The degree of commutativity of wreath products with infinite cyclic top group

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups $G$: the degree of commutativity $\text{dc}_S(G)$, with respect to a given finite generating set $S$, results from considering the fractions of commuting pairs of elements in increasing balls around $1_G$ in the Cayley graph $\mathcal{C}(G,S)$. We focus on restricted wreath products the form $G = H \wr \langle t \rangle$, where $H \ne 1$ is finitely generated and the top group $\langle t \rangle$ is infinite cyclic. In accordance with a more general conjecture, we show that $\text{dc}_S(G) = 0$ for such groups $G$, regardless of the choice of $S$. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox's main auxiliary result: in `reasonably large' homomorphic images of wreath products $G$ as above, the image of the base group has density zero, with respect to certain types of generating sets.


Introduction
Let G be a finitely generated group, with finite generating set S. For n ∈ N 0 , let B S (n) = B G,S (n) denote the ball of radius n in the Cayley graph C(G, S) of G with respect to S. Following Antolín, Martino and Ventura [1], we define the degree of commutativity of G with respect to S as dc S (G) = lim sup We remark that this notion can be viewed as a special instance of a more general concept, where the degree of commutativity is defined with respect to 'reasonable' sequences of probability measures on G, as discussed in a preliminary arXiv-version of [1] or, in more detail, by Tointon in [13].
If G is finite, the invariant dc S (G) does not depend on the particular choice of S, as the balls stabilise and dc(G) = dc S (G) simply gives the probability that two uniformly and randomly chosen elements of G commute.This situation was studied already by Erdős and Turán [4], and further results were obtained by various authors over the years; for example, see [5,6,8,9,11].For infinite groups G, it is generally not known whether dc S (G) is independent of the particular choice of S.
The degree of commutativity is naturally linked to the following concept of density, which is employed, for instance, in [2].The density of a subset X ⊆ G with respect to S is If the group G has sub-exponential word growth, then the density function δ S is biinvariant; compare with [2,Prop. 2.3].Based on this fact, the following can be proved, initially for residually finite groups and then without this additional restriction, even in the more general context of suitable sequences of probability measures; see [1,Thm. 1.3] and [13, Thms.1.6 and 1.17].
Theorem 1.1 (Antolín, Martino and Ventura [1]; Tointon [13]).Let G be a finitely generated group of sub-exponential word growth, and let S be a finite generating set of G. Then dc S (G) > 0 if and only if G is virtually abelian.Moreover, dc S (G) does not depend on the particular choice of S.
The situation is far less clear for groups of exponential word growth.In this context, the following conjecture was raised in [1].
Conjecture 1.2 (Antolín, Martino and Ventura [1]).Let G be a finitely generated group of exponential word growth and let S be a finite generating set of G.Then, dc S (G) = 0, irrespective of the choice of S.
In [1] the conjecture was already confirmed for non-elementary hyperbolic groups, and Valiunas [14] confirmed it for right-angled Artin groups (and more general graph products of groups) with respect to certain generating sets.Furthermore, Cox [3] showed that the conjecture holds, with respect to selected generating sets, for (generalised) lamplighter groups, that is for restricted standard wreath products of the form G = F ≀ t , where F = 1 is finite and t is an infinite cyclic group.Wreath products of such a shape are basic examples of groups of exponential word growth; in Section 2 we briefly recall the wreath product construction, here we recall that G = N ⋊ t with base group N = i∈Z F t i .In the present paper, we make a significant step forward in two directions, by confirming Conjecture 1.2 for an even wider class of restricted standard wreath products and with respect to arbitrary generating sets.
Theorem A. Let G = H ≀ t be the restricted wreath product of a finitely generated group H = 1 and an infinite cyclic group t ∼ = C ∞ .Then G has degree of commutativity dc S (G) = 0, for every finite generating set S of G.
