Arithmetic lattices in unipotent algebraic groups

Khalid Bou-Rabee; Daniel Studenmund

doi:10.1515/jgth-2019-0112

Article Publicly Available

Arithmetic lattices in unipotent algebraic groups

Khalid Bou-Rabee and Daniel Studenmund

Published/Copyright: November 8, 2019

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From the journal Journal of Group Theory Volume 23 Issue 2

Abstract

Fixing an arithmetic lattice Γ in an algebraic group G, the commensurability growth function assigns to each n the cardinality of the set of subgroups Δ with [Γ:Γ∩Δ][Δ:Γ∩Δ]=n. This growth function gives a new setting where methods of F. Grunewald, D. Segal and G. C. Smith’s “Subgroups of finite index in nilpotent groups” apply to study arithmetic lattices in an algebraic group. In particular, we show that, for any unipotent algebraic ℤ-group with arithmetic lattice Γ, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in p-s, where the degrees of the numerator and denominator are independent of p. This gives regularity results for the set of arithmetic lattices in G.

1 Introduction

Let G be an algebraic group defined over ℤ (an algebraic ℤ-group). Two subgroups Δ1 and Δ2 of G⁢(ℝ) are commensurable if their commensurability index

c(Δ1,Δ2):=[Δ1:Δ1∩Δ2][Δ2:Δ1∩Δ2]

is finite. An arithmetic lattice of G is a subgroup of G⁢(ℝ) that is commensurable with G⁢(ℤ). The first purpose of this article is to show that, when G is unipotent, the set of arithmetic lattices in G has a great deal of regularity. The second purpose is to bring attention to a new notion of quantifying commensurability.

The main tool of this article is the commensurability growth function

ℕ→ℕ∪{∞}

assigning to each n∈ℕ the cardinality

𝐜n(G(ℤ),G(ℝ)):=|{Δ≤G(ℝ):c(G(ℤ),Δ)=n}|.

We study 𝐜n⁢(G⁢(ℤ),G⁢(ℝ)) for groups G in the class 𝒰 of unipotent algebraic ℤ-groups, starting with the fact that 𝐜n⁢(G⁢(ℤ),G⁢(ℝ)) is finite for all n in Lemma 3.1. Note that the correspondence between G and G⁢(ℤ) is the Mal’cev correspondence [12] between unipotent ℤ-groups and torsion-free finitely generated nilpotent groups (see also [5]).

Our proofs follow the outline of F. Grunewald, D. Segal and G. C. Smith [6], who studied the subgroup growth function of torsion-free finitely generated nilpotent groups Γ, defined by an(Γ):={Δ≤Γ:[Γ:Δ]=n}, by decomposing the associated zeta function into a product of local functions (see also [4]). We overcome an additional technical hurdle in order to prove rationality of our local zeta functions. While Grunewald, Segal and Smith use a result of J. Denef [2] to prove rationality, we require rationality of an integral over a noncompact set, and therefore draw from work of A. MacIntyre [11] developed after [6]. See § 1.2 for more details.

In view of the relationship 𝐜n⁢(Γ,Γ)=an⁢(Γ), the function 𝐜n extends an to pairs of groups. While we have the simple relationship

𝐜n⁢(G⁢(ℤ),G⁢(ℝ))≥an⁢(G⁢(ℤ)),

it is clear from the results in this paper that this inequality is strict when G is unipotent. The difference is evident even in the one-dimensional case.

1.1 The setting

Our results about 𝐜n are stated in the language of zeta functions. Let G be an algebraic group and ℱ a family of subgroups of G⁢(ℝ). We associate to such a family the Dirichlet series

ζℱ(s)=∑Δ∈ℱc(G(ℤ),Δ)-s=∑n=1∞𝐜n(ℱ)n-s,

where 𝐜n(ℱ):=|{Δ∈ℱ:c(G(ℤ),Δ)=n}|.

Following [6], there are two questions one can ask about the sequence 𝐜n⁢(ℱ): (a) how fast does it grow, and (b) how regularly does it behave? Pursuing (a) is the study of commensurability growth, which we do not directly address here. We will address (b) by decomposing ζℱ into local parts and proving a local regularity theorem. The families we consider are the following:

ℒ(G):={all arithmetic lattices insideG},

ℒn(G):={Δ∈ℒ(G):c(Δ,G(ℤ))=n},

ℒ(p)(G):={Δ∈ℒ(G):c(Δ,G(ℤ))is a power ofp}.

