 from which we conclude that h (Γ ) 1np−1 n+2p p ≥ . hn(Γp) 2 2 For p = 2, the maximum of

µ1!µ2! 2 µ λ!µ11!(µ12!) (µ22/2)!2 12 is less localized than that of the corresponding term for p ≥ 3. In fact, the largest n terms with µ1 = b 2 c and µ1 = 0 diﬀer only by an exponential factor. Applying similar methods as in the case p ≥ 3, one ﬁnds that

1/4 O(n3/4 log2 n) hn(Γ2) = (n!) e ,

n 3/4 2 1/4 µ1 1 1/4 and that terms with µ22 < − n log n, µ1 < n , or < n are negligible. 2 µ11 4 Simultaneously increasing n, µ1, and µ11 by 1, we infer that h (Γ ) 1 n+1 2 ≥ n1/4, hn(Γ2) 5 while the substitutions n → n + 4, µ2 → µ2 + 2, and µ22 → µ22 + 2 imply h (Γ ) n n+4 2 ≥ . hn(Γ2) 3

To obtain an estimate for sn(Γp) for p ≥ 3, we start from the equation

n−1 X sn(Γp) = nhn(Γp) − sν(Γp)hn−ν(Γp). ν=1

We want to show that the sum is of lesser order of magnitude than the term nhn(Γp). We have n−1 n−1 1 X 1 X s (Γ )h (Γ ) ≤ νh (Γ )h (Γ ) nh (Γ ) ν p n−ν p nh (Γ ) ν p n−ν p n p ν=1 n p ν=1

X hν(Γp)hn−ν(Γp) ≤ . hn(Γp) 1≤ν≤n/2 17

1 2/3 2 n−1+ p O(n log n) Inserting our estimate for hn(Γp), the ν-th term of the last sum is ν e , 1 2/3 2 which is < n2 , provided that ν > n log n. For 1 ≤ ν ≤ 10p − 1, we use the estimate for hn+1(Γp)/hn(Γp) to obtain

X hν(Γp)hn−ν(Γp) X ν 1− 2 −ν −1+ 2 ≤ 6 h (Γ )(n p ) n p , h (Γ ) ν p 0<ν<10p n p 0<ν<10p 2/3 3 whereas for 10p ≤ ν ≤ n log n, iteration of the estimate for hn+2p(Γp)/hn(Γp) yields

X hν(Γp)hn−ν(Γp) X p−1 ν/2 (p−1)bν/(2p)c O(ν2/3 log2 ν) ≤ (ν!) 2p 2 n e hn(Γp) 10p≤ν≤n2/3 log3 n 10p≤ν≤n2/3 log3 n

− p−1 ν X n 2p 2νnp−1 ν 10p≤n2/3 log3 n X 1 < 2νnp−1−ν/9 . n ν≥10p We conclude that p−1 − p−1 n O(n2/3 log2 n) sn(Γp) ∼ nhn(Γp) = (n!) 2p 2 2p e . For p = 2, the same argument gives 1/4 O(n3/4 log2 n) sn(Γ2) ∼ nhn(Γ2) = (n!) e . The proof of (15) is complete. 2

5. The difference between growth functions

Our next example shows that the growth of different groups can be very similar while not coinciding completely. This is in some contrast to the stability result for finite groups: in [5] it is shown that for finite groups G1,G2, | Hom(G1,Sn)| ∼ | Hom(G2,Sn)| already implies that these functions are equal. k ¯ Example 5. For k ≥ 2, let ∆k = hx, y|[x , y] = 1i and ∆k = Ck ∗ C∞. Then we have ! Xnd/k s (∆ ) ∼ c(k)n(1−1/k)n exp − (1 − 1/k)n + + log(n) , (17) n k d d|k d

1 Proof. We have sn(∆k) = (n−1)! tn(∆k), where tn(∆k) denotes the number of homomorphisms ϕ : ∆k → Sn, such that imϕ acts transitively on [n], hence

