Residual dimension of nilpotent groups

1.1 Residual finiteness

We start with the case of infinite, finitely generated nilpotent groups N, where N/T⁢(N) is abelian.

The situation is more interesting when N/T⁢(N) has step length c>1 as seen in the following theorem.

For this theorem, we need to find an infinite sequence of elements {xi}i=1∞ such that the minimal finite order group Qi where there exists a surjective group morphism

has order approximately (log(∥xi∥)dimRFL⁡(N). We start the search for this sequence by demonstrating that we may assume that the nilpotent group N is torsion free. We then find an element x∈N such that there exists an infinite sequence of natural numbers {mi}i=1∞ for which our desired sequence of elements is given by {xmi}i=1∞. Any such element x will be a primitive element in the isolator of γc⁢(N), and associated to these elements, there exists a natural number invariant dimRFL⁡(N,x) which captures the lower asymptotic behavior of separating powers of x from the identity with surjective group morphisms to finite p-groups as one varies the prime number p. Letting RPN,x,i be the set of primes p such the order of a minimal finite p-group which witnesses the primitive element x has order pi, we take dimRFL⁡(N,x) to be the minimal i0 such that |RPN,x,i0|=∞. By maximizing dimRFL⁡(N,x) over all primitive elements in the isolator of γc⁢(N), we obtain the value dimRFL⁡(N).

For the next theorem, we say a finitely generated nilpotent group has tame residual dimension if the sets RPN,x,i have a natural density for all primitive elements x in the isolator of γc⁢(N) and indices i and if there exists a generating basis {zi}i=1h for the isolator of γc⁢(N) such that, for any primitive element g and prime p, there exists a surjective group morphism from N to a p-group of minimal order that witnesses both g and zi for some index i. This definition implies that we may associate a natural number invariant to any such primitive element x which takes into account the smallest index i, where we may apply the Prime Number Theorem relative to the set RPN,x,i. Moreover, the asymptotic behavior of separating powers of any primitive element from the identity using surjective group morphisms to finite p-groups as we vary the prime p is controlled by the asymptotic behavior of separating powers of elements of the given generating basis from the identity using surjective group morphisms to finite p-groups as we vary the prime p. We denote this value as dimRFU⁡(N,x), and note that our assumption implies that

By maximizing dimRFU⁡(N,zi) over elements of the generating basis, we obtain the value dimRFU⁡(N) which gives a degree for a polynomial in logarithm upper asymptotic bound for FN⁢(n) for this class of nilpotent groups. As Bou-Rabee’s example demonstrates, there exists a class of nilpotent groups whose residual finiteness is strictly slower than what is predicted by [6, Theorem 1.1], and the motivation in introducing the definition of tame residual dimension is to precisely capture this phenomenon. Thus, we have the following theorem.

We observe that the first statement of the above theorem is the upper bound produced for FN⁢(n) in [6, Theorem 1.1], and we claim no originality in including it in this article. We restate this theorem and give its proof to show the similarities and differences between the proof of the original upper bound and the proof of the improved upper bound in the case of finitely generated nilpotent groups that have tame residual dimension. In particular, we want to highlight how the geometry of finite p-quotients of a finitely generated nilpotent group changes as we vary the prime and how it contrasts with torsion-free quotients of the nilpotent group with one-dimensional center.

For this last class of nilpotent groups, we are able to apply our methods to completely characterize the asymptotic behavior of FN⁢(n). In addition to assuming that the sets RPN,x,i have a well-defined natural density for all primitive elements x and indices i, we assume that these sets RPN,x,i are finite when i<dimRFU⁡(N,x). In particular, we may ignore these sets of primes when calculating an asymptotic lower bound for FN⁢(n). Finitely generated nilpotent groups that satisfy these properties are said to have accessible residual dimension, and as a consequence, we have that dimRFL⁡(N)=dimRFU⁡(N) whose common value is denoted dimARF⁡(N).

1.2 Previous bounds found in the literature

To the author’s knowledge, the best previously known asymptotic lower bound for FN⁢(n) for the class of infinite, finitely generated nilpotent groups was log⁡(n) which was given by [1]. This lower bound does not use any of the structure of the nilpotent group other than it containing an infinite order element, and subsequently, the lower bound is not connected to the geometry of the nilpotent group in consideration. On the other hand, the lower bounds produced in this article reflect the step length of the nilpotent group and take into account the geometry of the pro-p completion as the prime p varies. In particular, this asymptotic lower bound represents a significant improvement in the understanding of FN⁢(n).

