On arithmetic properties of solvable Baumslag–Solitar groups

Laurent Hayez; Tom Kaiser; Alain Valette

doi:10.1515/jgth-2021-0069

Article Publicly Available

On arithmetic properties of solvable Baumslag–Solitar groups

Laurent Hayez , Tom Kaiser and Alain Valette

Published/Copyright: November 15, 2022

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From the journal Journal of Group Theory Volume 26 Issue 3

Abstract

For 0 < α ≤ 1 , we say that a sequence ( X k ) k > 0 of 𝑑-regular connected graphs has property D α if there exists a constant C > 0 such that diam ⁡ ( X k ) ≥ C ⋅ | X k | α . We investigate property D α for arithmetic box spaces of the solvable Baumslag–Solitar groups BS ⁡ ( 1 , m ) (with m ≥ 2 ): these are box spaces obtained by embedding BS ⁡ ( 1 , m ) into the upper triangular matrices in GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) and intersecting with a family M N k of congruence subgroups of GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) , where the levels N k are coprime with 𝑚 and N k ∣ N k + 1 . We prove that

if an arithmetic box space has D α , then α ≤ 1 2 ;
if the family ( N k ) k of levels is supported on finitely many primes, the corresponding arithmetic box space has D 1 / 2 ;
if the family ( N k ) k of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have D α for every α > 0 .

Moreover, we prove that if we embed BS ⁡ ( 1 , m ) in the group of invertible upper-triangular matrices T n ⁢ ( Z ⁢ [ 1 / m ] ) , then every finite index subgroup of the embedding contains a congruence subgroup. This is a version of the congruence subgroup property (CSP).

1 Introduction

Let 𝐺 be a finitely generated, residually finite group. If ( H k ) k > 0 is a decreasing sequence of finite index normal subgroups of 𝐺, with trivial intersection, and 𝑆 is a finite generating set of 𝐺, then the box space □ ( H k ) ⁢ G is the disjoint union of finite Cayley graphs

□ ( H k ) ⁢ G = ∐ k > 0 Cay ⁡ ( G / H k , S ) ;

here, by abuse of notation, we identify 𝑆 with its image in G / H k . Changing the generating set 𝑆 does not change the coarse geometry of the box space^[1], so we omit 𝑆 from the notation.

In the dictionary between group-theoretical properties of 𝐺 and metric properties of □ ( H k ) ⁢ G (see e.g. [6]), it is natural to look at the behaviour of the diameter of the Cayley graphs Cay ⁡ ( G / H k , S ) . Let 0 < α ≤ 1 . The box space □ ( H k ) ⁢ G satisfies property D α if there is some constant C > 0 such that, for every k > 0 ,

(1.1) diam ⁡ ( Cay ⁡ ( G / H k , S ) ) ≥ C ⁢ | G / H k | α .

Note that property D α is a coarse geometry invariant of the box space. The following is known.

Theorem 1.1

Let 𝐺 be a finitely generated, residually finite group.

(See [1, Corollary 1.7 and Lemma 5.1].) If some box space of 𝐺 has property D α for some α > 0 , then 𝐺 virtually maps onto ℤ.
(See [7, Theorem 3].) If 𝐺 maps onto ℤ, then for every 0 < α < 1 , there exists a box space of 𝐺 with property D α .
(See [7, Proposition 5].) The group 𝐺 is virtually cyclic if and only if some (hence any) box space of 𝐺 has property D 1 .∎

This paper considers the Baumslag–Solitar groups BS ⁡ ( n , m ) ( m , n > 0 ) with presentation

BS ⁡ ( n , m ) = ⟨ a , t ∣ t ⁢ a n ⁢ t - 1 = t m ⟩ .

They all map onto ℤ, by a ↦ 0 , t ↦ 1 . On the other hand, they are known to be residually finite if and only if n = 1 or n = m ; see [10, Theorem C]. It turns out that the solvable Baumslag–Solitar groups BS ⁡ ( 1 , m ) , with m ≥ 2 , have interesting box spaces. Indeed it is well known that BS ⁡ ( 1 , m ) may be viewed as a semi-direct product

BS ⁡ ( 1 , m ) = Z ⁢ [ 1 / m ] ⋊ Z ,

where the factor ℤ corresponds to the subgroup ⟨ t ⟩ acting by powers of 𝑚. We may identify this semi-direct product with the following subgroup G m of upper triangular matrices in GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) :

G m = { ( m k r 0 1 ) : k ∈ Z , r ∈ Z ⁢ [ 1 / m ] } .

The isomorphism is obtained by mapping 𝑎 to A = ( 1 1 0 1 ) and 𝑡 to T = ( m 0 0 1 ) . The associated embedding of BS ⁡ ( 1 , m ) into GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) is called the standard embedding.

In GL n ⁢ ( Z ⁢ [ 1 / m ] ) , we may define congruence subgroups. Let N > 0 be coprime with 𝑚. The principal congruence subgroup of level 𝑁 is the kernel M N of the reduction modulo 𝑁,

M N = ker ⁡ [ GL n ⁢ ( Z ⁢ [ 1 / m ] ) → GL n ⁢ ( Z / N ⁢ Z ) ] .

Definition 1.2

If 𝐺 is any subgroup of GL n ⁢ ( Z ⁢ [ 1 / m ] ) , and N > 0 is coprime to 𝑚, then the congruence subgroup G ⁢ ( N ) in 𝐺 is

G ⁢ ( N ) := G ∩ M N .

