Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.


Introduction
Studying infinite, finitely generated groups through their finite quotients is a common method in group theory.Groups in which one can distinguish elements using their finite quotients are called residually finite.Formally speaking, a group G is said to be residually finite if for every pair of distinct elements f, g ∈ G there exists a finite group Q and a surjective homomorphism π : G → Q such that π(f ) ̸ = π(g) in Q. Group properties of this type are called separability properties and are usually defined by what types of subsets we want to distinguish.In this article, we study conjugacy separability, meaning that we will study groups in which one can distinguish conjugacy classes using finite quotients.To be more specific, a group G is said to be conjugacy separable if for every pair of nonconjugate elements f, g ∈ G there exists a finite group Q and a surjective homomorphism π : G → Q such that π(f ) is not conjugate to π(g) in Q.
1.1.Motivation.One of the original reasons for studying separability properties in groups is that they provide an algebraic analogue to decision problems in finitely presented groups.To be more specific, if S ⊆ G is a separable subset such that S is recursively enumerable and where one can always effectively construct the image of S under the canonical projection onto a finite quotient of G, then one can then decide whether a word in the generators of G represents an element belonging to S simply by checking finite quotients.Indeed, it was proved by Mal'tsev [16], adapting the result of McKinsey [18] to the setting of finitely presented groups, that the word problem is solvable for finitely presented, residually finite groups in the following way.Given a finite presentation ⟨X | R⟩ and a word w ∈ F (X) where F (X) is the free group with the generating set X, one runs two algorithms in parallel.The first algorithm enumerates all the products of conjugates of the relators and their inverses and checks whether w appears on the list.On the other hand, the second algorithm enumerates all finite quotients of G and checks whether the image of the element of G represented by w is nontrivial.In other words, the first algorithm is looking for a witness of the triviality of w whereas the second algorithm is looking for a witness of the nontriviality of w.Using an analogous approach, Mostowski [20] showed that the conjugacy problem is solvable for finitely presented, conjugacy separable groups.In a similar fashion, finitely presented, LERF groups have solvable generalised word problem meaning that the membership problem is uniformly solvable for every finitely generated subgroup.In general, algorithms that involve enumerating finite quotients of an algebraic structure are sometimes called algorithms of Mal'tsev-Mostowski type or McKinsey's algorithms.
Given an algorithm, it is natural to ask how much computing power is necessary to produce an answer.In the case of algorithms of Mal'tsev-Mostowski type, one can measure their space complexity by the associated depth functions which we go into more detail.Given a residually finite group G with a finite generating set S, its residual finiteness depth function RF G,S : N → N quantifies how deep within the lattice of normal subgroups of finite index of G one needs to look to be able to decide whether or not a word of length at most n represents a nontrivial element.In particular, if w is a word in S of length at most n, then either G has a finite quotient of size at most RF G,S (n) in which the image of the element represented by w is nontrivial, and if there is no finite quotient of size less than or equal to RF G,S (n) in which the image of w is nontrivial, then w must represent the trivial word.In particular, we see that the the residual finiteness depth function of G with respect to the generating set S fully determines the size of finite quotients that McKinsey's algorithm needs to generate in order to give produce an answer.Since every finite group can be fully described by its Cayley table, we see that the space complexity of the word problem of G with respect to the generating set S can be bounded from above by (RF G,S (n)) 2 .Moreover, the notion of depth function can be generalised to different separability properties.In this note, we study conjugacy separability depth functions which denote as Conj G,S (n) which is a function that measures how deep within the lattice of normal subgroups of finite index one needs to go in order to be able to distinguish distinct conjugacy classes of elements of word length at most n with respect to the finite generating subset S. Just like in computational complexity, we study these functions up to asymptotic equivalence.See subsection 2.1 for the precise definitions of depth functions and the corresponding asymptotic notions.