Define the global and local commensurability zeta functions

ζG:=ζℒ⁢(G) and ζG,p:=ζℒ(p)⁢(G).

1.2 The main results

We start with an elementary example, proved in § 2.

Proposition 1.1.

Let G=Ga, the additive algebraic group, so that G⁢(Z)=Z and G⁢(R)=R. Then, formally,

ζG⁢(s)=ζ2⁢(s)ζ⁢(2⁢s).

The zeta functions introduced in [6] decompose into Euler products. Our first main result, proved in § 3.1, is an analogous decomposition of the commensurability zeta function ζG for G∈𝒰 .

Proposition 1.2.

If G∈U, then

ζG⁢(s)=∏pζG,p⁢(s)

as formal products over all primes of Dirichlet series.

The main content of this article, our next result, is proved in § 3.2.

Theorem 1.3.

Let G∈U; then the function ζG,p⁢(s) is rational in p-s, where the degrees of the numerators and denominators are bounded independently of the prime p.

Proposition 1.2 and Theorem 1.3 provide regularity results for the set of arithmetic lattices, ℒ⁢(G).

Corollary 1.4.

Let G∈U; then

𝐜n⁢(ℒ⁢(G))=∏𝐜piei⁢(ℒ⁢(G)), where n=∏piei is the prime factorization of n;
there exist positive integers l and k such that, for each prime p, the sequence (𝐜pi⁢(ℒ⁢(G)))i>l satisfies a linear recurrence relation over ℤ of length at most k.

Our proof of Theorem 1.3 uses [11, Theorem 22]. We require a stronger theorem than that used in [6] since we express the local commensurability zeta function as a p-adic integral over an unbounded set (see Proposition 3.12). To find this explicit formula, we constructed a parametrization of lattices that gives a closed form for the commensurability index function (see Lemma 3.10). We also discovered an explicit correspondence

{closed⁢K≤G⁢(ℚp):c⁡(G⁢(ℤp),K)=pk}↔{Δ≤G⁢(ℚ):c⁡(G⁢(ℤ),Δ)=pk}

that may be of independent interest (see Lemma 3.5).

We were moved to pursue this subject after reading the work of N. Avni, S. Lim and E. Nevo [1] on the related concept of commensurator growth. Commensurator growth in [1], subgroup growth in [6] and commensurability growth in this paper are all studied through their associated zeta functions. Associating zeta functions to growth functions in groups is an active area of research, centering around subgroup growth and representation growth. For background reading on these subjects, we recommend the references [10, 15, 8].

Notation

c(A,B)=[A:A∩B][B:A∩B].
𝒰 is the class of unipotent algebraic ℤ-groups.
𝒯 is the class of torsion-free finitely generated nilpotent groups.
|S| is the cardinality of the set S.
[x,y]=x-1⁢y-1⁢x⁢y=x-1⁢xy.
Gn=〈gn:g∈G〉 for n∈ℤ.
Z⁢(G) is the center of G.
Tn⁢(R) is the set of lower-triangular n×n matrices over a commutative ring R.
ζ⁢(s) is the Riemann zeta function.

2 The one-dimensional case

We begin with the integers. In this case, we can directly relate the commensurability zeta function with the classical zeta function.

Proposition 2.1.

Let G=Ga, the additive algebraic group, so that G⁢(Z)=Z and G⁢(R)=R. Then, formally,

ζG⁢(s)=ζ2⁢(s)ζ⁢(2⁢s).

Proof.

The subgroups of ℝ commensurable with ℤ are all of the form r⁢ℤ, where r∈ℚ*. Writing r=n/d in reduced form, we have c⁡(ℤ,r⁢ℤ)=n⁢d. Hence,

𝐜n⁢(ℒ⁢(G))=|{a/b:a,b∈ℕ,(a,b)=1,a⁢b=n}|.

From this, we get, for distinct primes p1,p2,p3,…,pn,

𝐜p1j1⁢p2j2⁢⋯⁢pnjn⁢(ℒ⁢(G))=𝐜p1j1⁢(ℒ⁢(G))⁢𝐜p2j2⁢(ℒ⁢(G))⁢⋯⁢𝐜pnjn⁢(ℒ⁢(G)).