One of the key ideas in [3] is to reduce the desired conclusion dc S (G) = 0, for the wreath products G = N ⋊ t under consideration, to the claim that the base group N has density δ S (N ) = 0 in G.We proceed in a similar way and derive Theorem A from the following density result, which constitutes our main contribution.
Theorem B. Let G = H ≀ t be the restricted wreath product of a finitely generated group H and an infinite cyclic group t ∼ = C ∞ .Then the base group N = i∈Z H t i has density δ S (N ) = 0 in G, for every finite generating set S of G.
The limitation in [3] to special generating sets S of lamplighter groups G is due to the fact that the arguments used there rely on explicit minimal length expressions for elements g ∈ G with respect to S. If one restricts to generating sets which allow control over minimal length expressions in a similar, but somewhat weaker way, it is, in fact, possible to simplify and generalise the analysis considerably.In this way we arrive at the following improvement of the results in [3, §2.2], for homomorphic images of wreath products.
Theorem C. Let G be a finitely generated group of exponential word growth of the form G = N ⋊ t , where Suppose further that S 0 is a finite generating set for H and that the exponential growth rates of H with respect to S 0 and of G with respect to S = S 0 ∪ {t} satisfy Then N has density δ S (N ) = 0 in G with respect to S.
For finitely generated groups G of sub-exponential word growth, the density of a subgroup of infinite index, such as N in G = N ⋊ t with t ∼ = C ∞ , is always 0; see [2].Thus Theorem C has the following consequence.Next we give a very simple concrete example to illustrate that the technical condition (1.1) in Theorem C is not redundant: the situation truly differs from the one for groups of sub-exponential word growth.It is not difficult to craft more complex examples.
Example 1.4.Let G = F × t , where F = x, y is the free group on two generators and t ∼ = C ∞ .Then F has density δ S (F ) = 1/2 > 0 in G for the 'obvious' generating set S = {x, y, t}.
Indeed, for every i ∈ Z we have and hence, for all n ∈ N,

This yields
We remark that in this example F and G have the same exponential growth rates: Furthermore, the argument carries through without any obstacles with any finite generating set S 0 of F in place of {x, y}.
Finally, we record an open question that suggests itself rather naturally.
Question 1.5.Let G be a finitely generated group such that dc S (G) > 0 with respect to a finite generating set S. Does it follow that there exists an abelian subgroup A ≤ G such that δ S (A) > 0?
For groups G of sub-exponential word growth the answer is "yes", as one can see by an easy argument from Theorem 1.1.An affirmative answer for groups of exponential word growth could be a step towards establishing Conjecture 1.2 or provide a pathway to a possible alternative outcome.At a heuristic level, an affirmative answer to Question 1.5 would fit well with the results in [12] and [13].
Notation.Our notation is mostly standard.For a set X, we denote by P(X) its power set.For elements g, h of a group G, we write g h = h −1 gh and [g, h] = g −1 g h .For a finite generating set S of G, we denote by l S (g) the length of g with respect to S, i.e., the distance between g and 1 in the corresponding Cayley graph C(G, S) so that , 0, 1}.We repeatedly compare the limiting behaviour of real-valued functions defined on cofinite subsets of N 0 which are eventually nondecreasing and take positive values.For this purpose we employ the conventional Landau symbols; specifically we write, for functions f, g of the described type, As customary, we use suggestive short notation such as, for instance, Acknowledgement.We thank two independent referees for detailed and valuable feedback.Their comments triggered us to improve the exposition and to sort out a number of minor shortcomings.In particular, this gave rise to Proposition 2.2.

Preliminaries
In this section, we collect preliminary and auxiliary results.Furthermore, we briefly recall the wreath product construction and establish basic notation.
2.1.Groups of exponential word growth.We concern ourselves with groups of exponential word growth.These are finitely generated groups G such that for any finite generating set S of G, the exponential growth rate This shows that lim Since lim α→0 + α α = 1, we conclude that lim A similar computation yields that the second sequence n n ⌈αn⌉ , n ∈ N, converges.Again directly, or by virtue of we conclude that also the second limit, for α → 0 + , is equal to 1.