1 2 k sn(∆k) = (π, σ) ∈ S :[π , σ] = 1, hπ, σi transitive . (n − 1)! n Suppose that πk has a cycle of length l. Let U ⊆ [n] be the set of all points contained in l-cycles of πk. Then U is the union of all m-cycles of π, where m ranges over all m integers satisfying (k,m) = l, hence U is invariant under π. On the other hand, by the k structure of CSn (π ), U is invariant under σ as well, hence, U = [n], l|n, and the cycle k n/l n type of π is l . The latter is the cycle type of a k-th power if and only if (k, l)| l . The number of pairs (π, σ) with l = 1 equals ¯ 2 k ¯ sn(∆k) (π, σ) ∈ S : π = 1, hπ, σi transitive = tn(∆k) = . n (n − 1)! k n! k For l ≥ 2, π can be chosen in (n/l)!ln/l ways. Since σ ∈ CSn (π ), σ can be chosen in at most k n/l |CSn (π )| = (n/l)!l ways, on the other hand, for any choice such that the projectionσ ¯ ∈ Sn/l is an n/l-cycle, k hπ , σi and a forteriori hπ, σi is transitive. Deﬁne Dl := {d|k : d = (ld, k)}, and note that k ∈ Dl. We have k π ∈ Sn : π = (12 . . . l)(l + 1l + 2 ... 2l) ...

X Y d−1 νd = ((d − 1)!l ) π ∈ Sn/l : π has νd d-cycles

νd,d∈D d∈Dl P dνd=n/l

Q d−1 νd X (n/l)! d∈D ((d − 1)!l ) = l Q νd d∈D νd!d νd,d∈D l P dνd=n/l

Increasing νk by d, and decreasing νd by k, a summand in the last expression is changed by a factor k−1 k k ((k − 1)!l )νd(νd − 1) ··· (νd − k + 1)d (νd − k + 1) d d−1 k ≥ d , (νm + 1)(νm + 2) ··· (νm + d)k ((d − 1)!l ) (νk + d) d/k which is ≥ 2, if νd > 2(νk + d) + k. Hence, the maximum is attained for some tuple d/k n 1/2 νd with νd ≤ 2(νk + d) + k for all d < k, and νk ≥ kl − 4n . Therefore n/kl k−1 √ k (n/l)! (k − 1)!l O( n log n) π ∈ Sn : π = (12 . . . l)(l + 1l + 2 ... 2l) ... = e . (n/kl)! k Putting the estimates together, we obtain n/kl X (n/l)! (k − 1)!lk−1 √ t (∆ ) = t (∆¯ ) + n! eO( n log n). n k n k (n/kl)! k l≥2 l(k,l)|n The summands are decreasing with respect to l, hence we obtain n/kl k−1 min √ ¯ (n/lmin)! (k − 1)!lmin O( n log n) sn(∆k) = sn(∆k) + n e (n/klmin)! k 19 where lmin is the least integer l ≥ 2 with l(k, l)|n. If this condition is satisﬁed for some l, it is also satisﬁed for all its non-trivial divisors, thus lmin is prime. This implies (18), while the asymptotic estimate (17) follows from this and [4, Theorem 1]. For the remaining claim, note that in the notation above, n prime implies l = n, hence transitivity is trivially ensured, and we get

2 k k |{(π, σ) ∈ Sn :[π , σ] = 1, π is an n-cycle, hπ, σi transitive}| 2 = |{(π, σ) ∈ Sn :[π, σ] = 1, π is an n-cycle}|, ¯ and therefore tn(∆k) = tn(∆k) + n!.

Set δ(n) = (1 − 1/k)/p, where p = p(n) is deﬁned as in the last example. Then δ(n) is less erratic than it might appear at ﬁrst sight, in fact, it is uniformly almost even, that is, for every > 0 there exists some N, such that |δ(n) − δ((n, N))| < , where Q n we understand that δ(1) = 0. Indeed, given > 0, set N = γ(k) p≤1/ p. If (n,γ(k)) is (n,N) divisible by some prime p ≤ 1/, p divides (n,N,γ(k)) as well, therefore, δ(n) = δ((n, N)). n If on the other hand, (n,γ(k)) has no such prime factor, then 0 < δ(n) ≤ , while δ((n, N)) = δ(1) = 0. The approximating functions δ((n, q)) have a similar group theoretical interpretation k q as δ itself. In fact, setting ∆k,q = hx, y|[x , y] = y = 1i, we have √ n!δ((n,q))(k − 1)!n/kpeO( n log n), δ((n, q)) > 0 s (∆ ) = s (C ∗ C ) + n k,q n k ∞ 0, δ((n, q)) = 0. This can be seen by an argument similar to (but simpler than) the one given above. As the next example shows, the situation changes when adding a further free factor to ¯ ∆k and ∆k. In particular, this demonstrates that the equivalence relation