And for any prime p, we have 𝐜pk⁢(ℒ⁢(G))=2. Hence, 𝐜n⁢(ℒ⁢(G))=2ωn, where ωn is the number of distinct primes dividing n. Following [7, p. 255], we compute

ζG⁢(s)=∑n=1∞2ωn⁢n-s=∏p⁢prime(1+∑k=1∞2pk⁢s)=∏p⁢prime(2⁢p-s1-p-s+1)=∏p⁢prime1-p-2⁢s(1-p-s)2=ζ2⁢(s)ζ⁢(2⁢s).∎

3 Unipotent algebraic groups

To any G∈𝒰, we fix an embedding of G⁢(ℝ) into a group Tn⁢(ℝ) such that

G⁢(ℚ)=Tn⁢(ℚ)∩G⁢(ℚ)

given by Kolchin’s theorem [9]. With this in hand, we first show that, for any unipotent algebraic ℤ-group G, the commensurability growth function takes values in ℕ.

Lemma 3.1.

Let G be in U. Then Ln⁢(G) is finite for any n∈N.

Proof.

Given n∈ℕ, for any Δ∈ℒn⁢(G) and for every h∈Δ, we have hn∈G⁢(ℤ). By [13, Exercise 7, p. 114], there exists a finitely generated subgroup Γ≤G⁢(ℚ) that contains every nth root of an element of G. Then every element in ℒn⁢(G) is a subgroup of Γ of index at most n[Γ:G(ℤ)]. Since Γ is finitely generated, it has finitely many subgroups of any given index, so ℒn⁢(G) is finite. ∎

As an immediate consequence of the proof of Lemma 3.1, we get the well-known fact that arithmetic lattices in G are contained in G⁢(ℚ).

Lemma 3.2.

If G is in U, then each element of L⁢(G) is a subgroup of G⁢(Q). ∎

Slight modifications of a technical idea in the proof of Lemma 3.1 will be used repeatedly throughout the rest of the paper, so we encapsulate them here. Throughout, for any Δ≤G⁢(ℝ) and n∈ℕ, we write Δ1/n for the group generated by elements g∈G⁢(ℝ) such that gn∈Δ.

Lemma 3.3.

Let G be in U and Γ=G⁢(Z) have nilpotence class c and Hirsch length d.

For any k∈ℕ, [Γ1/k:Γ] is finite.
We have (Γp-k)pk⁢c⁢(c+1)/2≤Γ.
For any prime p and any k∈ℕ, we have [Γp-k:Γ]≤pd⁢k⁢c⁢(c+1)/2.

Proof.

Item 1 follows from [13, Exercise 7, p. 114]. Item 2 follows from [13, Proposition 3, p. 113]. For the final statement, by [13, Exercise 7, p. 114], the group Γp-k is a finitely generated nilpotent group. As a finite extension of Γ, it has Hirsch length at most d, and hence

[Γp-k:(Γp-k)pk⁢c⁢(c+1)/2]≤pd⁢k⁢c⁢(c+1)/2.

Then, by item 2, it follows that we have bounded [Γp-k:Γ] as desired. ∎

3.1 The Euler decomposition

Proposition 3.4.

If G is in U, then

ζG⁢(s)=∏p⁢𝑝𝑟𝑖𝑚𝑒ζG,p⁢(s).

Proof.

Any group in ℒn⁢(G) is a subgroup of G⁢(ℤ)1/n. Hence, applying item 1 of Lemma 3.3, we see that Ω=〈ℒn⁢(G)〉 is finitely generated and contains each element of ℒn⁢(G) as a subgroup of finite index. Hence, Λn′=∩A∈ℒn⁢(G)A is a subgroup of finite index in Ω. Let Λn be the normal core of Λn′ in Ω.

Consider the group Γ:=Ω/Λn. It is a finite nilpotent group, and hence decomposes into a product of its Sylow p-subgroups:

Ω/Λn=∏pSp.

For any image A in Γ of an element Δ∈ℒn⁢(G), we get the decomposition

A=∏pAp,

where Ap are the Sylow p-subgroups of A and Ap≤Sp for every p. A similar statement is true for the image Q of G⁢(ℤ) in Γ; we have

Q=∏p⁢primeQp,where⁢Qp≤Sp.