For a sequence of integers in which each divides the next, one obtains a sequence of nested congruence subgroups and thus a box space of BS ⁡ ( 1 , m ) . Such box spaces deserve to be called arithmetic box spaces. We will study property D α for the arithmetic box spaces of BS ⁡ ( 1 , m ) through the standard embedding. From Theorem 1.1, we know that, for every 0 < α < 1 , there exists a box space of BS ⁡ ( 1 , m ) with property D α , but what about arithmetic box spaces? We will prove that box spaces with D α , for α > 1 2 , can be distinguished from arithmetic box spaces by coarse geometry. More precisely, we have the following result.

Theorem 1.3

For any m ≥ 2 , the following statements are true:

if an arithmetic box space □ ( G m ⁢ ( N k ) ) k ⁢ G m has property D α , then α ≤ 1 2 ;
there exists an arithmetic box space with property D 1 / 2 ;
there exists an arithmetic box space of G m without D α for any α ∈ ] 0 , 1 2 ] .

If □ ( H k ) ⁢ G , □ ( H k ′ ) ⁢ G are two box spaces of the same residually finite group 𝐺, we say that □ ( H k ′ ) ⁢ G covers □ ( H k ) ⁢ G if H k ′ ⊂ H k for every k > 0 . In this case, Cay ⁡ ( G / H k ′ , S ) is a Galois covering of Cay ⁡ ( G / H k , S ) .

The following proposition bridges Theorem 1.3 with part (2) of Theorem 1.1.

Proposition 1.4

Fix α < 1 . Any arithmetic box space of G m is covered by some box space with D α .

In addition, we study how property D α for an arithmetic box space depends on the prime factors of the 𝑁’s in the sequence of congruence subgroups ( M N ) N . In fact, if we denote by D ′ ⁢ ( P ) the analytic primitive density of the prime factors (see Section 3.2), we prove the following.

Theorem 1.5

Let □ ( G m ⁢ ( N k ) ) k ⁢ G m be an arithmetic box space, and let 𝑃 be the set of prime factors of the sequence ( N k ) k .

If | P | < + ∞ , then □ ( G m ⁢ ( N k ) ) k ⁢ G m has D 1 / 2 .
If D ′ ⁢ ( P ) > 0 , then □ ( G m ⁢ ( N k ) ) k ⁢ G m does not have D 1 / 2 .

Of course the choice of the standard embedding raises a natural question: how do the congruence subgroups in BS ⁡ ( 1 , m ) depend on the choice of the embedding 𝜌? Since congruence subgroups have finite index, this is related to a form of the congruence subgroup property (CSP), which we now recall.

Definition 1.6

Let 𝐺 be a subgroup of GL n ⁢ ( Z ⁢ [ 1 / m ] ) . We say that 𝐺 has the CSP if every finite index subgroup in 𝐺 contains G ⁢ ( N ) for some N > 0 .

Let T n ⁢ ( Z ⁢ [ 1 / m ] ) denote the group of upper triangular matrices in GL n ⁢ ( Z ⁢ [ 1 / m ] ) . We shall prove the following theorem.

Theorem 1.7

For every embedding

ρ : BS ⁡ ( 1 , m ) → T n ⁢ ( Z ⁢ [ 1 / m ] ) ,

the group ρ ⁢ ( BS ⁡ ( 1 , m ) ) has the CSP.

This result can be reformulated as follows. Taking the congruence subgroups ρ ⁢ ( G m ) ⁢ ( N ) as a neighbourhood basis for the identity gives a topology on ρ ⁢ ( G m ) (namely the congruence topology relative to 𝜌), and the completion ρ ⁢ ( G m ) ¯ with respect to this topology is a profinite group called the congruence completion of ρ ⁢ ( G m ) . As G m is residually finite, it also embeds in its profinite completion G m ^ which maps onto ρ ⁢ ( G m ) ¯ . The CSP for ρ ⁢ ( G m ) says that the map G m ^ → ρ ⁢ ( G m ) ¯ is an isomorphism.

Another immediate consequence of CSP is that if ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] ) is an embedding, then any box space of G m ≃ ρ ⁢ ( G m ) is covered by some arithmetic box space of ρ ⁢ ( G m ) .

When m = p ℓ is a prime power, we also propose another approach to CSP for BS ⁡ ( 1 , m ) : we view BS ⁡ ( 1 , p ℓ ) as a subgroup of the affine group G ⁢ ( Q ) , and this subgroup is commensurable to the group G ⁢ ( O S ) of 𝑆-integer points, with S = { p , ∞ } . We prove the following proposition.

Proposition 1.8

Let ℍ be a ℚ-algebraic subgroup of GL n , and let ρ : G → H be a ℚ-isomorphism. Then ρ ⁢ ( BS ⁡ ( 1 , p ℓ ) ) has CSP.

As for the structure of the paper, we will start in Section 2 by recalling some well-known facts about elementary number theory, especially concerning the multiplicative order of 𝑚 in Z / N ⁢ Z .

We then study property D α for solvable Baumslag–Solitar groups in Section 3. We begin with some metric aspects, where we estimate the diameter of the arithmetic box spaces of BS ⁡ ( 1 , m ) , which is what we use to prove Theorem 1.3 and Proposition 1.4 (see Theorem 3.6). To conclude Section 3, we investigate the role of prime numbers in property D α , and we prove Theorem 1.5 (see Theorem 3.8).

Finally, in Section 4, we prove Theorem 1.7.

Remark

We will use the Bachmann–Landau notations O , o , Ω , Θ for the asymptotics, as defined by Knuth [8]. Let f , g be real-valued functions.