−A Γ ≈ ∆ :⇔ sn(∆) = (1 + O(n ))sn(Γ) for all A ∈ N is not stable under free multiplication. Example 6. Let G 6= 1 be a cyclic group, finite or infinite. Then there exist constants a1, a2,..., such that ! ¯ X −1/m sn(∆k ∗ G) ≈ sn(∆k ∗ G) 1 + aνn , ν≥1 where m = [k, |G|] if G is finite, and m = k if G = C∞. Moreover, denote by l the k least integer such that there exists some π ∈ Sl with π 6= 1. Then we have aν = 0 for ¯ ν < lm(1 − 1/k), and alm(1−1/k) = hl(∆k) − hl(∆k) > 0.

−1/m Proof. The existence of an asymptotic expansion for hn(∆k ∗G) in terms of n follows ¯ from the existence of an asymptotic series for | Hom(G, Sn)| and h(∆k), together with Example 5; cf. [5, Theorem 5] and [4, Theorem 1]. From this, we obtain the existence 20 of an asymptotic series for sn(∆k ∗ G) using the equation n−1 X sn(∆k ∗ G) = nhn(∆k ∗ G) − hν(∆k ∗ G)sn−ν(∆k ∗ G) (20) ν=1 together with the fact that −2+ 1 + 1 h (∆ ∗ G)h (∆ ∗ G) n k |G| ν k n−ν k . (21) hn(∆k ∗ G) ν ¯ ¯ By the deﬁnition of l, we have hν(∆k) = hν(∆k) for ν < l, while hl(∆k) > hl(∆k). k In fact, the assignment x 7→ π, y 7→ 1, where π ∈ Sl satisﬁes π 6= 1, extends to a ¯ homomorphism ϕ : ∆k → Sl, which does not factor over ∆k. Hence, subtracting (20) from the corresponding equation for ∆,¯ all summands with ν < l or ν > n − l vanish, while summands with l < ν < n − l can be estimated via (21), and we obtain ¯ ¯ sn(∆k ∗ G) − sn(∆k ∗ G) = nhn(∆k ∗ G) − nhn(∆k ∗ G) − nhl(∆k ∗ G)hn−l(∆k ∗ G) ¯ ¯ −(l+1)(2−1/k−1/|G|) + nhl(∆k ∗ G)hn−l(∆k ∗ G) + O(n ). (22) Using this equation for G = 1 and the result of Example 5, we deduce ¯ ¯ −(l+1)(1−1/k) hn(∆k) − hn(∆k) = (hl(∆k) − hl(∆k))hn−l(∆k) + O(n hn(∆k)) h (∆ ) = (h (∆ ) − h (∆¯ )) n k + O(n−(l+1)(1−1/k)h (∆ )). l k l k nl(1−1/k) n k Putting this back into (22), and using (21) together with the assumption |G| ≥ 2, we obtain nh (∆¯ ∗ G) s (∆ ∗ G) = s (∆¯ ∗ G) + (h (∆ ) − h (∆¯ )) n k (1 + O(n−1/2)) n k n k l k l k nl(1−1/k) s (∆¯ ∗ G) = s (∆¯ ∗ G) + (h (∆ ) − h (∆¯ )) n k (1 + O(n−1/2)), n k l k l k nl(1−1/k) ¯ that is, aν = 0 for ν < lm(1 − 1/k) and alm(1−1/k) = (hl(∆k) − hl(∆k)) > 0, as claimed.

6. Counting finite index subgroups up to conjugation

There are many interesting instances, where the fundamental group π1(X) of some manifold X is large (both in the sense of structure [8], and as concerns its subgroup growth). From the point of view of the covering theory of such spaces, counting conjugacy classes of ﬁnite index subgroups is more natural than counting individual subgroups. However, from an asymptotic viewpoint, these two counting problems turn out to be closely related. For a ﬁnitely generated group Γ, denote by cn(Γ) the number of conjugacy classes of index n subgroups in Γ.