We compute

Q∩A=∏p⁢prime(Ap∩Qp).

Hence,

c⁡(G⁢(ℤ),Δ)=c⁡(Q,A)=∏p⁢primec⁡(Ap,Qp).

Further, c⁡(Ap,Qp) is the greatest power of p that divides n. Since any such element of ℒn⁢(G) arises from such a decomposed A, we have

𝐜n⁢(ℒ)=∏pk∥n𝐜pk⁢(ℒ),

and hence the Euler decomposition for the commensurability zeta function above holds. ∎

3.2 p-adic formulation and the proof of Theorem 1.3

Let G∈𝒰. Fix a Mal’cev basis (x1,…,xn) for G⁢(ℤ) so that

1<〈x1〉<〈x1,x2〉<⋯<〈x1,…,xn〉=G⁢(ℤ)

is a central series with infinite cyclic successive quotients. Elements in G⁢(ℚp) may be identified with the set of all “p-adic words” of the form

𝐱⁢(𝐚)=x1a1⁢⋯⁢xnan,where⁢𝐚∈ℚpn.

Note that 𝐱 parametrizes G⁢(ℤ) when restricted to 𝐚∈ℤn. Let λ, κ, and μ(i) for i≥2 be polynomials defined over ℚ as in [6, p. 197] for the 𝒯 group G⁢(ℤ) so that, for 𝐚∈ℤn, we have

𝐱⁢(𝐚)k=𝐱⁢(a1,a2,a3,…,an)k=𝐱⁢(λ⁢(𝐚,k)),

𝐱⁢(𝐚1)⁢𝐱⁢(𝐚2)⁢⋯⁢𝐱⁢(𝐚i)=𝐱⁢(μ(i)⁢(𝐚1,𝐚2,…,𝐚i)),

[𝐱⁢(𝐚1),𝐱⁢(𝐚2)]=𝐱⁢(κ⁢(𝐚1,𝐚2)).

We denote the closure of a subset S of G⁢(ℚp) to be S¯ or S-. For any Γ∈ℒ⁢(G), we have that Γ¯ is the pro-p completion of Γ [16, Theorem 4.3.5].

Lemma 3.5.

For each k, the closure map gives a one-to-one correspondence between elements of Lpk⁢(G) and closed subgroups H of G⁢(Qp) with

c⁢(H,G⁢(ℤp))=pk.

Proof.

Let Γ=G⁢(ℤ)p-k. Note that Γ contains every subgroup in ℒpk⁢(G). Moreover, by item 2 of Lemma 3.3, we have that ΓpN≤G⁢(ℤ) for N=k⁢c⁢(c+1)/2, where c is the nilpotence class of G. It follows that any Δ∈ℒpk⁢(G) contains ΓpN+k. Note that ΓpN+k is a normal subgroup of Γ and the quotient group Γ/ΓpN+k is a finite p-group. It follows that, for each Δ∈ℒpk⁢(G), there is some ℓ∈ℕ such that [Γ:Δ]=pℓ.

Let K be the closure of Γ in G⁢(ℚp). Then K is a finitely generated pro-p group. Any closed subgroup H≤G⁢(ℚp) satisfying c⁢(H,G⁢(ℤp))=pk is a finite-index subgroup of G⁢(ℤp)p-k=K. The closure map gives an index-preserving bijection between subgroups of Γ with index a power of p and finite-index subgroups of K. This completes the proof. ∎

Define Gi=〈x1r1,…,xiri:r1,…,ri∈ℚp〉. The following is an extension of the notion of good basis from [6] to the G⁢(ℚp) setting.

Definition 3.6.

Let H be a closed subgroup of G⁢(ℚp). An n-tuple (h1,…,hn) of elements in H is called a good basis if 〈h1,…,hi〉-=H∩Gi for each i=1,…,n.

Lemma 3.7.

Let hi∈Gi∖Gi-1 for i=1,…,n, and let H=〈h1,…,hn〉-. Then the following statements hold.

c⁡(H,G⁢(ℤp))<∞.
(h1,…,hn) is a good basis for H if and only if
(3.1)[hi,hj]∈〈h1,…,hi-1〉- 𝑓𝑜𝑟⁢ 1≤i<j≤n.
Suppose equation (3.1) holds, and let h1′,…,hn′∈H. Then (h1′,…,hn′) is a good basis for H if and only if there exists ri∈ℤp*, wi∈〈h1,…,hi-1〉- such that hi′=wi⁢hiri for 1≤i≤n.