O ⁢ ( g ) denotes the set of all 𝑓 such that there exist C , N 0 > 0 such that
| f ⁢ ( x ) | ≤ C ⁢ g ⁢ ( x ) for all ⁢ x ≥ N 0 .
In some computations, we will write f = O ⁢ ( g ) as a shortcut, with the understanding that, for 𝑥 large enough, 𝑓 is of order at most 𝑔 and that one could do the same computations with C ⋅ g for 𝑥 large enough.
o ⁢ ( g ) denotes the set of all 𝑓 such that lim x → ∞ ⁡ f ⁢ ( x ) g ⁢ ( x ) = 0 . As above, we will write f = o ⁢ ( g ) .
Ω ⁢ ( g ) denotes the set of all 𝑓 such that there exist C , N 0 > 0 such that
f ⁢ ( x ) ≥ C ⁢ g ⁢ ( x ) for all ⁢ x ≥ N 0 .
Again, we write f = Ω ⁢ ( g ) .
Θ ⁢ ( g ) denotes the set of all 𝑓 such that there exist C , C ′ , N 0 > 0 such that
C ⁢ g ⁢ ( x ) ≤ f ⁢ ( x ) ≤ C ′ ⁢ g ⁢ ( x ) for all ⁢ x ≥ N 0 .
We also use the notation f = Θ ⁢ ( g ) .

2 Some elementary number theory

We gather some technical lemmas. First we recall some facts about greatest common divisors and lowest common multiples, which will be respectively denoted by gcd and lcm .

Proposition 2.1

Let μ , a 1 , … , a n ∈ N .

gcd ⁡ ( μ ⋅ a 1 , μ ⋅ a 2 ) = μ ⋅ gcd ⁡ ( a 1 , a 2 ) ;
gcd ⁡ ( a 1 , … , a n ) = gcd ⁡ ( gcd ⁡ ( a 1 , … , a n - 1 ) , a n ) ;
gcd ⁡ ( a 1 , gcd ⁡ ( a 2 , a 3 ) ) = gcd ⁡ ( gcd ⁡ ( a 1 , a 2 ) , a 3 ) ;
gcd ⁡ ( a 1 , a 2 ) ⋅ lcm ⁡ ( a 1 , a 2 ) = a 1 ⋅ a 2 ;
lcm ⁡ ( a 1 , … , a n ) = lcm ⁡ ( lcm ⁡ ( a 1 , … , a n - 1 ) , a n ) .∎

The following lemma generalises Proposition 2.1 4.

Lemma 2.2

Let a 1 , … , a n ∈ N ; then

lcm ⁡ ( a 1 , … , a n ) = a 1 ⁢ ⋯ ⁢ a n gcd ⁡ ( a 1 ⁢ ⋯ ⁢ a n - 1 , a 1 ⁢ ⋯ ⁢ a n - 2 ⁢ a n , … , a 2 ⁢ ⋯ ⁢ a n ) .

Proof

Decomposing the a i ’s into a product of primes using the fundamental theorem of arithmetic, the result is straightforward from the definition of the lcm and the gcd . ∎

We now recall the ideal structure of the ring Z ⁢ [ 1 / m ] .

Lemma 2.3

Let 𝐼 be a proper subgroup in Z ⁢ [ 1 / m ] . The following are equivalent:

𝐼 is an ideal in Z ⁢ [ 1 / m ] ;
there exists N > 1 such that I = N ⁢ Z ⁢ [ 1 / m ] .

Moreover, Z ⁢ [ 1 / m ] / N ⁢ Z ⁢ [ 1 / m ] ≃ Z / N ⁢ Z

Proof

The non-trivial direction follows from general results about localisations of rings, viewing Z ⁢ [ 1 / m ] as the localisation of ℤ with respect to powers of 𝑚: the map I → I ∩ Z provides a bijection between ideals of Z ⁢ [ 1 / m ] and ideals 𝐽 in ℤ such that 𝑚 is not a zero-divisor in Z / J (see [4, Proposition 2 in Section 11.3]). Finally, observing that Z + N ⁢ Z ⁢ [ 1 / m ] = Z ⁢ [ 1 / m ] , we have, by a classical isomorphism theorem,

Z ⁢ [ 1 / m ] / N ⁢ Z ⁢ [ 1 / m ] = ( Z + N ⁢ Z ⁢ [ 1 / m ] ) / N ⁢ Z ⁢ [ 1 / m ] ≃ Z / ( Z ∩ N ⁢ Z ⁢ [ 1 / m ] ) = Z / N ⁢ Z . ∎

Lemma 2.3 allows us to work with Z / N ⁢ Z , which has a familiar ring structure. We will write Z / N ⁢ Z × for the multiplicative group of Z / N ⁢ Z . We denote by ord m ⁡ ( N ) the multiplicative order of 𝑚 in Z / N ⁢ Z × . We define the following function.

Definition 2.4

Let m , N ∈ N be such that gcd ⁡ ( m , N ) = 1 . Write

m ord m ⁡ ( N ) = μ ⁢ N + 1 for some ⁢ μ ∈ N ,

and let the function η N : N → N be defined by

η N ⁢ ( k ) = { 1 if ⁢ k = 1 , N k - 1 gcd ⁡ ( μ , N ) if ⁢ k ≥ 2 .

Lemma 2.5

Let m , N ∈ N be such that gcd ⁡ ( m , N ) = 1 . Write

m ord m ⁡ ( N ) = μ ⁢ N + 1 for some ⁢ μ ∈ N .

Then ord m ⁡ ( N 2 ) = ord m ⁡ ( N ) ⋅ N gcd ⁡ ( μ , N ) , and more generally,

ord m ⁡ ( N k ) = ord m ⁡ ( N ) ⋅ η N ⁢ ( k ) for all ⁢ k ≥ 1 .

Proof

The case k = 1 being obvious, we consider k = 2 and set β = ord m ⁡ ( N ) . The smallest positive integer 𝜆 that satisfies m λ ≡ 1 ( mod N 2 ) is λ = β ⁢ N gcd ⁡ ( μ , N ) . Indeed, β ∣ λ ; thus λ = β ⁢ λ ~ for some λ ~ ∈ N . From the fact that

( m β ) λ ~ ≡ 1 + λ ~ ⁢ μ ⁢ N ( mod N 2 ) ,

we see that λ ~ = N gcd ⁡ ( μ , N ) .