µ+o(1) Proposition. Let Γ be a ﬁnitely generated group such that sn(Γ) = (n!) for some constant µ > 0. Then, for every ε > 0, −1 µ +ε cn(Γ) = n sn(Γ) + O (n!) 2 . In particular, almost all subgroups are self-normalizing. 21

Proof. Let ∆ ≤ Γ be a subgroup of index n. Then the number of conjugates of ∆ in Γ is given by (Γ : NΓ(∆)), whence the inequality −1 cn(Γ) ≥ n sn(Γ). On the other hand, X 1 X c (Γ) ≤ n−1s (Γ) + s/ (∆0), n n d n/d d|n (Γ:∆0)=d dnormal subgroup ∆ of ∆0 is determined by the isomorphism type of ∆0/∆, together with the images of the generators of ∆0 in ∆0/∆. By [9, Corollary], there are at most n ( 2 +o(1))(log(n/d)/ log 2)2 2 27 ≤ nlog n, n ≥ n d 0 isomorphism types of groups of order n/d, hence

n 2 s/ (∆0) ≤ 1+d(r−1)nlog n ≤ ern, n ≥ n , n/d d 0 since (n/d)d is maximal for d = n/e. It follows that for large n

−1 rnX cn(Γ) ≤ n sn(Γ) + e sd(Γ) d|n d

−1 o(1)X µ = n sn(Γ) + (n!) (d!) d|n d

−1 µ +o(1) ≤ n sn(Γ) + (n!) 2 .

As immediate consequences of this proposition, together with [7, Theorem 2] and [4, Theorem 1], we obtain the following.

Example 7. (i) Let Γ be a surface group of rank d ≥ 3. Then cn(Γ) satisﬁes the asymptotic expansion ∞ d−2 X −ν cn(Γ) ≈ 2(n!) 1 + Cν(d)n (n → ∞), ν=d−2 where the coeﬃcients Cν(d) are as in [7, formula (13)].

(ii) Let Γ = G1 ∗ · · · ∗ Gs ∗ Fr be a free product of finite groups Gi of order mi and a free group of rank r, and suppose that χ(Γ) < 0. Then cn(Γ) satisfies the asymptotic expansion ∞ X ˜ −ν/mΓ cn(Γ) ≈ LΓΦΓ(n) 1 + Dν(Γ)n , ν=1 22 ˜ with coefficients Dν(Γ) as given in [4, formula (29)], 2 ! X smi/2(Gi) L := (m . . . m )−1/2 exp − , Γ 1 s 2m i i 2|mi s ! r−1 n s (G ) −χ(Γ)n X X d i d/mi Φ (n) := (2πn) 2 exp n , Γ e d i=1 d|mi d

7. Large oscillations of hn(Γ)

Despite the fact that, in some of the examples above, we were not able to determine the size of hn(Γ) with asymptotic precision, we still believe that, in all cases up to now, this function behaves rather smoothly, a behaviour like hn(Γ) ∼ nc appears likely. hn−1(Γ) However, our next example shows that this cannot be expected for arbitrary groups. Before we can formulate a precise statement, we need the following deﬁnitions. A function f : N → C is called 2-multiplicative, if k ! X i Y i f ei2 = f(2 ), ei ∈ {0, 1}. i=0 0≤i≤k ei=1 Clearly, a 2-multiplicative function is determined by its values on powers of 2. A function g : N → C is called uniformly almost periodic, if for every > 0 there exists an integer q, and a q-periodic function g, such that |g(n) − g(n)| < for all integers n.

Example 8. Let Z2 be the additive group of 2-adic integers. Then we have hn(Z2) = f(n)g(n), where f is the 2-multiplicative function with f(2i) = 2−i, and g is uniformly almost periodic. Moreover, we have h ( ) n Z2 = Ω nlog n/(2 log 2)−2 . hn−1(Z2) Finally, 2 2 sn(Z2 ∗ Z2) ∼ n!f(n) g(n) .