The first asymptotic upper bound for FN⁢(n) in the literature was (log⁡(n))h⁢(N) and was given by [1, Theorem 0.2]. This upper bound partially reflects the structure of N in that h⁢(N) gives the dimension of the real Mal’tsev completion of N. However, the bound (log⁡(n))ψRF⁢(N) reflects the geometry of the different quotients of the real Mal’tsev completion which has one-dimensional center in which a given primitive central element of N does not vanish. In particular, the original bound given in [1] can be improved as soon as the center of the Mal’tsev completion has dimension greater than 1 which can be seen in the above theorem.

When the nilpotent group has tame residual dimension, the upper bounds can be improved even further. This improved bound reflects the number theoretic properties of these nilpotent groups in relation to how finite p-quotients behave as the prime p varies. In particular, these finite p-quotients are related to the distribution of primes p for which a polynomial F⁢(T) with integer coefficients has a solution mod⁡p as we will see in Bou-Rabee’s example. As a consequence, the upper bound for residual finiteness for nilpotent groups with tame residual dimension reflects more strongly how the geometry of the pro-p completion of the given nilpotent group varies with the prime p and how their geometry differs from the Mal’tsev completion than what is seen in the literature or in the first asymptotic upper bound given in this article.

1.3 Partial recovery of the asymptotic behavior of FN⁢(n)

It is important to note that we are unable to fully recover the asymptotic behavior of FN⁢(n) for a general finitely generated nilpotent group N; however, if we could show that the sets RPN,x,i have positive natural density or are finite, we would be able to fully characterize FN⁢(n). For reasons related to the structure of nilpotent linear algebraic groups defined over ℚ, we believe this to be the case and go into detail in the next few paragraphs.

Up to passing to a finite index subgroup, every torsion-free, finitely generated nilpotent group N arises as the group of integral points of a connected linear algebraic group 𝐍 defined over ℚ where the image of N under the inverse of the exponential map provides a ℤ-structure for the Lie algebra 𝔫 of 𝐍. Since 𝐍 satisfies strong approximation, the smooth mod⁡p reduction of 𝐍, denoted 𝐍p, is well defined for all but finitely many primes p, and for such primes p, the image of N under the natural reduction group morphism is isomorphic to 𝐍p⁢(𝔽p), where 𝐍p⁢(𝔽p) is the group of 𝔽p-points of 𝐍p. In particular, the Lie algebra of 𝐍p, denoted by 𝔫p, is given by the mod⁡p reduction of the ℤ-structure on 𝔫. Thus, if there exists a normal connected algebraic subgroup 𝐇p⁢⊴⁢𝐍p such that (𝐍p/𝐇p)⁢(𝔽p) witnesses x and where dim𝔽p⁡(𝐍p/𝐇p)=i for all primes p∈RPN,x,i, we may reduce the study of separating powers of x to the study of quotients of 𝔫p in which a nonzero vector X∈𝔫 does not vanish as we vary the prime p. At this point, we would use Lie theoretic methods and basic algebraic geometry as a way to study Lie quotients of 𝔫p for different primes p. This perspective gives more traction to study the sets RPN,x,i and provides intuitive justification for the sets RPN,x,i having a natural density.

For p∈RPN,x,i, where i<dimRFU⁡(N,x), we have that any finite p-group P that witnesses x and where |P|=pi may have step length less than that of N. These primes would correspond to finite quotients of the ℤ-structure on 𝔫 which lose the structure of N. The idea is that the structure constants of the ℤ-structure may vanish mod p; however, since there are finitely many structure constants, we would have that the Lie algebra 𝔫p would receive the full Lie bracket structure of 𝔫 for all but finitely many primes p. That should imply that the set RPN,x,i is finite when i<dimRFU⁡(N,x) for any primitive element x.

1.4 Plan of paper

In Section 2, we introduce the necessary background and conventions for this paper. Section 3 expounds on the example provided to by Khalid Bou-Rabee. Section 4 introduces and defines the constants dimRFL⁡(N), dimRFU⁡(N), dimARF⁡(N) for infinite, finitely generated nilpotent groups. Sections 5, 6 and 7 provide proofs of Theorem 1.2, Theorem 1.3 and Theorem 1.4, respectively. Section 8 finishes with the computation of FN⁢(n) when N is a Filiform nilpotent group.