The same arguments can be applied to show that ord m ⁡ ( N k ) = β ⁢ N k - 1 gcd ⁡ ( μ , N ) for k ≥ 2 . ∎

Lemma 2.6

Let k , N ∈ N . Then η N ⁢ ( k ) ≥ N k - 2 .

Proof

If k = 1 , then 1 ≥ N - 1 . If k ≥ 2 , observe that gcd ⁡ ( μ , N ) ≤ N implies

N k - 1 gcd ⁡ ( μ , N ) ≥ N k - 2 . ∎

Denote by P ⊂ N the set of prime numbers. The following lemma gives us a formula to compute the order of 𝑚 in Z / N ⁢ Z for any N ∈ N . It is an immediate consequence of the Chinese remainder theorem.

Lemma 2.7

For every N ∈ N , which we write N = p 1 β 1 ⁢ p 2 β 2 ⁢ ⋯ ⁢ p n β n with p i ∈ P and β i ∈ N for every i ∈ { 1 , … , n } , we have

ord m ⁡ ( N ) = ord m ⁡ ( p 1 β 1 ⁢ p 2 β 2 ⁢ ⋯ ⁢ p n β n ) = lcm ⁡ ( ord m ⁡ ( p 1 β 1 ) , ord m ⁡ ( p 2 β 2 ) , … , ord m ⁡ ( p n β n ) ) .

Lemma 2.8

Let 𝑃 be a finite set of primes, not dividing 𝑚. There exists a constant C ⁢ ( m , P ) > 0 such that, for every integer 𝑁 with all prime factors in 𝑃, we have

ord m ⁡ ( N ) N ≥ C ⁢ ( m , P ) .

Proof

Write N = p 1 β 1 ⁢ ⋯ ⁢ p k β k , with p i ∈ P , all different, β i > 0 and η p i defined as in Definition 2.4. In addition, we define the set

Π k := { ord m ( p 1 ) η p 1 ( β 1 ) ⋯ ord m ( p k - 1 ) η p k - 1 ( β k - 1 ) , ord m ⁡ ( p 1 ) ⁢ η p 1 ⁢ ( β 1 ) ⁢ ⋯ ⁢ ord m ⁡ ( p k - 2 ) ⁢ η p k - 2 ⁢ ( β k - 2 ) ⁢ ord m ⁡ ( p k ) ⁢ η p k ⁢ ( β k ) , ⋮ ord m ( p 2 ) η p 2 ( β 2 ) ⋯ ord m ( p k ) η p k ( β k ) } ,

containing all possible products with k - 1 factors, each one of ord m ⁡ ( p i ) ⁢ η p i ⁢ ( β i ) , i = 1 , … , k . Using Lemmas 2.2, 2.5 and 2.7, we obtain

ord m ⁡ ( N ) = lcm ⁡ ( ord m ⁡ ( p 1 β 1 ) , ord m ⁡ ( p 2 β 2 ) , … , ord m ⁡ ( p k β k ) ) = ord m ⁡ ( p 1 ) ⁢ η p 1 ⁢ ( β 1 ) ⁢ ⋯ ⁢ ord m ⁡ ( p k ) ⁢ η p k ⁢ ( β k ) gcd ⁡ ( Π k ) ;

thus

(2.1) ord m ⁡ ( N ) N = ord m ⁡ ( p 1 ) ⁢ η p 1 ⁢ ( β 1 ) ⁢ ⋯ ⁢ ord m ⁡ ( p k ) ⁢ η p k ⁢ ( β k ) N ⋅ gcd ⁡ ( Π k ) .

Moreover, ord m ⁡ ( p i ) ≥ 1 for every 𝑖, and using Lemma 2.6 on each η p i ⁢ ( β i ) , we obtain, from equation (2.1),

ord m ⁡ ( N ) N ≥ p 1 β 1 - 2 ⁢ ⋯ ⁢ p k β k - 2 p 1 β 1 ⁢ ⋯ ⁢ p k β k ⋅ gcd ⁡ ( Π k ) = 1 p 1 2 ⁢ ⋯ ⁢ p k 2 ⋅ gcd ⁡ ( Π k ) > 0 .

So we may take C ⁢ ( m , P ) as the minimum of the 1 p 1 2 ⁢ ⋯ ⁢ p k 2 ⋅ gcd ⁡ ( Π k ) ’s taken over all subsets { p 1 , … , p k } of 𝑃. ∎

The following material will be used in the proof of Proposition 4.10.

Definition 2.9

Let N ∈ N , and let N = ∏ i = 1 r p i β i be its decomposition in prime factors. The dominant prime of 𝑁 is the factor p I in 𝑃 such that p I β I ≥ p i β i for all 𝑖. The factor p I β I is called the dominating factor.

Lemma 2.10

Fix s ∈ ( Z ⁢ [ 1 / m ] ) × , with s > 0 . There exist infinitely many integers N > 0 , coprime to 𝑚, such that 𝑠 has odd order in the multiplicative group ( Z ⁢ [ 1 / m ] / N ⁢ Z ⁢ [ 1 / m ] ) × ≃ ( Z / N ⁢ Z ) × .

Proof

If s = 1 , take any 𝑁 coprime with 𝑚. So we assume s ≠ 1 , and replacing 𝑠 by s - 1 if necessary, we assume s > 1 . Let 𝑃 be the set of primes dividing 𝑚, and set r = | P | . Write s = a 1 a 2 with a 1 > a 2 > 0 , coprime, and all their prime factors in 𝑃. For k ∈ N , set a 1 k - a 2 k = N k ⁢ q k , where N k is the maximal factor coprime to 𝑚. Observe that if a prime in 𝑃 divides q k , it cannot simultaneously divide a 1 and a 2 since they are coprime.