Statement of the results.
Not much is known about the asymptotic behaviour of the function Conj G,S (n) for different classes of groups.The first result of this kind was by Lawton, Louder, and McReynolds [14] who showed that if G is a nonabelian free group or the fundamental group of a closed oriented surface of genus g ≥ 2, then Conj G (n) ⪯ n n 2 .For the class of finitely generated nilpotent groups, the second named author and Deré [7] showed that if G is a finite extension of a finitely generated abelian group, then Conj G (n) ⪯ (log(n)) d for some natural number d, and when G is a finite extension of a finitely generated nilpotent group that is not virtually abelian, then there exist natural numbers d 1 and Finally, in [10], the authors of this note gave upper bounds for Conj A≀B (n) of wreath products of conjugacy separable groups A and B which generalises Remeslennikov's classification of conjugacy separable wreath products [21].However, when applied directly to wreath products of abelian groups, the formulas given in [10] produce rather coarse upper bounds.Applying [10,Theorem C] to the lamplighter group F 2 ≀ Z, one can then demonstrate that its conjugacy depth function can be bounded from above by the function 2 n n 2 . Similarly, applying [10, Theorem C] to the group Z ≀ Z one can show that the conjugacy depth function is bounded from above by n n n 2 . In this note, we only focus on conjugacy depth functions of wreath products of finitely generated abelian groups.This restriction allows us to use more effective methods to obtain much better upper bounds than those presented in [10].Additionally, we are able to use methods from commutative algebra to produce lower bounds and in the case of the lamplighter group, we fully determine the asymptotic equivalence class of its conjugacy depth function.
Before stating our main results, we introduce some notation.Letting f, g : N → N be nondecreasing functions, we write f ⪯ g if there is a constant C ∈ N such that f (n) ≤ Cg(Cn) for all n ∈ N. If f ⪯ g and g ⪯ f , we then write f ≈ g.
This first theorem addresses the asymptotic behavior of conjugacy separability of wreath products of the form A ≀ B where A is a finite abelian group and B is an infinite, finitely generated abelian group.For the statements of our theorem, we say the torsion free rank of a finitely generated abelian group A is the largest such natural number k such that A ∼ = Z k ⊕ Tor(A) where Tor(A) is the subgroup of finite order elements of A.
Theorem 1.1.Let A be a finite abelian group, and suppose that B is an infinite finitely generated abelian group.If the torsion free rank of B is 1, then As a corollary, we are able to compute the precise asymptotic behaviour of conjugacy separability for lamplighter groups.
Corollary 1.2.Let F q be the finite field with q elements.Then This next theorem addresses the asymptotic behavior of conjugacy separability of A ≀ B when A and B are both infinite, finitely generated abelian groups.
Theorem 1.3.Let A be an infinite, finitely generated abelian group, and suppose that B is an infinite finitely generated abelian group.If B has torsion free rank 1, then By combining Theorem 1.1 and Theorem 1.3, the following corollary gives the best known result for asymptotic behaviour of conjugacy separability of wreath products of finitely generated abelian groups with an infinite acting group.
Corollary 1.4.Let A be a finitely generated abelian group, and suppose that B is an infinite, finitely generated abelian group.
Suppose that A is a finitely generated abelian group.If B has torsion free rank 1, then If B has torsion free rank k > 1, then Suppose that A is finite.If B has torsion free rank 1, then If B has torsion free rank k > 1, then This last theorem applies Theorem 1.1 and Theorem 1.3 to provide exponential lower bounds for conjugacy separable wreath products A ≀ B where Z ≤ Z(B) or where B has an infinite cyclic subgroup as a retract.
Theorem 1.5.Let A be a nontrivial finitely generated abelian group, and suppose that G is a conjugacy separable finitely generated group with separable cyclic subgroups that contains an infinite cyclic group as a retract or satisfies Z ≤ Z(B).If A is finite, then

Outline of the paper.
In Section 2, we recall standard mathematical notions and concepts that will be used throughout the paper.In particular, in subsection 2.1, we recall the notions of word length, depth functions and associated asymptotic notions.In subsection 2.2, we recall the basic terminology of wreath products of groups.Finally, in subsection 2.3, we recall the notion of Laurent polynomial rings and show that groups of the form R ≀ Z, were R is a commutative ring, can be realised as R[x, x −1 ] ⋊ Z where R[x, x −1 ] is the ring of Laurent polynomials over the R and Z acts on R[x, x −1 ] via multiplication by x.We finish this section by giving a criterion for conjugacy for such groups purely in terms of commutative algebra.
In Section 3 we use methods from commutative algebra to produce lower bounds for the conjugacy depth functions by constructing infinite sequences of pairs of non-conjugate elements that require quotients that are at least exponentially large in the word lengths of the nonconjugate pair of elements in order to remain non-conjugate.
In Section 4, we use combinatorial methods together with the conjugacy criterion for wreath products of abelian groups to construct upper bounds for wreath products of abelian groups.
Finally, in Section 5 we combine the lower bounds obtained in Section 3 together with the upper bounds constructed in Section 4 to prove Theorem 1.5.We then proceed to apply our methods to give lower bounds on the conjugacy depth function for wreath products where the acting group may not necessarily be abelian.

Preliminaries
We denote F p as the finite field of p elements where p is prime.We denote Sym(n) as the symmetric group on n letters.For x, y ∈ G, we write x ∼ G y if there exists an element z ∈ G such that zxz −1 = y and suppress the subscript when the group G is clear from context.Whenever the given group is abelian, we will use additive notation.
We say that a subgroup H ≤ G is conjugacy embedded in G if for every f, g ∈ H we have that f ∼ H g if and only if f ∼ G g. Following the definition, one can easily check that the relation of being conjugacy embedded is transitive.That means if A ≤ B ≤ C such that A is conjugacy embedded in B and B is conjugacy embedded in C, then A is conjugacy embedded in C.
Given a group G, we say that a subgroup R ≤ G is a retract of G if there exists a surjective homomorphism ρ : G → R such that ρ ↾ R = id R .The following remark is a natural consequence of the definition of being a retract.
The next lemma allows us to reduce the study of conjugacy in a semidirect product of abelian groups A ⋊ B to conjugacy in A ⋊ (B/K) where K is the kernel of the action of B on A. Since wreath products are a special type of a semidirect product, this lemma will be useful throughout the article.Finally, in this lemma, we will be using additive notation, with B acting on A by multiplication, i.e. for b 1 , b 2 ∈ B and a ∈ A we write In particular, for a 1 , a 2 ∈ A and b 1 , b 2 ∈ B, we write Lemma 2.2.Suppose that A and B are finitely generated abelian groups, and suppose that a 1 , a 2 ∈ A and b ∈ B. Then where K is the kernel of the action of B on A.
Proof.For this proof, we denote the action of b ∈ B on a ).Thus, we may assume that (a 1 , b) ∼ A⋊(B/K) (a 2 , b).Suppose that there exists (x, y) Thus, we have Hence, we must have that k = 0. Therefore, we have (x, y)(a 1 , b)(x, y) −1 = (a 2 , b) giving our claim.
Given an abelian group B, we will use Tor(B) to denote When B is a finitely generated, it it easy to see that Tor(B) is a characteristic subgroup which provides a splitting B = Z k ⊕ Tor(B) for some k ∈ N. We say that k is the torsion-free rank of B. By fixing a splitting, we define τ : B → Tor(B) and ϕ : B → Z k as the associated retractions.Then every element b ∈ B can be uniquely expressed as b = τ (b) + ϕ(b).We say that τ (b) is the torsion part of b and ϕ(b) is the torsion-free part of b.Whenever we say the torsion part or torsion-free part of an element of B, we are saying that with respect to some fixed splitting of the above form.
To ease notation, we will view direct sums of groups over some indexing set as finitely supported functions on the indexing set with range in the index groups.More precisely, if is a direct sum of copies of a group A indexed by a set I, then for f ∈ G we will write f (i) to denote the i-th coordinate of f .In particular, elements in G correspond to functions f : I → A where f (i) = 1 for all but finitely many elements in I.The support of f which is the set of elements on which f is not trivial will be denoted as The range of f will be denoted as