2.1 Notation and conventions

We let lcm⁡{r1,…,rm} be the lowest common multiple of {r1,…,rm}⊆ℤ with the convention that lcm⁡(a)=|a| and lcm⁡(a,0)=0. We let gcd⁡(r1,r2) be the great common divisor of the integers r1 and r2 with the convention that gcd⁡(r,0)=|r|. We write ℕ for the set of natural numbers excluding 0 and denote by ℙ the set of prime numbers.

We denote by ∥g∥S the word length of g in G with respect to the finite generating subset S, and when the subset S is clear from context, we write ∥g∥. We denote the identity of G by 1 and denote by {1} the trivial group. We let OrdG⁢(x) be the order of x as an element of G and denote the cardinality of a finite group G by |G|. For a normal subgroup H⁢⊴⁢G, we set πH:G→G/H to be the natural projection and write x¯=πH⁢(x) when H is clear from context. When given a nonempty subset X⊂G, we denote by 〈X〉 the subgroup generated by the elements of X. We define [x,y]=x-1⁢y-1⁢x⁢y. For x1,…,xk∈G, we define [x1,…,xk]=[x1,[x2,…,xk]], where [x2,…,xk] is inductively defined. For nonempty subsets A,B⊂G, we define [A,B] as the subgroup generated by the subset {[a,b]∣a∈A,b∈B}.

We denote the center of G by Z⁢(G) and denote by γi⁢(G) the i-th term of the lower central series of G. For a finitely generated nilpotent group N, we denote by h⁢(N) its Hirsch length and by T⁢(N) the subgroup generated by finite order elements. For natural numbers m, we let Gm=〈gm∣g∈G〉 and write πm for the natural projection πm:G→G/Gm.

2.2 Residual finiteness

Following [1], we define the depth function DG:G\{1}→ℕ∪{∞} of the finitely generated group G as

with the understanding that DG⁢(g)=∞ if no such finite group exists.

For a residually finite, finitely generated group G with finite generating subset S, we define the associated complexity function FG,S:ℕ→ℕ as

For any residually finite, finitely generated group G with finite generating subsets S1 and S2, we have that FG,S1⁢(n)≈FG,S2⁢(n) (see [1, Lemma 1.1]). Henceforth, we will suppress the choice of finite generating subset.

2.3 Nilpotent groups

For more details of the following discussion, see [4, 7]. We define γ1⁢(G)=G and inductively define γi⁢(G)=[γi-1⁢(G),G]. We call the subgroup γi⁢(G) the i-th term of the lower central series of G.

For a subgroup H≤N of a nilpotent group, we define the isolator of H in N as

It is well known that HN is a subgroup of N for all H≤N when N is a torsion-free, finitely generated nilpotent group. Additionally, if H⁢⊴⁢N, then HN⁢⊴⁢N. As a natural consequence, N/HN is torsion free. When the group N is clear from context, we will simply write H.

The torsion subgroup of a finitely generated nilpotent group N is defined as

It is well known that T⁢(N) is a finite-characteristic subgroup of N. Moreover, if N is an infinite, finitely generated nilpotent group, then N/T⁢(N) is a torsion-free, finitely generated nilpotent group.

When given a torsion-free, finitely generated nilpotent group, we may refine the series {γi⁢(N)}i=1c to obtain a normal series {Hi}i=1h⁢(N) where Hi/Hi-1≅ℤ for all i. The number of terms in this series is known as the Hirsch length of N and is denoted by h⁢(N). In particular, the Hirsch length may be computed as

We choose x1∈N such that H1≅〈x1〉, and for each 2≤i≤h⁢(N), we choose xi∈Hi such that Hi/Hi-1≅〈πHi-1⁢(xi)〉. Any generating subset chosen in this way will be called a Mal’tsev basis. Via [5, Lemma 8.3], we may uniquely represent each element of N with respect to this generating subset as g=∏i=1h⁢(N)xiαi, where αi∈ℤ. The values {αi}i=1h⁢(N) are referred to as the Mal’tsev coordinates of g. Whenever we reference a Mal’tsev basis, we suppress reference to the series {Hi}i=1h⁢(N).

The next proposition and its proof can be originally found in [6, Lemma 3.8]. It relates the Mal’tsev coordinates of an element g to its word length with respect to the generating subset given by the Mal’tsev basis.

2.4 Density

For A⊂ℙ, we define the natural density of A in ℙ as

when the limit exists. When this limit exists for a set A, we say that A has a natural density.