Proposition 2.2.Let G be a finitely generated group of exponential word growth, with finite generating set S. Then there exists a non-decreasing unbounded function q : N → R ≥0 such that q ∈ o(n) and In this notation, we seek a non-decreasing unbounded function q : N → R ≥0 such that, simultaneously, ( We show below that for every m ∈ N, From this we see that there is an increasing sequence of non-positive integers c(m), m ∈ N, such that, for each m, c(m) ≥ m 2 and ∀n ∈ N ≥c(m) : Setting q 1 (n) = ⌊n/m⌋ for n ∈ N with c(m) ≤ n < c(m + 1) and we arrive at a function q : N → R ≥1 meeting the requirements (2.2).It remains to establish (2.3).Let m ∈ N and put In the following we deal repeatedly with sums of the form for k ∈ N, and using subadditivity, we obtain From our set-up, we deduce that Since which tends to infinity as l → ∞.This proves (2.3).
In [10, Lemma 2.2] Pittet seems to claim that from which Proposition 2.2 could be derived much more easily.However, we found the explanations in [10] not fully conclusive and thus opted to work out an independent argument.Naturally, it would be interesting to establish a more effective version of Proposition 2.2, if possible.

Wreath products.
We recall that a group G = H ≀ K is the restricted standard wreath product of two subgroups H and K, if it decomposes as a semidirect product G = N ⋊ K, where the normal closure of H is the direct sum N = k∈K H k of the various conjugates of H by elements of K; the groups N and K are referred to as the base group and the top group of the wreath product G, respectively.Since we do not consider complete standard wreath products or more general types of permutational wreath products, we shall drop the terms "restricted" and "standard" from now on.Throughout the rest of this section, let be the wreath product of a finitely generated subgroup H and an infinite cyclic subgroup t ∼ = C ∞ .Every element g ∈ G can be written uniquely in the form where ρ(g) ∈ Z and g = i∈Z (g |i ) t i ∈ N with 'coordinates' g |i ∈ H.The support of the product decomposition of g is finite and we write Furthermore, it is convenient to fix a finite symmetric generating set S of G; thus G = S , and g ∈ S implies g −1 ∈ S. We put d = |S| and fix an ordering of the elements of S: here W determines and is determined by the function In this situation the standard process of collecting powers of t to the right yields where σ = σ S,W is short for the negative1 cumulative exponent function We define the itinerary of g associated to the S-expression (2.6) as the pair and we say that It(S, W ) has length l, viz. the length of the word W .For the purpose of concrete calculations it is helpful to depict the functions ι W and σ S,W as finite sequences.The term 'itinerary' refers to (2.7), which indicates how g can be built stepwise from the sequences ι W and σ S,W ; see Example 2.4 below.In particular, g is uniquely determined by the itinerary It(S, W ) = (ι, σ) and, accordingly, we refer to g as the element corresponding to that itinerary.But unless G is trivial and S is empty, the element g has, of course, infinitely many S-expressions which in turn give rise to infinitely many distinct itineraries of one and the same element.
For discussing features of the exponent function σ S,W , we call maxi It(S, W ) = max(σ S,W ) and mini It(S, W ) = min(σ S,W ) the maximal and minimal itinerary points of It(S, W ). Later we fix a representative function W : G → X 1 , . . ., X d , g → W g which yields for each element of G an Sexpression of shortest possible length.In that situation we suppress the reference to S and refer to as the W-itinerary, the maximal W-itinerary point and the minimal W-itinerary point of any given element g.
To illustrate the terminology we discuss a concrete example.
Figure 1.An illustration of the itinerary of g in (2.9) associated to the S-expression in (2.8); the support of g is also indicated .