Proof.

Since hi∈Gi∖Gi-1 for i=1,…,n, we have

hi=x1ri⁢1⁢x2ri⁢2⁢⋯⁢xnri⁢n,where⁢ri⁢j∈ℚp.

For any such nonzero ri⁢j, we have ri⁢j=u⁢p-k, where k∈ℤ and u∈ℤp*. If all such k are nonpositive, then we appeal to [6, Lemma 2.1]. Otherwise, let k be the maximal integer that appears in this way. Then

H≤〈x1p-k,x2p-k,…,xnp-k〉-≤G⁢(ℤp)p-k.

By continuity of taking powers, G⁢(ℤp)p-k is the closure of G⁢(ℤ)p-k in G⁢(ℚp). Thus, by applying item 2 of Lemma 3.3, we have that there exists m∈ℕ such that Hm≤G⁢(ℤp). It follows that G⁢(ℤp)∩H must have the same dimension as G, and so item 1 follows. Items 2 and 3 follow from small modifications of the proof of [6, Lemma 2.1]. ∎

Definition 3.8.

For Δ∈ℒ⁢(G), define 𝔘⁢(Δ) to be the collection of pairs (A,B), where A∈Tn⁢(ℚp), B∈Tn⁢(ℤp), such that if ai denotes the ith row of A and bi denotes the ith row of B,

(𝐱⁢(a1),…,𝐱⁢(an)) is a good basis for Δ¯,
(𝐱⁢(b1),…,𝐱⁢(bn)) is a good basis for Δ¯∩G⁢(ℤp).

Combining suitable 𝔘⁢(Δ) into one set, we define

𝔘p:=⋃k=1∞(⋃Δ∈ℒpk⁢(G)𝔘(Δ)).

The next proposition shows that 𝔘p is Lp-defined in the sense of [11] (that is, it can be defined using first-order logic, p-norms and field operations). Note that |x|p≥|y|p can be stated in LP (see the example in [11, p. 71]).

Lemma 3.9.

Let (A,B)∈Tn⁢(Qp)×Tn⁢(Zp). Then (A,B)∈Up if and only if each of the following holds.

det⁡(A)≠0 and the rows a1,…,an of the matrix A satisfy
𝑓𝑜𝑟⁢ 1≤i<j≤n,there exist⁢Yi⁢j1,…,Yi⁢ji-1∈ℤp⁢such thatκ⁢(ai,aj)=μ(i-1)⁢(λ⁢(a1,Yi⁢j1),…,λ⁢(ai-1,Yi⁢ji-1)).
det⁡(B)≠0 and the rows b1,…,bn of the matrix B satisfy
𝑓𝑜𝑟⁢ 1≤i<j≤n,there exist⁢Yi⁢j1,…,Yi⁢ji-1∈ℤp⁢such thatκ⁢(bi,bj)=μ(i-1)⁢(λ⁢(b1,Yi⁢j1),…,λ⁢(bi-1,Yi⁢ji-1)).
Moreover, let 𝐲:ℤpn→G⁢(ℚp) be the function defined by
𝐲(𝐝):=𝐱(a1)d1𝐱(a2)d2⋯𝐱(an)dn.
Then there exists vectors c1,…,ci∈ℤpn such that, for all i=1,…,n,
𝐲⁢(ci)=𝐱⁢(bi).
|det⁡(B)|p is maximal over all B satisfying 2.

Proof.

Let (A,B)∈𝔘⁢(Δ). Then conditions 1, 2 are satisfied by Lemma 3.7. To see condition 3, note that |det(B)|p=[G(ℤ):G(ℤ)∩Δ], while any matrix B′ satisfying condition 2 determines a subgroup of G⁢(ℤp)∩Δ¯ and hence satisfies |det(B′)|p≤[G(ℤ):G(ℤ)∩Δ].

Let (A,B)∈𝔘p. Let H be the closed subgroup of G⁢(ℚp) generated by 𝐱⁢(ai), where ai are the rows or A, and let Δ be the subgroup in ℒpℓ⁢(G) corresponding to H given by Lemma 3.5. The conditions above ensure (A,B)∈𝔘⁢(Δ). ∎

Our parametrization of arithmetic lattices gives a nice formula for the commensurability index.