Asymptotic notions and depth functions.
Let G be a finitely generated group equipped with a finite generating subset S. We define the word length of an element g ∈ G with respect to S as where |w| denotes the word length of w in F (S).We use B G,S (n) to denote the ball of radius n centered around the identity with respect to the finite generating subset S. When the finite generating subset is clear from context, we will instead write B G (n).
The conjugacy separability depth function of G is defined in the following way.Let f, g ∈ G be a pair of elements such that f ̸ ∼ G g.The conjugacy depth of the pair (f, g), denoted CD G (f, g), is given by n) and f ̸ ∼ G g}.We note that Conj G,S (n) depends on the choice of the finite generating subset S.However, one can easily check that the asymptotic behaviour does not.It is well known that a change of a finite generating subset is a quasi-isometry.In particular, if S 1 , S 2 ⊂ G are two finite generating subsets of a group G, then ∥ • ∥ S 1 ≈ ∥ • ∥ S 2 .The same holds for depth functions.For non-decreasing functions f, g : [14] for more details.As we are only interested in the asymptotic behaviour of the above defined functions, we will suppress the choice of generating subset whenever we reference the depth functions or the word-length.
Let G be a conjugacy separable group with a finitely generated conjugacy embedded subgroup R. We now relate the conjugacy depth of a pair of nonconjugate elements r 1 , r 2 ∈ R as elements of R with the conjugacy depth of r 1 , r 2 as elements of G.
Lemma 2.3.Let G be a finitely generated conjugacy separable group with a finitely generated conjugacy embedded subgroup R ≤ G. Then for r 1 , r 2 ∈ R where r 1 ≁ R r 2 , we have We conclude this subsection with the following lemma which relates Conj G (n) with Conj R (n) where R is a retract of a finitely generated conjugacy separable group G.
Lemma 2.4.Suppose that G is a finitely generated conjugacy separable group with a finitely generated subgroup R ≤ G such that R is a retract.Then R is conjugacy separable and conjugacy embedded into G.Moreover, we have Proof.Let ρ : G → R be the corresponding retraction.We start by showing there is a finite generating set Following the previous paragraph together with remark 2.1, we see that ).We note that this inequality holds for all r 1 , r 2 ∈ R where r 1 ≁ R r 2 , and since