It is clear that, for odd 𝑘, 𝑠 will also have odd order in

( Z ⁢ [ 1 / m ] / ( a 1 k - a 2 k ) ⁢ Z ⁢ [ 1 / m ] ) × ,

hence also in ( Z ⁢ [ 1 / m ] / N k ⁢ Z ⁢ [ 1 / m ] ) × . It is therefore enough to prove that the map k ↦ N k takes infinitely many values on odd integers. This will follow immediately from the following.

Claim

There is an infinite family of odd integers 𝑘 such that q k ≤ ( a 1 2 ⁢ r - a 2 2 ⁢ r ) r .

To prove the claim, take 𝑘 odd with k > 2 ⁢ r . By the pigeonhole principle, there are i , j ∈ { 0 , 1 , … , r } with i < j such that the dominant primes in q k - 2 ⁢ i and q k - 2 ⁢ j are the same. We have that

( s k - 2 ⁢ i - 1 ) - ( s k - 2 ⁢ j - 1 ) = s k - 2 ⁢ j ⁢ ( s 2 ⁢ ( j - i ) - 1 ) = a 1 k - 2 ⁢ j a 2 k - 2 ⁢ i ⁢ ( a 1 2 ⁢ ( j - i ) - a 2 2 ⁢ ( j - i ) ) .

Write p i β i for the dominating factor of q k - 2 ⁢ i . Set β = min ⁡ { β i , β j } and p = p i ; say that β = β i (otherwise, replace 𝑖 by 𝑗). We see that p β divides the numerator on the left; hence it divides the numerator on the right. We note that 𝑝 does neither divide a 1 nor a 2 ; hence it must divide a 1 2 ⁢ ( j - i ) - a 2 2 ⁢ ( j - i ) , which is bounded by ( a 1 2 ⁢ r - a 2 2 ⁢ r ) . So p β ≤ ( a 1 2 ⁢ r - a 2 2 ⁢ r ) , and it follows that q k - 2 ⁢ i ≤ ( a 1 2 ⁢ r - a 2 2 ⁢ r ) r . ∎

3 Property D α for solvable Baumslag–Solitar groups

3.1 Metric aspects of solvable Baumslag–Solitar groups

We study the diameter of arithmetic box spaces of BS ⁡ ( 1 , m ) according to equation (1.1). In this section, we will always assume that gcd ⁡ ( m , N ) = 1 . We recall that every element of BS ⁡ ( 1 , m ) ( m > 1 ) admits a unique normal form of the type t - i ⁢ a ℓ ⁢ t j with i , j ≥ 0 , ℓ ∈ Z , and ℓ can be a multiple of 𝑚 only if either 𝑖 or 𝑗 is zero. Indeed, one can rewrite t ⁢ a as a m ⁢ t , t ⁢ a - 1 as a - m ⁢ t , a ⁢ t - 1 as t - 1 ⁢ a m , and a - 1 ⁢ t - 1 as t - 1 ⁢ a - m , and the result follows.

The normal form of a word is usually not the geodesic form, and we want to estimate how well the normal form approximates the geodesic form.

Proposition 3.1

Proposition 3.1 ([2, Proposition 2.1])

There exist constants C 1 , C 2 , D 1 , D 2 > 0 such that, for any ω = t - i ⁢ a ℓ ⁢ t j ∈ BS ⁡ ( 1 , m ) with ℓ ≠ 0 , we have

C 1 ⁢ ( i + j + log ⁡ | ℓ | ) - D 1 ≤ ∥ ω ∥ ≤ C 2 ⁢ ( i + j + log ⁡ | ℓ | ) + D 2 ,

where ∥ ⋅ ∥ is the word metric with respect to { a ± 1 , t ± 1 } . Moreover, we may take C 2 = D 2 = m .

Let A = ( 1 1 0 1 ) , T = ( m 0 0 1 ) ∈ GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) , and denote by G m the subgroup of GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) generated by 𝐴 and 𝑇. In the previous section, we saw that BS ⁡ ( 1 , m ) ≃ G m ≤ GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) .

As mentioned before, BS ⁡ ( 1 , m ) ≃ G m is a finitely generated, residually finite group that surjects onto ℤ so that Theorem 1.1 applies, and we know that, for every 0 < α < 1 , there exists a box space of G m with property D α . However, we are interested in specific box spaces of G m , namely the arithmetic box spaces, or in other words, box spaces of the form □ ( G m ⁢ ( N k ) ) k ⁢ G m . To this end, we start by studying the quotients G m / G m ⁢ ( N ) and then explore how the diameters evolve.

Proposition 3.2

Let N ∈ N be such that gcd ⁡ ( m , N ) = 1 . Then

G m / G m ⁢ ( N ) ≃ Z / N ⁢ Z ⋊ m Z / ord m ⁡ ( N ) ⁢ Z , | G m / G m ⁢ ( N ) | = N ⋅ ord m ⁡ ( N ) ,

where Z / ord m ⁡ ( N ) ⁢ Z acts on Z / N ⁢ Z by multiplication by 𝑚.

Proof

Consider reduction modulo 𝑁,

φ : G m → GL 2 ⁢ ( Z / N ⁢ Z ) , w = ( m k x 0 1 ) ↦ ( [ m k ] [ x ] [ 0 ] [ 1 ] ) .

The image of 𝜑 is clearly isomorphic to Z / N ⁢ Z ⋊ m Z / ord m ⁡ ( N ) ⁢ Z and of order N ⋅ ord m ⁡ ( N ) . Moreover, we have

w = ( m k x 0 1 ) ∈ G m ⁢ ( N ) ⇔ m k ≡ 1 ( mod N ) and x ∈ N Z [ 1 / m ] ⇔ φ ⁢ ( w ) = 1 .