An alternative S-expression for the same element g is It has length 18 and is based on the semigroup word In this case, the itinerary associated to the S-expression (2.10) is There is a natural notion of a product of two itineraries, and it has the expected properties.We collect the precise details in a lemma.
Lemma and Definition 2.5.In the general set-up described above, suppose that I 1 = (ι 1 , σ 1 ) and I 2 = (ι 2 , σ 2 ) are itineraries, of lengths l 1 and l 2 , associated to Sexpressions W 1 , W 2 for elements g 1 , g 2 ∈ G. Then W = W 1 W 2 is an S-expression for g = g 1 g 2 ; the associated itinerary has length l = l 1 + l 2 and its components are given by We refer to I as the product itinerary and write Conversely, if I is the itinerary of some element g ∈ G associated to some Sexpression of length l and if l 1 ∈ [0, l] Z , there is a unique decomposition I = I 1 * I 2 for itineraries I 1 of length l 1 and I 2 of length l 2 = l − l 1 ; the corresponding elements g 1 , g 2 ∈ G satisfy g = g 1 g 2 .
Proof.The assertions in the first paragraph are easy to verify from and the observation that, for k ∈ Conversely, let I be the itinerary of an element g, associated to some S-expression W = l k=1 X ι(k) of length l, and let l 1 ∈ [0, l] Z .Then W decomposes uniquely as a product W 1 W 2 of semigroup words of lengths l 1 and l−l 2 , namely for . These are S-expressions for elements g 1 , g 2 and g = g 1 g 2 .Moreover, W 1 and W 2 give rise to itineraries I 1 , I 2 such that I = I 1 * I 2 .Since W 1 and I 1 , respectively W 2 and I 2 , determine one another uniquely, the decomposition Lemma 2.6.Let G = H ≀ t be a wreath product as in (2.4), with generating set S as in (2.5).Put Then the following hold.
Let g ∈ G with itinerary I, associated to an S-expression of length l S (g).Then, for every j ∈ Z with mini(I) − m S ≤ j ≤ maxi(I) + m S , the elements h = gu t j+ρ(g) , = u t j g ∈ G satisfy ρ(h) = ρ( ) = ρ(g) and the 'coordinates' of h, are given by Furthermore, l S (h) ≤ l S (g) + D and l S ( ) ≤ l S (g) + D.
Proof.We write I = (ι, σ) for the given itinerary of g, and l denotes the length of I.
from this inclusion the two inequalities follow readily.
(ii) In addition we now have l = l S (g).The first assertions are very easy to verify.We justify the upper bound for l S (h); the bound for l S ( ) is derived similarly.
Since mini(I) − m S ≤ j ≤ maxi(I) + m S and since itineraries proceed, in the second coordinate, by steps of length at most r S ≤ m S , there exists k Next we decompose the itinerary I as the product We denote by g 1 = g 1 t −σ(k+1) and g 2 = g 2 t σ(k+1)+ρ(g) the elements corresponding to I 1 and I 2 so that g = g 1 g 2 = g 1 g 2 t σ(k+1) t ρ(g) .Moreover, we observe from |j−σ(k+1)| ≤ m S + r S that has length l 3 ≤ l S (u) + 2 l S t j−σ(k+1) ≤ D. Our choice of k guarantees that the support of g 2 t σ(k+1) does not overlap with {j} = supp(u t j ); compare with (i).Thus g 2 t σ(k+1) and u t j , both in the base group, commute with one another.This gives h = gu t j+ρ(g) = g 1 g 2 t σ(k+1) u t j t ρ(g) = g 1 u t j g 2 t σ(k+1) t ρ(g) = g 1 t −j+σ(k+1) ut j−σ(k+1) g 2 = g 1 g 3 g 2 , and we conclude that l S (h) ≤ l 1 + l 2 + l 3 ≤ l + D = l S (g) + D.