Note that, since Z2 is a topological group, we have to adjust our previous definitions slightly. For a topologically finitely generated group Γ we denote by sn(Γ) the number of open subgroups of index n, while Hom(Γ,Sn) denotes the set of continuous homomorphisms Γ → Sn. Here, Sn is equipped with the discrete topology, that is, we consider locally constant homomorphisms only. As in the discrete case, we write 1 hn(Γ) = n! | Hom(Γ,Sn)|. Finally, the free product is to be understood in the cate- gory of profinite groups as well. With these conventions, the transformation formula (1) is still valid. This can be seen either by repeating the proof, or by the fact that sn(Γ) = sn(Γ/N) for some open normal subgroup N. 23

Proof. The quantity | Hom(Z2,Sn)| equals the number of elements of 2-power order in Sn, that is, the number of elements with all cycle lengths powers of 2. Hence,

X n! | Hom( ,S )| = . (23) Z2 n Q νe eν!2 ν eν ≥0 ν P eν 2ν =n

P ν To estimate this sum, assign to each partition n = ν eν2 of n into powers of 2 a list P 0 ν 0 of integers ak as follows. Start with the partition n = ν eν2 satisfying eν ∈ {0, 1}. Reconstruct from this the partition eν in a series of steps as follows: let ν0 be the 0 0 0 0 0 greatest index with eν > eν. Then replace eν0 by eν0 − 1, and eν0−1 by eν0−1 + 2, in this case we say that a splitting took place at ν0. Now repeat this step with a possibly 0 smaller value of ν0, until eν ≤ eν for all ν. We claim that at each stage of the algorithm, 0 0 we have eν ≥ eν for all ν > ν0, implying that the algorithm terminates with eν = eν for all ν. At the beginning of the algorithm our claim is true, so assume that it is true at 0 0 0 a certain stage. For ν > ν0 we have eν = eν. If eν0 > eν0 + 1, decreasing eν0 by 1 does 0 not change ν0, and therefore does not aﬀect our claim. If eν0 = eν0 + 1, ν0 will decrease 0 in the next step, while in this step equality eν0 = eν0 is achieved. Since ν0 is decreasing 0 during the algorithm, we have eν ≤ 1 for ν ≤ ν0 − 2, that is,

X 0 ν eν2 ν≤ν0−2 equals the least non-negative remainder of n mod 2ν0−1, and therefore

X 0 ν X ν eν2 ≤ eν2 , ν≤ν0−2 ν≤ν0−2

0 which implies eν0−1 ≥ eν0−1. Moreover, equality in the latter inequality implies equality 0 0 in the former. Hence, denoting the old value of ν0 by ν0, we see that eν = eν for all 0 0 ν with ν0 < ν ≤ ν0, while either eν0 > eν0 , or the algorithm had already terminated with eν0 . P ν Since the algorithm is deterministic, we can assign to a partition eν2 a sequence (ak), where ak is the number of times a splitting takes place at ν0 = k. Note that a 0 sequence (ak) is a splitting sequence, if and only if 2ak+1 − ak + ek ≥ 0 for all k, and in 0 0 this case 2ak+1 − ak + ek = ek, where ek is the 2-adic representation of n. Hence, with the latter convention, we can express | Hom(Z2,Sn)| in terms of the ak via X n! | Hom(Z2,Sn)| = 0 . (24) Q 0 k(2ak+1−ak+ek) k(2ak+1 − ak + ek)!2 (ak) ∀k:2a −a +e0 ≥0 k+1 k k

We want to show that, for every > 0, there is some k0, such that the contribution of all terms with ak 6= 0 for some k > k0 is less than n!f(n). Since the summand associated with the sequence ak = 0 equals n!f(n), it then follows that we may neglect all term with ak 6= 0 for some k > k0. If ak = 0 for all k, then on the right-hand side of (24), the factorials in the denominator vanish, hence it suﬃces to establish the same 24 property for the sum X n! 0 0 Q k(2ak+1−ak+ek) (2a0 − a1 + e0)! k 2 (ak) ∀k:2a −a +e0 ≥0 k+1 k k X n! = 0 , 0 Q (k−2)ak+ek (2a0 − a1 + e0)! k 2 (ak) ∀k:2a −a +e0 ≥0 k+1 k k for which, however, existence of such a k0 is clear, since taking k0 suﬃciently large, the (k−2)ak presence of a factor 2 6= 1 with k > k0 already forces the entire sum to be small. Hence, in (23), we may restrict the sum to all partitions eν, which are obtained from the 0 0 partition eν with eν ∈ {0, 1} without any splitting at places k > k0. These partitions correspond to partitions of the least non-negative remainder n mod 2k0 , and we obtain