So ker ⁡ ( φ ) = G m ⁢ ( N ) , and the result follows from the first isomorphism theorem. ∎

Example 3.3

Consider BS ⁡ ( 1 , 2 ) and N = 5 . Then ord 2 ⁡ ( 5 ) = 4 , and

G 2 / G 2 ⁢ ( 5 ) ≃ Z / 5 ⁢ Z ⋊ 2 Z / 4 ⁢ Z .

The Cayley graph of the quotient is the graph drawn below, using a = ( 1 , 0 ) and t = ( 0 , 1 ) as generators. Note that one still needs to identify the bottom line with the upper line, and the line to the left with the line to the right.

Thanks to the familiar structure of the quotient G m / G m ⁢ ( N ) and [2, Proposition 2.1], we are able to estimate the diameter of arithmetic box spaces of BS ⁡ ( 1 , m ) .

Lemma 3.4

Let N ≥ 2 . Then

diam ⁡ ( Cay ⁡ ( G m / G m ⁢ ( N ) ) ) = Θ ⁢ ( ord m ⁡ ( N ) ) .

More precisely, there exists a constant C m > 0 such that

1 3 ⋅ ord m ⁡ ( N ) ≤ diam ⁡ ( Cay ⁡ ( G m / G m ⁢ ( N ) ) ) ≤ C m ⋅ ord m ⁡ ( N ) .

Proof

Let m ≥ 2 , and consider BS ⁡ ( 1 , m ) ≃ G m ⊂ GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) . Recall that

G m / G m ⁢ ( N ) ≃ Z / N ⁢ Z ⋊ m Z / ord m ⁡ ( N ) ⁢ Z

so that

diam ⁡ ( G m / G m ⁢ ( N ) ) = diam ⁡ ( Z / N ⁢ Z ⋊ m Z / ord m ⁡ ( N ) ⁢ Z ) ≥ diam ⁡ ( Z / ord m ⁡ ( N ) ⁢ Z ) .

Since the Cayley graph of Z / ord m ⁡ ( N ) ⁢ Z is a cycle, we can roughly estimate the diameter to obtain

diam ⁡ ( G m / G m ⁢ ( N ) ) ≥ 1 3 ⁢ ord m ⁡ ( N ) .

For the second inequality, let ( [ x ] , [ k ] ) ∈ Z / N ⁢ Z ⋊ m Z / ord m ⁡ ( N ) ⁢ Z be an element realising the diameter. We rewrite ( [ x ] , [ k ] ) as ( m k x 0 1 ) ⁢ G m ⁢ ( N ) . The induced metrics are always smaller in a quotient; thus

∥ ( m k x 0 1 ) ⁢ G m ⁢ ( N ) ∥ G m / G m ⁢ ( N ) ≤ ∥ ( m k x 0 1 ) ∥ G m .

Recall that any word ω ∈ BS ⁡ ( 1 , m ) can be written in normal form as ω = t - i ⁢ a ℓ ⁢ t j . In the quotient G m / G m ⁢ ( N ) , the situation is even simpler as, by the semi-direct product structure, every word can be written as

A ℓ ⁢ T j with ⁢ 0 ≤ ℓ < N ⁢ and ⁢ 0 ≤ j < ord m ⁡ ( N ) .

With ℓ = x and j = k , we identify ( m k x 0 1 ) with the element A x ⁢ T k in normal form in G m . If x = 0 , we get T k and

∥ T k ∥ G m = k < ord m ⁡ ( N ) .

Assume that x ≠ 0 . From Proposition 3.1, we obtain

(3.1) ∥ ( m k x 0 1 ) ∥ G m = ∥ A x ⁢ T k ∥ G m ≤ 2 ⁢ m ⁢ ( k + log ⁡ x + 1 ) .

Note that, since m ord m ⁡ ( N ) ≥ N (equivalently log ⁡ ( N ) ≤ ord m ⁡ ( N ) ⋅ log ⁡ ( m ) ) and moreover log ⁡ x ≤ log ⁡ ( N ) , equation (3.1) becomes

∥ ( m k x 0 1 ) ∥ G m ≤ 2 ⁢ m ⁢ ( 2 + log ⁡ ( m ) ) ⁢ ord m ⁡ ( N ) .

Setting C m := 2 ⁢ m ⁢ ( 2 + log ⁡ ( m ) ) , we obtain

diam ⁡ ( Cay ⁡ ( G m / G m ⁢ ( N ) ) ) ≤ C m ⋅ ord m ⁡ ( N ) . ∎

Proposition 3.5

An arithmetic box space □ ( G m ⁢ ( N k ) ) k ⁢ G m has property D α if and only if

ord m ⁡ ( N k ) = Ω ⁢ ( N k α 1 - α ) .

Proof

Using Lemma 3.4,

□ ( G m ⁢ ( N k ) ) k ⁢ G m ⁢ has ⁢ D α ⇔ diam ⁡ ( G m / G m ⁢ ( N k ) ) = Ω ⁢ ( | G m / G m ⁢ ( N k ) | α ) ⇔ ord m ( N k ) = Ω ( N k α ⋅ ord m ( N k ) α ) ⇔ ord m ⁡ ( N k ) = Ω ⁢ ( N k α 1 - α ) . ∎

We present here the main structure theorem for the arithmetic box spaces of BS ⁡ ( 1 , m ) .