Proofs of Theorems A and B
First we explain how Theorem A follows from Theorem B. The first ingredient is the following general lemma.Lemma 3.1 (Antolín, Martino and Ventura [1, Lem.3.1]).Let G = S be a group, with finite generating set S. Suppose that there exists a subset X ⊆ G satisfying Then G has degree of commutativity dc S (G) = 0.
The second ingredient comes from [3, §2.1],where Cox shows the following.If G = H ≀ t is the wreath product of a finitely generated group H = 1 and an infinite cyclic group t , with base group N , and if S is any finite generating set for G, then The idea behind Cox' proof is as follows.For g ∈ G N , the centraliser C G (g) is cyclic and the 'translation length' of g with respect to S is uniformly bounded away from 0. The latter means that there exists τ S > 0 such that Consequently, for g ∈ G N the function n → |C G (g) ∩ B S (n)| is bounded uniformly by a linear function, while G has exponential word growth.Thus, Theorem B implies Theorem A, and it remains to establish Theorem B. Throughout the rest of this section, let be the wreath product of a finitely generated subgroup H and an infinite cyclic subgroup t , just as in (2.4).The exceptional case H = 1 poses no obstacle, hence we assume H = 1.We fix a finite symmetric generating set S = {s 1 , . . ., s d } for G and employ the notation established around (2.5).Finally, we recall that G has exponential word growth and we write λ = λ S (G) > 1 for the exponential growth rate of G with respect to S; see (2.1).We start by showing that the subset of N consisting of all elements with suitably bounded support is negligible in the computation of the density of N .Proposition 3.2.Fix a representative function W which yields for each element of G an S-expression of shortest possible length and let q : N → R ≥1 be a non-decreasing unbounded function such that q ∈ o(log n).
Then the sequence of sets The proof of Proposition 3.2 is preceded by some preparations and two auxiliary lemmata.We keep in place the set-up from Proposition 3.2.For i ∈ Z, we write H i = H t i .Using the notation established in Section 2.2, we accumulate the 'coordinates' of elements of S in a set Then S i is a finite symmetric generating set of H i for each i ∈ Z. Indeed, every element h ∈ H satisfies h = h = h |0 and can thus be written in the form based upon a suitable itinerary I = (ι, σ) of length l.We conclude that H = S 0 and consequently H i = S i for i ∈ Z; the generating systems inherit from S the property of being symmetric.Moreover, we have It is convenient to split the analysis of the set R q (n) from Proposition 3.2 into two parts.First we take care of elements whose 'coordinates' fall within sufficiently small balls around 1 in H, with respect to the generating set S 0 .Lemma 3.3.In addition to the set-up above, let f : N → R >0 be a non-decreasing unbounded function such that f ∈ o(n/q(n)).
Then the sequence of subsets Proof.Let C = C(S) ∈ N be as is in Lemma 2.6(i) and choose C ′ ∈ N such that λ C ′ > λ S 0 (H).Then we have, for all sufficiently large n ∈ N, From f ∈ o(n/q(n)) we obtain where C = C(S) ∈ N continues to denote the constant from Lemma 2.6(i).Let I = (ι, σ), viz.I g = (ι g , σ g ), denote the W-itinerary of g.Then .
By successively cancelling sub-products of adjacent factors that evaluate to 1 and have maximal length with this property (in an orderly fashion, from left to right, say), we arrive at a 'reduced' product expression , that picks out a subsequence of factors.In particular, this means that, for and moreover we have l S (g) By means of suitable perturbations, we aim to produce from g a collection of ℓ distinct elements ġ(1), . . ., ġ(ℓ) which each carry sufficient information to 'recover' the initial element g.We proceed as follows.For each choice of j ∈ [1, ℓ] Z we decompose the itinerary I for g into a product I = I j,1 * I j,2 of itineraries of length κ(j) and l S (g)− κ(j); compare with Lemma 2.5.Then g = g j,1 g j,2 , where g j,1 , g j,2 denote the elements of G corresponding to I j,1 , I j,2 .From g ∈ R q (n) it follows that maxi(I j,1 ) − mini(I j,1 ) and maxi(I j,2 )−mini(I j,2 ) are bounded by q(n); in particular, ρ(g j,1 ) ∈ [−q(n), q(n)] Z .