n! | Hom(Z2,Sn)| n! 0 ≤ ≤ (1 + ) 0 . k0 Q ke k0 Q ke (n mod 2 )! 2 k | Hom( 2,S k0 )| (n mod 2 )! 2 k k>k0 Z n mod 2 k>k0 Hence, Y −ke0 | Hom(Z2,Sn)| = n! 2 k g(n), k where g is uniformly almost periodic, which implies our ﬁrst claim. Next, let n = 2a be a power of two. Then we have a−1 hn( 2) Y Z 2−a 2k = 2a(a−3)/2 nlog n/(2 log 2)−2. hn−1( 2) Z k=1 This proves our second claim. In order to establish the last claim, observe that, if 2ν is the greatest power of 2 dividing n, we have ν−1 hn( 2) Y Z 2−ν 2k 1. (25) hn−1( 2) Z k=1

Our last assertion on sn(Z2∗Z2) follows from this estimate together with the asymptotics for hn(Z2). Indeed, in order to transfer asymptotic information from hn to sn, we have to show that, as n → ∞, 2 2 X | Hom(Z2,Sk)| | Hom(Z2,Sn−k)| 2 → 0. | Hom( 2,Sn)| 0 blog2 nc to this sum is at most −2 −2 2 X n 2 n 2log n n2log n n−1, k blog2 nc blog2 nc

8. Problems

The previous examples might have left the reader with the impression that, for most groups Γ, the function sn(Γ) behaves extremely chaotically, and that results like [4, Theorem 1] might not indicate a general phenomenon, but are rather characteristic of a specific class of groups, in this case free products of finite groups. Indeed, we believe that the phenomena displayed in our examples are typical for many large groups. On the other hand, our observations still seem to point at certain rather general regularity properties; this has lead us to pose a number of problems aiming at a better under- standing of the relationship between growth and structural information. Let Γ be a one-relator group with d generators. As Example 1 demonstrates, the function sn(Γ) can grow considerably faster than sn(Fd−1). Our first problem is concerned with the relationship between these two functions.

Problem 1. (i) Is it true that sn(Γ) ≥ (1 + o(1))sn(Fd−1)? (ii) Does the limit log s (Γ) γ(Γ) := lim n n→∞ n log n always exist? (iii) What are the possible values for γ(Γ), as Γ ranges over all d-generator one- relator groups? In particular, is γ(Γ) always rational? Does there exist a sequence of groups Γn, such that γ(Γn) converges from above? Can γ(Γ) become equal to d? (As seen in Example 1, γ(Γ) may approach d arbitrarily well from below.)

Let Γ be a one-relator group. The number α(Γ) introduced in Section 2 is an interesting if somewhat mysterious structural invariant of Γ; it measures optimal approximation of Γ by some free product of cyclic groups. In part (ii) of Example 1, the growth of Γ is completely determined by such virtually free images, and one may ask for which one-relator groups Γ this is the case. In the latter context, the relationship between α(Γ) and the limit γ(Γ) is of importance. This is the topic of our next problem. Problem 2. (i) Does there exist an algorithm to compute α(Γ)? (ii) We always have γ(Γ) ≥ d − α(Γ). The reverse inequality is almost certainly wrong; but is there an upper bound for γ(Γ) + α(Γ) − d?

(iii) Deﬁne β(Γ) = sup γ(Γvf ), where Γvf ranges over all virtual free images of Γ. We have γ(Γ) ≥ β(Γ) ≥ d − α(Γ). When does equality hold? Are the diﬀerences γ(Γ) − β(Γ), β(Γ) + α(Γ) − d bounded from above?

In Examples 3 and 4, the argument proceeds as follows: one ﬁrst obtains an approximate c+o(1) asymptotic formula for hn(Γ) of the form hn(Γ) = (n!) with some constant c. This in turn enables one to estimate the contribution to

X hk(Γ)hn−k(Γ) hn(Γ) 0

As indicated in Section 2, the computation of | Hom(Γ,Sn)| via character theory for a group Γ can be seen as a discrete and non-abelian analogue of the circle method in additive number theory. In analogy to Waring’s problem we pose the following.