Theorem 3.6

For any m ≥ 2 , the following statements hold:

if an arithmetic box space □ ( G m ⁢ ( N k ) ) k ⁢ G m has property D α , then α ≤ 1 2 ;
there exists an arithmetic box space with property D 1 / 2 ;
there exists an arithmetic box space of G m without D α for any α ∈ ] 0 , 1 2 ] ;
fix α < 1 ; every arithmetic box space of G m is covered by some box space with D α .

Proof

(1) If □ ( G m ⁢ ( N k ) ) k ⁢ G m has property D α , using N k ≥ ord m ⁡ ( N k ) and Proposition 3.5, we get

N k = Ω ⁢ ( N k α 1 - α ) ,

which forces α ≤ 1 2 .

(2) Let ( N k ) k ⊂ N be the sequence defined by N k = ( m 2 - 1 ) k . Clearly, it holds N k ∣ N k + 1 for every 𝑘. We apply Lemma 2.5 with N = m 2 - 1 so that ord m ⁡ ( m 2 - 1 ) = 2 and μ = 1 . We thus obtain

ord m ⁡ ( N k ) = 2 ⋅ ( m 2 - 1 ) k - 1 for all ⁢ k ≥ 1 ,

i.e. ord m ⁡ ( N k ) = Ω ⁢ ( N k ) . By Proposition 3.5, the box space □ ( G m ⁢ ( N k ) ) k ⁢ G m has property D 1 / 2 .

(3) We consider the sequence ( N k ) k defined by N k = m 2 k - 1 , and we prove that the arithmetic box space □ ( G m ⁢ ( N k ) ) k ⁢ G m does not have property D α for any α ∈ ] 0 , 1 2 ] . It is straightforward that N k ∣ N k + 1 for every 𝑘, and ord m ⁡ ( N k ) = 2 k . We have

lim k → ∞ ⁡ ord m ⁡ ( N k ) N k α 1 - α = lim k → ∞ ⁡ 2 k ( m 2 k - 1 ) α 1 - α = 0 , i.e. ord m ⁡ ( N k ) = o ⁢ ( N k α 1 - α ) .

By Proposition 3.5, this shows that the arithmetic box space □ ( G m ⁢ ( m 2 k - 1 ) ) k ⁢ G m does not have property D α for any α ∈ ] 0 , 1 2 ] .

(4) We adapt the proof of [7, Theorem 3]. Pick an integer D > 0 with D D + 1 ≥ α . Let □ G m ⁢ ( N k ) ⁢ G m be any arithmetic box space of G m . Define

n k = ord m ⁡ ( N k ) ⋅ N k D and M k = { ( m ℓ ⁢ n k r 0 1 ) : ℓ ∈ Z , r ∈ N k ⁢ Z ⁢ [ 1 / m ] } .

It is readily checked that M k is a subgroup and, because ord m ⁡ ( N k ) ∣ n k , that M k is normal in G m and is contained in G ⁢ ( N k ) . As n k ∣ n k + 1 , we have that □ ( M k ) ⁢ G m is a box space which covers □ ( G m ⁢ ( N k ) ) ⁢ G m .

It remains to check that □ ( M k ) ⁢ G m has property D α . But G / M k maps onto the cyclic group Z / n k ⁢ Z , so we have

diam ⁡ ( G / M k ) ≥ diam ⁡ ( Z / n k ⁢ Z ) ≥ n k 3 .

On the other hand,

| G / M k | α = n k α N k α = ord m ( N k ) α N k α ⁢ ( D + 1 ) ≤ ord m ( N k ) N k D = n k .

This concludes the proof. ∎

3.2 Density results

A natural question after encountering the constructions of Theorem 3.6 2 and 3 is “how many arithmetic box spaces of BS ⁡ ( 1 , m ) have D 1 / 2 ”? In the following paragraphs, we give a partial answer to this question.

Let ( N k ) k ⊂ N be such that N k ∣ N k + 1 for every k > 0 , and denote by P k the set of prime factors of N k . Moreover, we define the set of prime factors of the sequence ( N k ) k by

P := ⋃ k = 1 + ∞ P k .

Before stating our main result from this section, we need to introduce some definitions about the density of prime numbers. We follow Powell [11] for the terminology.

Definition 3.7

Let P ⊂ P be a subset of the prime numbers. The natural primitive density of 𝑃 is (if the limit exists)

d ′ ⁢ ( P ) := lim N → ∞ ⁡ | { p ≤ N : p ∈ P } | | { p ≤ N : p ∈ P } | .

The analytic primitive density of 𝑃 is (if the limit exists)

D ′ ⁢ ( P ) = lim s → 1 + ⁡ ∑ p ∈ P 1 p s ∑ p ∈ P 1 p s .

If 𝑃 is finite, then d ′ ⁢ ( P ) = D ′ ⁢ ( P ) = 0 . Suppose now that D ′ ⁢ ( P ) > 0 . In this case, we see that ∑ p ∈ P 1 p = + ∞ ; otherwise, D ′ ⁢ ( P ) would be equal to 0. Observe that

∏ p ∈ P ( 1 - 1 p ) = 0 ⇔ ∑ p ∈ P ln ⁡ ( 1 - 1 p ) = - ∞ .

But, using that ln ⁡ ( 1 + x ) ≤ x for x > - 1 ,

∑ p ∈ P ln ⁡ ( 1 - 1 p ) ≤ - ∑ p ∈ P 1 p = - ∞ .

Therefore, we obtain

(3.2) ∏ p ∈ P ( 1 - 1 p ) = 0

if D ′ ⁢ ( P ) > 0 .

Theorem 3.8

Let □ ( G m ⁢ ( N k ) ) k ⁢ G m be an arithmetic box space, and let 𝑃 be the set of prime factors of the sequence ( N k ) k .

If | P | < + ∞ , then □ ( G m ⁢ ( N k ) ) k ⁢ G m has D 1 / 2 .
If D ′ ⁢ ( P ) > 0 , then □ ( G m ⁢ ( N k ) ) k ⁢ G m does not have D 1 / 2 .