We define with C = C(S) as above; see Figure 2 for a pictorial illustration, which features an additional parameter τ that we introduce in the proof of Lemma 3.4.
(ii) Let j ∈ [1, ℓ] Z , and write G 1 = supp(g j,1 ), G 2 = supp(g j,2 ).Lemma 2.6(i) implies that the sets G 1 and G 2 − ρ(g j,1 ) = supp t ρ(g j,1 ) g j,2 lie wholly within the interval subject to the standard conventions min ∅ = +∞ and max ∅ = −∞ in special circumstances; see Figure 2 for a pictorial illustration.In contrast, gaps between two elements in G 1 or two elements in G 2 are strictly less than q(n) + 2C ≤ τ .Consequently, we can identify the two components in (3.4) and thus G 1 and G 2 − ρ(g j,1 ), without any prior knowledge of j or g j,1 , g j,2 .Therefore, for each i ∈ Z the ith coordinate of g satisfies and hence g can be recovered from ġ(j).
For the proof of Proposition 3.2 we now make a more careful choice of the nondecreasing unbounded function f : N → R >0 , which entered the stage in Lemma 3.3: we arrange that with C = C(S) as in Lemma 2.6(i).For instance, we can take f = f α for any real parameter α with 0 < α < 1, where Indeed, since q(n) ∈ o(log n) and q(n) ≥ 1 for all n ∈ N, each of these functions satisfies Proof of Proposition 3.2.We continue with the set-up established above; in particular, we make use of the refined choice of f .In view of Lemma 3.3 it remains to show that We define a map 3) and Lemma 3.4(i).From Lemma 3.4(ii) we deduce that , and hence, by submultiplicativity, Remark 3.5.Proposition 3.2 can be established much more easily under the extra assumption that H has sub-exponential word growth.Indeed, in this case, one can prove that for any non-decreasing unbounded function q : N → R >1 such that q ∈ o(n); the proof is similar to the one of Lemma 4.1 below.If we assume that H is finite, it is easy to see that there exists α ∈ R >0 such that Next we establish Theorem B, using ideas that are similar to those in the proof of Proposition 3.2: again we work with perturbations of a given element g in such a manner that the original element can be retrieved easily.We begin with some preparations to establish an auxiliary lemma.
Fix a representative function W which yields for each element of G an S-expression of shortest possible length, and fix an element u ∈ H {1}. Consider g ∈ N with W-itinerary I = (ι, σ), viz.I g = (ι g , σ g ).We put For the time being, we suppose that We decompose the itinerary for g as I = I 1 * I 2 * I 3 , where I 1 , I 2 , I 3 have lengths k + , k − − k + , l S (g) − k − ; compare with Lemma 2.5.
(ii) As in the discussion above suppose that k + = k + W,g and k − = k − W,g satisfy k + ≤ k − ; the other case k − < k + can be dealt with similarly.We have to check that g can be recovered from g(J), if we are allowed to use one of the parameters σ + , σ − .Indeed, from −ρ(g(J)) = 2 σ + − σ − + 4C we deduce that in such a case both, σ + and σ − are available to us.Furthermore, Lemma 2.6(i) gives allows us to recover ẋ(J), ẏ(J) and ż(J) via (3.5) and ẋ Using (3.7), we recover g = ẋ(J) ẏ(J) ż(J).
Proof of Theorem B. We continue within the set-up established above; in particular, we employ the J-variants g(J) of elements g ∈ N for two-element subsets J ⊆ [σ − g , σ + g ] Z , with respect to a fixed representative function W and a chosen element u ∈ H {1}.
