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 <aj P where χ = j 1/aj − d + 1 is the Euler characteristic of Ca1 ∗ · · · ∗ Cad , and 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.