Wreath products.
For groups A and B, we denote the restricted wreath product of A and B, written as A ≀ B, by where B acts on b∈B A via left multiplication on the coordinates.An element f With a slight abuse of notation, we will use Following the given notation, if H ≤ A and K ≤ B, we will use H K to denote the subset Keeping this notation in mind, the wreath product H ≀ K can then be naturally identified with the subgroup Lemma 2.5.Let A, B be finitely generated abelian groups, and suppose that R A ≤ A and Proof.Let ρ A : A → R A and ρ B : B → R B be the associated retraction maps.We define a map ρ : We see that ρ is a surjective homomorphism and ρ| R = id R , and thus, it follows that R is a retract of A ≀ B. We finish by noting that Remark 2.
Suppose that b ∈ B and f ∈ A B is a function with a finite support.We say that f is minimal with respect to b if all elements of supp(f ) lie in distinct cosets of ⟨b⟩ in B. We will say that an element The following lemma is a special case of [10, Lemma 5.13].
Lemma 2.6.Let A, B be finitely generated groups, and suppose that b ∈ B and f : Then there exists a constant C independent of b and f and f ′ ∈ A B such that the following hold: The following statement and its proof which provides a conjugacy criterion for wreath products of abelian groups follows from [10, Lemma 5.14].
Lemma 2.7.Let A, B be abelian groups, and let G = A ≀ B be their wreath product.Let In particular, there exists an element c ∈ B such that One interpretation of Lemma 2.7 is that by ensuring that we are only working with reduced elements of A ≀ B, we only need to worry about them being conjugate by an element from B.
Let A be a finite abelian group and let B be a finitely generated abelian group of torsion free rank at least 1.This next lemma allows to reduce the study of asymptotic lower bounds for conjugacy separability of groups of the form A ≀ B to that of groups of the form F p ≀ Z. Lemma 2.8.Let A be a finite abelian group where p | |A|, and let B be an infinite, finitely generated abelian group.The group F p ≀ Z is conjugacy embedded in the group A ≀ B and Proof.Since Z is a retract of B, we have that A ≀ Z is a retract of A ≀ B. We then have by Lemma 2.4 that Conj A≀Z (n) ⪯ Conj A≀B (n).Thus, we may assume that B ∼ = Z.
We now demonstrate that where s 1 , s 2 ∈ Z.Moreover, we may assume are both reduced.We claim that and given that Z is abelian, we then have s = s 1 = s 2 .By Lemma 2.7, we have there exists a b t such that b t • supp(f 1 ) = supp(f 2 ) and f 1 (b t x) = f 2 (x).However, that is equivalent to f 1 b s and f 2 b s being conjugate in F p ≀ Z as desired.
For the second part of the statement, we first show that F p ≀ Z can be realised as an undistorted subgroup of A ≀ B. If Z/p e Z = ⟨a⟩, we then see that Z/pZ = ⟨a p e−1 ⟩.Letting X pe = {a, a p e−1 b} ⊆ L p e and X p = {a p e−1 , b} ⊆ L p , it then follows that L p e = ⟨X pe ⟩ and L p = ⟨X p ⟩.One can easily check that for any x ∈ L p that ∥x∥ Xp = ∥x∥ X p e , and subsequently, B Lp,Xp (n) ⊆ B L p e ,X p e (n).Now suppose that x, y ∈ B Fp≀Z (n) are not conjugate.We then have that f, g ∈ B A≀Z (n), and since As a consequence of the above inequality, and the previous paragraph, we see that The next lemma, which is a direct consequence of [6,Theorem 3.4], relates the size of the support of its function part and the size of the elements in the range of the function with the word length of an element.We omit the proof in order to avoid having to introduce more technical notation, we encourage a curious reader to inspect [6, Theorem 3.4] and prove check that the statement indeed holds.Lemma 2.9.Let A, B be finitely generated groups and let G = A ≀ B be their wreath product.Then there exists a constant C > 0 such that if g = f b where f ∈ A B and b ∈ B, then Given a wreath product A ≀ B with a surjective homomorphism π : B → B, we denote π : A ≀ B → A ≀ B as the canonical extension of π to all of A ≀ B given by where K = ker(π).Note that since the group A is abelian and the function f ∈ A B is finitely supported, the above sum is well defined.Similarly, if π : A → A is a surjective homomorphism, we let π : A ≀ B → A ≀ B as the natural extension of π to all of A ≀ B.

Wreath products and Laurent polynomial rings.
Much of the following discussion, which includes undefined notation and terms, can be found in [2,8,13].Given a commutative ring R, we will write R[x] to denote the ring of polynomials in the variable x with coefficients in R, and we will use R[x, x −1 ] to denote the ring of Laurent polynomials over R.
We first note that R[x, x −1 ] is the localisation of the ring R[x] on the set S = {x m |m ∈ N}.We then have that the ideals of R[x, x −1 ] are in one-to-one correspondence with ideals of R[x] that don't intersect the set S. In particular, for any ideal I ⊂ R[x] where I ∩ S = ∅, we have that R[x, x −1 ]/(S −1 I) = S −1 (R[x]/I).We finish by observing that the maximal ideals of R[x, x −1 ] can be written as I = (f ) where f is an irreducible polynomial not in S.
We now focus on the following representation of R ≀ Z as a semidirect product of the ring R[x, x −1 ] and Z. First, let us define a function P : R Z → R[x, x −1 ] given by One can easily check in the context of finitely supported functions that P is a bijection and for any r ∈ R, f, g ∈ R Z , and m ∈ Z that the following holds: (i) P (rf ) = rP (f ), (ii) P (f + g) = P (f ) + P (g), (iii) P (m • f ) = x m P (f ).We will use these three equalities without mention.
Lemma 2.10.Let R be either the ring Z or F p where p is prime.The group is ring of Laurent polynomials with addition and for t ∈ Z, we have t Proof.Let φ : R ≀ Z → R[x, x −1 ] ⋊ Z be the map given by φ (f m) = (P (f ), m).It is then easy to see that this map is an isomorphism.
The following lemma allows us to understand finite quotients of R ≀ Z in terms of the cofinite ideals of R[x, x −1 ].For the following lemma, we identify R[x, x −1 ] with the normal subgroup of R ≀ Z given by elements of the form (P, 0) where Lemma 2.11.Let R be either the ring Z or F p where F p be the field with p elements.Let . We note for (P, 0) ∈ M that (0, m) (P, 0) (0, −m) = (x m P, 0) ∈ M since M is normal.In particular, we have that M is closed under multiplication by x m in R[x, x −1 ] for all m ∈ Z.Additionally, for (P 1 , 0), (P 2 , 0) ∈ M we have that (P 1 , 0)(P 2 , 0) = (P 1 + P 2 , 0).That implies M is closed under addition.Since multiplying P by r is the same as adding r copies of P and given that M is a subgroup of R[x, x −1 ] with addition as its group operation, we must have that (rP, 0) ∈ M .Thus, for (P, 0) ∈ M and a general element m∈Z a m x m of R[x, x −1 ], we may write Thus, M is an ideal in R[x, x −1 ].Moreover, the second part of the statement immediately follows.
The following lemma gives the explicit expression for the conjugacy class of an arbitrary element of R ≀ Z. Lemma 2.12.Let R be either the ring Z or F p where F p is the field with p elements.For (P, m) ∈ R ≀ Z, its conjugacy class is given by and ℓ ∈ Z be arbitrary.We then write Since Q was arbitrary, we may replace it by −Q allowing us to write From here, our statement is clear.