Let q : N → R ≥1 be a non-decreasing unbounded function such that q ∈ o(log n).We make use of the decomposition where R q (n) = R W,q (n) is defined as in Proposition 3.2 and R ♭ q (n) = R ♭ W,q (n) denotes the corresponding complement in N ∩ B S (n).Let D ′ ∈ N be as in Lemma 3.6(i).
Below we show that (3.11) |B This bound and submultiplicativity yield Together with Proposition 3.2 we deduce from (3.10) that N has density zero: properly as a limit.
From this observation and Lemma 3.6(iii) we conclude that which is the bound (3.11) we aimed for.

Proof of Theorem C
Throughout this section let G denote a finitely generated group of exponential word growth of the form G = N ⋊ t , where (a) the subgroup t is infinite cyclic; (b) the normal subgroup N = H t i | i ∈ Z is generated by the t -conjugates of a finitely generated subgroup H N ; (c) the t -conjugates of this group H commute elementwise: H t i , H t j = 1 for all i, j ∈ Z with H t i = H t j .Suppose further that S 0 = {a 1 , . . ., a d } ⊆ H is a finite symmetric generating set for H and that the exponential growth rates of H with respect to S 0 and of G with respect to S = S 0 ∪ {t, t −1 } satisfy This is essentially the setting of Theorem C; for technical reasons we prefer to work with symmetric generating sets.Our ultimate aim is to show that δ S (N ) = 0.
Using the commutation rules recorded in (c), it is not difficult to see that every g ∈ N admits S-expressions of minimal length that take the special form For the following we fix, for each g ∈ N , expressions as described and we use subscripts to stress the dependency on g: we write σ − g , σ + g and w g,i for i ∈ [σ − g , σ + g ] Z , where necessary.The notation is meant to be reminiscent of the one introduced in Definition 2.3, but one needs to keep in mind that we are dealing with a larger class of groups now.Lemma 4.1.In addition to the general set-up described above, let q : N → R >0 be a non-decreasing unbounded function such that q ∈ o(n).Then the sequence of sets This allows us to bound the number of possibilities for the elements w g,i (a 1 , . . ., a d ) in an S-expression of the form (4.2) for g ∈ R q (n) and, writing q(n) = 2⌊q(n)⌋ + 1, we obtain ≤ n + q(n) q(n) M q(n) (µ + ε) n , and hence (4.4) We notice that q ∈ o(n) implies q ∈ o(n).Thus Lemma 2.1 implies that n+q(n) q(n) M q(n) grows sub-exponentially, and the term on the right-hand side of (4.4) tends to 0 as n tends to infinity.
Proof of Theorem C. We continue to work in the notational set-up introduced above.In addition we fix a non-decreasing unbounded function q : N → R ≥0 such that q ∈ o(n) and (4.5) see Proposition 2.2.As in the proof of Theorem B, we make use of a decomposition where R q (n) is defined as in Lemma 4.1 and R ♭ q (n) denotes the corresponding complement in N ∩ B S (n).
In view of Lemma 4.1 it suffices to show that (4.6) It is enough to consider sufficiently large n so that n > q(n) holds.For every such n and g ∈ R ♭ q (n), with chosen minimal S-expressions (4.2) and (4.3), we have σ − = σ − g < −q(n) or σ + = σ + g > q(n), hence gt −q(n) , gt q(n) ∩ B S (n − q(n)) = ∅.
As each of the right translation maps g → gt −q(n) and g → gt q(n) is injective, we conclude that |R ♭ q (n)| ≤ 2|B S (n − q(n))|, and thus (4.6) follows from (4.5).

Corollary 1 . 3 .
Let G = A ⋊ t be a finitely generated group, where A is abelian and t ∼ = C ∞ .Then A has density δ S (A) = 0 in G, with respect to any finite generating set of G that takes the form S = S 0 ∪ {t} with S 0 ⊆ A.

Figure 3 .
Figure 3.A schematic illustration of the decomposition g = xyz.

Figure 4 .
Figure 4.A schematic illustration of the support components of g(J).