Lemma 3.10.

Let Δ∈L⁢(G). Then, for any (A,B)∈U⁢(Δ), we have

c⁡(Δ¯,G⁢(ℤp))=|det⁡(B)2⁢det⁡(A)-1|p-1.

Proof.

A direct application of the results of [6, p. 198] shows

[G(ℤp):G(ℤp)∩Δ¯]=|det(B)|p-1.

Let C be the matrix whose ith row ci satisfies 𝐲⁢(ci)=𝐱⁢(bi), where 𝐲 is the function defined in Lemma 3.9. Note that C∈Tn⁢(ℤp). Because Δ¯=𝐲⁢(ℤpn), it follows as above from [6, p. 198] that

[Δ¯:G(ℤp)∩Δ¯]=|det(C)|p-1.

Using the fact that the rows of B are coordinate vectors of a good basis, computation shows that bi=(qi⁢1,…,qi⁢(i-1),ai⁢i⁢ci⁢i,0,…,0), where qi⁢j∈ℚp for all j<i and ai⁢i and ci⁢i are the respective diagonal elements of A and C. It follows that

|det⁡(B)|p-1=|det⁡(A⁢C)|p-1.

The desired result follows. ∎

For n∈ℕ, let ν be the Haar measure on ℚpn normalized so that ν⁢(ℤpn)=1.

Lemma 3.11.

For a matrix M∈Tn⁢(Qp), let f(M):=∏i=1nm11⋯mi⁢i. Let Δ∈L⁢(G). Then U⁢(Δ) is an open subset of Tn⁢(Qp)×Tn⁢(Zp) and

ν⁢(𝔘⁢(Δ))=(1-p-1)2⁢n⁢|f⁢(A)⁢f⁢(B)|p

for any (A,B)∈U⁢(Δ).

Proof.

For Δ∈ℒ⁢(G) and a ring R, let ℳR⁢(Δ) be the set of matrices M in Tn⁢(R) with rows m1,…,mn such that 𝐱⁢(m1),…,𝐱⁢(mn) are a good basis for Δ¯. Notice that 𝔘⁢(Δ)=ℳℚp⁢(Δ)×ℳℤp⁢(G⁢(ℤ)∩Δ). Hence, that 𝔘⁢(Δ) is an open subset of Tn⁢(ℚp)×Tn⁢(ℤp) is a straightforward application of the first part of the proof of [6, Lemma 2.5, p. 198].

Moreover, the second part of the proof of [6, Lemma 2.5, p. 198] and Fubini’s theorem give

∫𝔘⁢(Δ)1⁢d⁢ν=∫ℳℚp⁢(Δ)×ℳℤp⁢(G⁢(ℤ)∩Δ)1⁢d⁢ν=∫ℳℚp⁢(Δ)∫ℳℤp⁢(G⁢(ℤ)∩Δ)1⁢d⁢ν=∫ℳℚpΔ)(1-p-1)n⁢|f⁢(B)|p⁢d⁢ν=(1-p-1)2⁢n⁢|f⁢(A)⁢f⁢(B)|p.∎

Proposition 3.12.

Let f be as in Lemma 3.11. We have

ζG,p⁢(s)=1(1-p-1)2⁢n⁢∫(A,B)∈𝔘p|f⁢(A)⁢f⁢(B)|p-1⁢|det⁡(B)2⁢det⁡(A)-1|ps⁢d⁢ν.

Proof.

By Lemma 3.11, for any (A,B)∈𝔘⁢(Δ), the integrand evaluates to

ν(𝔘(Δ))-1c(Δ,G(ℤ))-s.