Lower bounds
In this section, we construct asymptotic lower bounds for conjugacy separability for the groups F p ≀ Z and Z ≀ Z. which we divide into two subsections.The first subsection goes over the lower bounds for Conj Fp≀Z (n).The second subsection constructs lower bounds for Conj Z≀Z (n).

Lower bounds for Conj A≀B (n)
where A is a finite abelian group.In this section, we provide asymptotic lower bounds for Conj A≀B (n) when A is a finite abelian group and B is an infinite, finitely generated abelian group by finding asymptotic bounds for Conj Fp≀Z (n) .Proposition 3.1.Let A be a finite abelian group and B be an infinite, finitely generated abelian group.Then 2 n ⪯ Conj A≀B (n).
Proof.By Lemma 2.8, we may assume that A ∼ = F p for some prime and that B ∼ = Z.We need to find an infinite sequence of pairs of elements {f where C > 0 is some constant.
Let {q i } ∞ i=1 be an enumeration of the set of primes greater than p such that p is a primitive root mod q i .In this case, it is well known that ψ q i (x) = q i −1 i=1 x i is an irreducible polynomial over F p .Let Let us consider the quotient F p [x, x −1 ]/(ψ q i (x)) ⋊ (Z/q i Z)) with the associated projection map π i .We then see that and that π i (f i ) = (0, 0) and π i (g i ) = (x − 1, 0) ̸ = (0, 0).
It follows that π(f i ) ≁ π(g i ).Subsequently, we see that f i and g i are not conjugate in F p ≀ Z and that CD Fp≀Z (f i , g i ) ≤ q i p q i −1 .
To finish, we will demonstrate that p q i ≤ CD Fp≀Z (f i , g i ) for all i.In other words, we need to show that if N ⊴ f.i.F p ≀ Z is given such that |(F p ≀ Z)/N | < p q i , then f i ∼ g i mod N .Suppose that such a normal finite index subgroup N is given.We note by Lemma 2.11 that . Thus,for the purpose of the proof, we may assume that N ∼ = J ⋊ tZ for some t ∈ N. If J N = F p [x, x −1 ], then (F p ≀ Z)/N is a finite abelian group.In particular, we have that F p [x, x −1 ] ≤ ker(π N ), and thus, π N (f i ) = π N (g i ).Hence, we may assume that J N is a proper ideal in Since F[x, x −1 ] is a localisation of a principal ideal domain, it is also a principal ideal domain.Therefore, there exists a polynomial P ∈ F p [x] such that J N = (P ).Thus, we note that one of the following cases must hold: gcd(x . We see that we may ignore the first two cases, as in both we have that p q i ≤ |F p ≀ Z/N |. For the third case, we have that x − 1 ∈ J N .Therefore, we have π N (f i ) = (x q i − 1 mod J N , q i mod t) = ((x − 1)ψ q i (x) mod J N , q i mod t) = (0, q i mod t).
For the last case, we may assume that gcd(x q i − 1, P ) = 1.Let us recall that, following Lemma 2.12, we can write the conjugacy class of f i as In order for f i ∼ g i mod N , we need to have The above is equivalent to Using basic algebra, we see that the above is equivalent to Thus, we have that f i ∼ g i mod N if and only if x − 1 ∈ (x q i − 1) mod (P ).
We see that f i ∼ g i mod N, and therefore, From the construction of the elements f i , g i , it can be easily seen that there is a constant C ′ such that q i ≤ ∥f i ∥ ≤ C ′ q i and q i ≤ ∥g i ∥ ≤ C ′ q i .There, we have that p n ≤ Conj Fp≀Z (C ′ n).Hence, we may write