Problem 4. (i) Let w(x1, . . . , xk) be a word and let l ≥ 1 be an integer. Deﬁne the one-relator group Γw,l by

Γw,l = x11, . . . , x1k, x21, . . . , x2k, . . . , xlk|w(x11, . . . , x1k) ··· w(xl1, . . . , xlk) = 1 .

Prove that there is some constant l0, such that for all l > l0 the limit `(Γw,l) exists and is 1 or 2; and, more precisely, `(Γw,l) = 2 if and only if w is trivial as a word over C2, that is, for each xi, the sum of exponents of xi is even.

(ii) Let w(x1, . . . , xk) be a word, and let χ be an irreducible character of Sn. Deﬁne the Fourier-coeﬃcient αχ(w) to be −k X αχ(w) := n! χ(w(g1, . . . , gk)).

g1,...,gk∈Sn

Show that there are constants δ > 0 and n0, such that for all n > n0 and all 1−δ irreducible characters χ of Sn we have |αχ(w)| < χ(1) .

(iii) Let v(x1, . . . , xd−2) be a non-trivial word. Show that the conclusion of Exam- ple 2 remains valid for the words w1(x1, . . . , xd) = v(x1, . . . , xd−2)[xd−1, xd] and 2 2 w2(x1, . . . , xd) = v(x1, . . . , xd−2)xd−1xd.

According to Pride [8], a finitely generated group Γ is called large, if it contains a finite index subgroup ∆, which maps homomorphically onto a non-abelian free group. c Obviously, for a large group we have sn(Γ) > n! with some c > 0 and infinitely many n. For a group Γ, define real numbers γ±(Γ) via log s (Γ) γ+(Γ) := lim sup n n→∞ n log n log s (Γ) γ−(Γ) := lim inf n . n→∞ n log n 27

Then we ask the following. Problem 5. (i) Does there exist a group Γ with γ+(Γ) > 0, which is not large in the sense of Pride? (ii) For Γ a group with γ+(Γ) > 0, do we necessarily have that γ−(Γ) > 0, or at least that sn(Γ) > 0 for all but ﬁnitely many n? Does γ(Γ) exist for all large groups? (iii) Let Γ be a 2-generator one-relator group. Does there exist an algorithm to decide whether Γ is large or not?

(iv) Let Γ be a large group. Is the function sn(Γ) monotonically increasing for n suﬃciently large?

References

[1] A. Dress and T. Müller, Decomposable functors and the exponential principle, Adv. in Math. 129 (1997), 188–221. [2] A. Lubotzky, Subgroup Growth, lecture notes prepared for the conference Groups ’93 Galway/St. Andrews, University College, Galway. [3] T. Müller, Combinatorial aspects of finitely generated virtually free groups, J. London Math. Society 44, 75–94. [4] T. Müller,Subgroup growth of free products, Invent. math. 126 (1996), 111–131. [5] T. Müller, Finite group actions and asymptotic expansion of eP (z), Combinatorica 17 (1997), 523–554. [6] T. Müller, Enumerating representations in finite wreath products, Adv. in Math. 153 (2000), 118–154. [7] T. Müller and J.-C. Puchta, Character theory of symmetric groups and subgroup growth of surface groups, J. London Math. Soc. (2) 66 (2002), 623–640. [8] S. J. Pride, The concept of largeness in group theory, in: Word problems II, North Holland Publishing Company, 1980, 299–335. [9] L. Pyber, Enumerating finite groups of given order, Annals of Math. 137 (1993), 203–220. [10] O. Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Sem. Hamburg 5 (1927), 161–183. [11] E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. 17 (1967), 296–304.

Thomas W. M¨uller,School of Mathematical Sciences, Queen Mary & Westﬁeld College, University of London, Mile End Road, E1 4NS London, UK ([email protected])

Jan-Christoph Schlage-Puchta, Mathematisches Institut, Albert-Ludwigs-Universit¨at, Eckerstr. 1, 79104 Freiburg, Germany ([email protected])