By decomposing 𝔘p into disjoint open sets,

𝔘p=⋃k=1∞(⋃Δ∈ℒpk⁢(G)𝔘⁢(Δ)),

using Lemma 3.11, we compute

1(1-p-1)2⁢n⁢∫𝔘p(|f⁢(A)⁢f⁢(B)|p-1⁢|det⁡(B)2⁢det⁡(A)-1|ps)⁢d⁢ν

=∑k=1∞(∑Δ∈ℒpk⁢(G)(∫𝔘⁢(Δ)ν(𝔘(Δ))-1c(Δ,G(ℤ))-sdν))

=∑k=1∞(∑Δ∈ℒpk⁢(G)(∫𝔘⁢(Δ)ν⁢(𝔘⁢(Δ))-1⁢p-k⁢s⁢d⁢ν))

=∑k=1∞(∑Δ∈ℒpk⁢(G)p-k⁢s)=∑k=1∞𝐜pk⁢(ℒ⁢(G))pk⁢s=ζG,p⁢(s).∎

We are now ready to prove our main result.

Proof of Theorem 1.3.

We first show that the local commensurability zeta functions converge for sufficiently large s. Given Δ∈ℒpk⁢(G), for any γ∈Δ, we have γpk∈G⁢(ℤp) so Δ≤Γk:=(G(ℤ))p-k. Then, by item 3 of Lemma 3.3, there exists D, depending only on G, such that |Γk:G(ℤ)|≤pD⁢k. Hence, any Δ∈ℒpk⁢(G) is a subgroup of Γk of index at most p(1+D)⁢k, giving

(3.2)𝐜pk⁢(ℒ⁢(G))≤sp(1+D)⁢k⁢(Γk)≤sp(1+D)⁢k⁢(N),

where sm⁢(G) is the subgroup growth function (the number of subgroups of index at most m) and N is the free nilpotent group of class and rank equal to that of G⁢(ℤ). It follows from [6] that sp(1+D)⁢k⁢(N)≤pα⁢k for some fixed α that does not depend on p. Hence, inequality (3.2) gives that ζG,p⁢(s) is finite for s>M, where M does not depend on p.

The integrand in Proposition 3.12 and 𝔘p are both Lp-definable in the sense of [11] (see also [2]) by Lemma 3.9. Thus, by [11, Theorem 22], we have that ζG,p⁢(s) is a rational function with numerator and denominator degrees bounded by a constant depending only on G. ∎

4 Parting remarks

At present, we are unable to precisely compute closed forms of the zeta functions for unipotent groups of dimension greater than one. Even the two-dimensional case is too difficult for us. However, we can compute an explicit formula for 𝐜n in the two-dimensional case. Here we work globally over ℚ rather than locally as we did in § 3.2.

Consider the case G⁢(ℤ)=ℤ2. Let 𝔙n be the set of pairs of nonnegative matrices

(A,C)=((a11a120a22),(c11c120c22)),

where ai⁢j∈ℚ, ci⁢j∈ℤ, 0≤a12<a22 and c12<c22, satisfying the following two properties:

det⁡(C2)⁢det⁡(A)=n,
if D∈GL2⁡(ℚ) satisfies D⁢A∈GL2⁡(ℤ), then |det⁡(D)|≥det⁡(C).

Then 𝐜n⁢(ℒ⁢(G))=|𝔙n|. The bijection is defined as follows: Suppose Δ≤G⁢(ℚ) is a lattice with c⁢(ℤ2,Δ)=n. The rows of A give an ordered generating set S for Δ≤ℚ2. The rows of C give the coordinates of a generating set of Δ∩ℤ2 with respect to S.

We can simplify this expression. Define D:ℚ→ℕ to be the denominator of a rational number in reduced form. Eliminating the ci⁢j’s gives the formula

𝐜n(ℒ(G))=|{(a11,a12,a22)∈ℚ≥03:a12<a22,a11a22(lcm(D(a12/a22),D(a11))D(a22))2=n}|.

We do not have such a simple expression for the Heisenberg group. Computing nice expressions for these examples would, at the very least, give an indication of whether the results in [3, 14] hold for commensurability zeta functions.

Communicated by John S. Wilson

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1405609

Award Identifier / Grant number: DMS-1547292

Funding statement: K. Bou-Rabee supported in part by NSF grant DMS-1405609. D. Studenmund supported in part by NSF grant DMS-1547292.

Acknowledgements

We are grateful to Benson Farb, Tasho Kaletha, Michael Larsen, and Andrew Putman for their conversations and support. In particular, Benson Farb provided helpful comments on an early draft. We are very grateful to an anonymous referee for useful comments and corrections on an earlier draft.

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Received: 2019-08-08

Revised: 2019-10-01

Published Online: 2019-11-08

Published in Print: 2020-03-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2019-0112