3.2.
Lower bounds for Conj A≀B (n) when A and B are infinite.In this subsection, we provide asymptotic lower bounds for Conj A≀B (n) where A and B are infinite, finitely generated abelian groups.We start with the group Z ≀ Z as seen in Proposition 3.3.Before we start, we have the following lemma.Proof.We note that m = ℓ gcd(m, n) for some integer ℓ.Therefore, x m ≡ x ℓ gcd(m,n) ≡ 1 mod (x gcd(m,n) − 1).
We now come to the last proposition of this section.
Proposition 3.3.Let A and B be infinite, finitely generated abelian groups.Then Proof.Let us first note that we may choose splittings of A and B as direct sums A ≃ Z k ⊕ Tor(B) and B ≃ Z d ⊕ Tor(B).Since we assumed that both A, B are infinite, we see that d, k > 0. In particular, A contains an element a of an infinite order such that ⟨a⟩ is an retract of A and B contains an element b of an infinite order such that ⟨b⟩ is an retract of B. By Lemma 2.5 we see that the subgroup Z ≀ Z ≃ ⟨a⟩ ≀ ⟨b⟩ is a retract of A ≀ B and Conj Z≀Z (n) ⪯ Conj A≀B (n).Hence, we may assume that A ≀ B ∼ = Z ≀ Z.
We need to find an infinite sequence of pairs of nonconjugate elements {f i , g i } such that log(Cmax{∥f i ∥, ∥g i ∥}) Cmax{∥f i ∥,∥g i ∥} < CD Z≀Z (f i , g i ) for some C > 0. For ease of writing, we denote Let {q i } be an enumeration of the primes, and let α(i) = lcm(1, • • • , q i − 1).Finally, let k i be the smallest integer such that α(i) ≤ 2 k i .We define the elements f i , g i ∈ Z ≀ Z as To see that f i is not conjugate to g i , we set k i be the ideal in Z[x, x −1 ] given by (2 k i , x 2 k i − 1), and let We note by Lemma 2.11 that In particular, J ⋊ (N ∩ Z) is a normal subgroup in Z ≀ Z.Similarly, N ∩ Z = bZ for some b ∈ Z.Therefore, we denote N ′ = J N ⋊ bZ.Thus, it follows that N ′ is a finite index normal subgroup of Z ≀ Z where N ′ ≤ N .In particular, if Therefore, we may assume that (Z ≀ Z)/N takes the form (Z[x, x −1 ]/J ) ⋊ (Z/bZ) where J is a cofinite ideal and b is an integer.
Following Lemma 2.2, we see that f i N ′′ ≁ g i N ′′ for N ′′ = J ⋊ (bZ + K) where K ≤ Z is the preimage under the projection modulo b of the kernel of the action of Z/bZ on Z[x, x −1 ]/J N .Letting b 0 ≥ 0 be such that b 0 Z = bZ + K, we note that N ′′ is a finite index normal subgroup where Therefore, by the above discussion, we may assume that where Z/bZ acts faithfully on Z[x, x −1 ]/J .
We now show we may assume that Z/bZ acts freely on Z[x, x −1 ]/J .Suppose that there are polynomials ρ(x), λ(x) ∈ Z[x, x −1 ]/J such that for some 0 ≤ m < b.Since x m is a unit, we may cancel and write ρ(x) + J N = λ(x) + J N which gives our claim.
Let ℓ be the multiplicative order of x + J in Z[x, x −1 ]/J .We claim that ℓ = b.By definition, we have that b is the smallest integer such that In particular, we have that x b • 1 = 1 mod J .Thus, we have that ℓ | b.If ℓ ⪇ b, we then have that x ℓ ρ(x) = ρ(x) mod J for all ρ(x) ∈ Z[x, x −1 ].However, that implies Z/bZ doesn't act faithfully on Z[x, x −1 ]/J which is a contradiction.Therefore, we have that ℓ = b.
Since Z/bZ acts freely and transitively on the set of powers of If d < q i , then d | α(i), and subsequently, Hence, f i = g i mod N .Therefore, we may assume that d ≥ q i .By Lemma 2.12, we may write the conjugacy class of f i as Thus, we have that f i ∼ g i if and only if which is equivalent to Since x ℓ − 1 + Q can be any Laurent polynomial, we have that f i ∼ g i if and only if By Lemma 3.2, we have that Therefore, we may write the conjugacy class of f i in (Z ≀ Z)/N as where 0 ≤ t < k i .Therefore, f i N ∼ g i N if and only if Recall that α(i) = exp{υ(q i − 1)}, where υ : N → N is the second Chebyshev's function.The Prime Number Theorem [25, 1.2] then implies that there are constants Following the definition of k i , we see that there are constants From the construction of the elements f i , g i , it can be easily seen that there is a constant C ′ such that where n i = max{∥f i ∥, ∥g i ∥}.Following the previous discussion, we see that there are constants In particular, we see that q i ≤ log(Cn i ) for some C > 0. Therefore, q

Upper bounds
The aim of this section is to construct upper bounds for the conjugacy depth function of a wreath product A ≀ B of finitely generated abelian groups.The idea is to show that we can always find a quotient of the acting group B such that Lemma 2.7 can be used to demonstrate that the images of the elements are not conjugate and provide asymptotic bounds on the size of this quotient.Recall that one of the assumptions of Lemma 2.7 is that we are working with reduced elements, i.e. the elements of the supports lie in distinct cosets of the acting element.Thus, in order to ensure we are working with reduced elements, 4.1 we show how to construct a finite quotient of the acting group that separates finite subsets and infinite cyclic subgroups.Subsection 4.2 then deals with the conditions that Lemma 2.7 uses to establish non-conjugacy.In particular, we show that if a quotient of a finitely generated abelian group is of sufficient size, then certain finite subsets do not become translates of each other in the quotient.Finally, subsection 4.3 combines these methods to construct a finite quotient preserving non-conjugacy of our given non-conjugate elements and gives an upper bound on its size in terms of their word lengths.
Before we proceed, we recall some notation.If B is a finitely generated abelian group, we by fixing a splitting may write B = Z k ⊕ Tor(B) where Tor(B) is the subgroup of finite order elements of B and k is the torsion-free rank of B. Letting ϕ : B → Z k and τ : B → Tor(B) denote the natural projections associated to the fixed splitting, we may then write every x ∈ B uniquely as x = ϕ(x) + τ (x) where we refer to ϕ(x) as the torsion-free part of x and τ (x) as the torsion part of x.When given a vector b = (b 1 , . . ., b k ) ∈ Z k , we denote gcd(b) = gcd(b 1 , . . ., b k ).Given two real numbers a < b, we let [a, b] denote closed interval from a to b.Given two vectors v, w ∈ R k , we denote their dot product as v • w.Finally, for a finite group T , we denote its exponent as exp(T ).

Simultaneous cosets.
In this subsection, we study effective separability of cosets of cyclic subgroups in finitely generated abelian groups.Given an infinite, finitely generated abelian group G, an element b ∈ G, and a finite subset S ⊆ B G (ℓ), we give an upper bound in terms of ∥b∥ and ℓ on the size of a finite quotient of the group G such that each pair of cosets of the cyclic subgroup generated by b corresponding to two distinct elements in S remain distinct.In the following arguments, we use the observation that s 1 ⟨b⟩ = s 2 ⟨b⟩ if and only if s −1 1 s 2 ∈ ⟨b⟩.The following lemma is important for the proof of Lemma 4.5.Then for every s ∈ S, we have that Furthermore, if π(s) ∈ ⟨π(b)⟩, then π(s) = tπ(b) for the smallest integer t with respect to the absolute value such that s = tb.In particular, |t| ≤ m.

Proof. Observe that the map
Finally, suppose that π(s) ∈ ⟨π(b)⟩ and that π(s) = aπ(b) in Z/mZ where a ∈ Z is the smallest such value with respect to the absolute value.Following the previous argument, it follows that ab ∈ [−c|b| − 1, c|b|], and therefore, we have s = ab in Z.
To deal with the higher-dimensional cases, we first prove two technical lemmas.This first lemma gives bounds of lengths of a free generating basis for the kernel of the linear map given by the dot product with a vector in terms of size of the entries of the vector.For this lemma, when given vectors v 1 , . . ., v k ∈ Z n , we denote ⟨v 1 , . . ., v k ⟩ as the subgroup generated by the set {v 1 , . . ., v k }.Then there are vectors λ (1) , . . ., λ (k−1) ∈ Z k such that ker(φ b ) = ⟨λ (1) , . . ., λ (k−1) ⟩ and ∥λ (i) ∥ ≤ 2 k−1 ∥b∥ for all i.
By construction, we have that b (1) , . . ., b (k−1) ∈ ker(φ b ).Let Λ i be the maximal subgroup of Z k of rank i that contains b (1) , . . ., b (i) .Since the vectors b (1) , . . ., b (k−1) are linearly independent over R, we immediately see that Now assume that we already have a set of generators for Λ i−1 which we denote as λ (1) , . . ., λ (i−1) .By construction, the elements {λ (1) , . . ., λ (i−1) } satisfy Λ j = ⟨λ 1 , . . ., λ j ⟩ for all j < i where we see that Λ i /L i is a finite cyclic group.Furthermore, a preimage of some of its generator must be contained within the i-dimensional parallelogram given by the vectors λ (1) , . . ., λ (i−1) , b (i) .In particular, we see that One can then easily check that This next lemma implies any vector in Z k whose entries have greatest common denominator as 1 is a part of a free base of Z k .This lemma also shows that there exists an matrix T ∈ GL k (Z) which sends the vector to an element of the canonical basis and gives a bound on how much the matrix T stretches the unit cube in R k in terms of the size of the entries of the vector.
To finish the proof, we recall that for all i.In particular, this means that for every i, j ∈ {1, . . ., k} we have the (i, j)-th entry of T which we denote as T i,j satisfies |T i,j | ≤ 2 k−1 ∥b∥.Therefore, we have The following corollary is not consequential for this paper, but we feel it an interesting result in its own right.Then there are elements λ (1) , . . ., λ (k−1) ∈ Z k such that the set {b, λ (1) , . . ., λ (k−1) } is a free base of Z k and ∥λ (i) ∥ ≤ 2 k ∥b∥.(2) m > 2 k kCn 2 ; (3) the homomorphism π : Z k → (Z/mZ) k given by reducing every mod m is injective on the set S. Then for every s ∈ S we have that π(s) ∈ ⟨π(b)⟩ if and only if s ∈ ⟨b⟩.
Furthermore, if π(s) ∈ ⟨π(b)⟩, then there is an integer t ∈ Z such that s = tb and |t| ≤ m/c.

Proof. Set b
where {e 1 , . . ., e k } is the canonical free basis of Z k .Since T is an automorphism of Z d , we see that T (s) ∈ ⟨T (b)⟩ = ⟨ce 1 ⟩ if and only if s ∈ ⟨b⟩.We note that Set m = 2cl where l ∈ N is the smallest natural number such that cl > 2 k−1 kCn 2 , and denote K = mZ k ≤ Z k .By construction, we have that m ≤ 2 k+1 kCn 2 .Therefore, we see that the projection π  defined as the identity on Tor(B) and as the coordinate-wise projection on Z k is injective on the set S and for every s ∈ S we have that π(s) ∈ ⟨π(b)⟩ if and only if s ∈ ⟨b⟩.
The main argument of the proof when k = 1 is analogous to the case when k ≥ 2, but instead of Lemma 4.5 one would use Lemma 4.1.For this reason, we leave proof in the case when k = 1 as an exercise.
Denote e = exp(Tor(B)), and suppose that m > 0 and π : Z k ⊕ Tor(B) → (Z/mZ) k ⊕ Tor(B) are as in the statement of the lemma.Assuming that π(s) ∈ ⟨π(b)⟩ for some s ∈ S, there is some t ∈ N such that π(s) = tπ(b) which we pick to be as small possible.In particular, we see that t ≤ gcd(m, e) = m.
We write:

Translations.
We say that an ordered list X In this case, we say that σ realises a translation of X onto Y .
We have the following lemma which gives conditions of when two sets in a product of groups are translates of each other in terms of translations of their images in the projection onto the factor groups.
Proof.It is straightforward to see that X is a translate of Y then |X| = |Y |.Thus, we may assume that |X|, |Y | = n for some n ∈ N. As mentioned above, we have that X is a translate of Y if and only if there is σ ∈ Sym(n) such that That is equivalent to saying that σ realises a translation of (π 1 (x 1 ), . . ., π 1 (x n )) onto (π 1 (y 1 ), . . ., π 1 (y n )) and that σ realises a translation of the list (π 2 (x 1 ), . . ., π 2 (x n )) onto the list (π 2 (y 1 ), . . ., π 2 (y n )).This next lemma tells us there exists a constant ℓ such that when given two finite subsets X, Y in Z k whose coordinates of each element have absolute value at most ℓ, then X and Y are translations of each other if and only if their images are translations in the group (Z/4ℓZ) k where we reduce each coordinate mod 4ℓ. for some ℓ ∈ N, and suppose that c ≥ 4ℓ Let π : Z k → (Z/cZ) k be the homomorphism given by reducing each coordinate mod c.Then the following are equivalent: (1) π(X) = (π(x 1 ), . . ., π(x n )) is a translate of π(Y ) = (π(y 1 ), . . ., π(y n )) in (Z/cZ) k .
(2) X is a translate of Y in Z k .Furthermore, for all σ ∈ Sym(n), we have that σ realises a translation of π(X) onto π(Y ) if and only if σ realises a translation of X onto Y .
Proof.We will only prove the 'furthermore part of the statement since the first part follows from it.

Finite base groups.
Let A be a finite abelian group, and let B be an infinite, finitely generated abelian group.Using Lemma 4.6 and Lemma 4.2, the next proposition demonstrates when given nonconjugate elements x, y in a s A ≀ B that there exists a finite quotient B of B such that the images of x and y in A ≀ B remain non-conjugate.Moreover, this lemma gives a bound of the size of the quotient of B in terms of the word lengths of x and y.

Lemma 4 . 1 .
Let b ∈ Z satisfy b ∈ [−n, n] and S ⊆ Z be a subset such that S ⊆ [−Cn, Cn] for some constant C > 0. Suppose that c is a natural number where |b|c > CN , and let m = 2|b|c.Finally, let π : Z → Z/mZ be the natural projection.
For a vector b ∈ Z k , this next lemma gives bounds on the size of the integer m we reduce entries in Z k mod m to preserves cosets of the infinite cyclic subgroup generated by b in terms of the size of the entries in b.

Lemma 4 . 5 .
Let k > 1 and n ∈ N be fixed.Let S ⊆ Z k and b = (b 1 , . . ., b k ) ∈ Z k satisfy b ∈ B Z k (n)and where S ⊆ B Z k (Cn) for some C > 0. Suppose m is an integer satisfying the following:(1) m is divisible by gcd(b);

Lemma 4 . 6 .
Let B be a finitely generated infinite abelian group of torsion free rank k.Let b ∈ B B (n) and S ⊂ B B (Cn) be given for some constant C > 0. If k = 1, assume that m ∈ N satisfies m ≥ 2Cn and where both ∥ϕ(b)∥ and exp(T ) divide m.If k ≥ 2, assume that m ∈ N is such that m ≥ k2 k Cn 2 and both c = gcd(ϕ(b)) and exp(Tor(B)) divide m.Then the homomorphism π : Z k ⊕ Tor(B) → (Z/mZ) k ⊕ Tor(B)

Proposition 4 . 10 .
Let A be an abelian group and B be an infinite, finitely generated abelian group.Let f, g : B → A be finitely supported functions and b ∈ B an element such that f b, gb ∈ B A≀B (n) and f b ̸ ∼ A≀B gb.Then there exists a surjective homomorphism π : B → B to a finite group such that π(f b) ̸ ∼ π(gb) in A ≀ B.Moreover, there exists a constant C > 0 such that if B has torsion free rank 1, then we have | B| ≤ Cn, and if B is of torsion-free rank k > 1, we then have | B| ≤ Cn 2k .
1, m] where π K is the reduction of each coordinate mod m.In particular, since S ⊆ k i=1 [−m + 1, m], for any s ∈ S we have that T (s) mZ k ⊆ ⟨e 1 ⟩K if and only if T (s) ∈ ⟨e 1 ⟩.Now suppose that s ∈ ⟨b ′ ⟩, i.e.T (s) ∈ ⟨e 1 ⟩.In this case, we may retract onto the first coordinate and assume that we are working in Z.The rest of the statement then follows by Lemma 4.1.This next lemma extends Lemma 4.5 to when the abelian group has torsion.