Proof

In view of Proposition 3.5, we must study the asymptotics of ord m ⁡ ( N k ) N k .

(1) By Lemma 2.8, there exists a constant C ⁢ ( m , P ) such that

ord m ⁡ ( N k ) N k ≥ C ⁢ ( m , P ) , i.e. ord m ⁡ ( N k ) = Ω ⁢ ( N k ) .

Proposition 3.5 applies to show that □ ( G m ⁢ ( N k ) ) k ⁢ G m has D 1 / 2 .

(2) Assume now D ′ ⁢ ( P ) > 0 ; pick N = p 1 β 1 ⁢ ⋯ ⁢ p k β k with p 1 , … , p k ∈ P . We have ord m ⁡ ( N ) ≤ φ ⁢ ( N ) = | ( Z / N ⁢ Z ) × | , where 𝜑 denotes Euler’s totient function. Then

ord m ⁡ ( N ) N ≤ ∏ i = 1 k φ ⁢ ( p i β i ) p i β i = ∏ i = 1 k ( 1 - 1 p i ) .

In view of equation (3.2), we then get ord m ⁡ ( N k ) = o ⁢ ( N k ) , which proves the second part of the theorem thanks to Proposition 3.5. ∎

3.3 Open questions

The following open questions are related to the previous theorem.

If we assume that D ′ ⁢ ( P ) > 0 , does the associated arithmetic box space have D α for some α ∈ ] 0 , 1 2 [ or not?
What happens in the case | P | = + ∞ and D ′ ⁢ ( P ) = 0 ?
Given α ∈ ] 0 , 1 2 ] , can we create an arithmetic box space with exactly D α ?

4 CSP for BS ⁡ ( 1 , m )

For a subring 𝐴 of ℚ, we define three subgroups of GL n ⁢ ( A ) :

T n ⁢ ( A ) , the subgroup of upper triangular matrices;
U n ⁢ ( A ) , the unipotent subgroup, i.e. the subgroup of T n ⁢ ( A ) consisting of matrices with 1’s down the diagonal;
D n ⁢ ( A ) , the subgroup of diagonal matrices.

The map Δ : T n ⁢ ( A ) → D n ⁢ ( A ) taking a matrix to its diagonal is a surjective group homomorphism with kernel U n ⁢ ( A ) . We will also need the set N n ⁢ ( A ) of upper triangular nilpotent matrices, i.e. upper triangular matrices with 0’s down the diagonal. Note that T n ⁢ ( A ) = 1 n + N n ⁢ ( A ) .

4.1 Representations of 𝐴 into U n ⁢ ( A )

We recall Chernikov’s theorem (see [13, Theorem 4.10]): if 𝑁 is a torsion-free nilpotent group, for every k ≥ 1 , the map N → N : x ↦ x k is injective.

The first lemma discusses one-parameter subgroups in U n ⁢ ( A ) .

Lemma 4.1

For every g ∈ U n ⁢ ( A ) , there exists a unique homomorphism

α : A → U n ⁢ ( A ) such that α ⁢ ( 1 ) = g .

Proof

The existence follows from [13, Corollary 10.25]: if g = 1 n + X , with X ∈ N n ⁢ ( A ) , then for r ∈ A ,

α ⁢ ( r ) = ( 1 n + X ) r = ∑ k = 0 ∞ ( r k ) ⁢ X k = 1 n + r ⁢ X + r ⁢ ( r - 1 ) 2 ⁢ X 2 + ⋯

(note that the sum is finite as X n = 0 ). For the uniqueness, let 𝛽 be another homomorphism with β ⁢ ( 1 ) = g ; for r ∈ A , write r = a b with a , b ∈ Z , b > 0 and a , b coprime. Then

α ⁢ ( r ) b = α ⁢ ( a ) = g a = β ⁢ ( a ) = β ⁢ ( r ) b ,

so α ⁢ ( r ) = β ⁢ ( r ) by Chernikov’s theorem. ∎

Definition 4.2

For a subgroup 𝐻 of U n ⁢ ( A ) , the isolator of 𝐻 is

I ⁢ ( H ) = { g ∈ U n ⁢ ( A ) : g k ∈ H ⁢ for some ⁢ k ≥ 1 } .

By [13, Theorem 3.25], I ⁢ ( H ) is a subgroup of U n ⁢ ( A ) . Clearly, H ⊂ I ⁢ ( H ) .

Lemma 4.3

For m ≥ 2 , let α : Z ⁢ [ 1 / m ] → U n ⁢ ( Z ⁢ [ 1 / m ] ) be an injective homomorphism. Then I ⁢ ( α ⁢ ( Z ⁢ [ 1 / m ] ) ) is abelian and the exponent of the group

I ⁢ ( α ⁢ ( Z ⁢ [ 1 / m ] ) ) / α ⁢ ( Z ⁢ [ 1 / m ] )

is finite.

Proof

The proof is in three steps.

(1) By Lemma 4.1, there exists a unique homomorphism α ~ : Q → U n ⁢ ( Q ) that extends 𝛼. We show that I ( α ( Z [ 1 / m ] ) ⊂ α ~ ( Q ) , from which the first statement will follow. For g ∈ I ⁢ ( α ⁢ ( Z ⁢ [ 1 / m ] ) ) , there exist k ≥ 1 and r ∈ Z ⁢ [ 1 / m ] such that g k = α ⁢ ( r ) . Then

α ~ ⁢ ( r k ) k = α ~ ⁢ ( r ) = α ⁢ ( r ) = g k ;

hence g = α ~ ⁢ ( r k ) by Chernikov’s theorem.

(2) Let us show that there exists some non-zero x 0 ∈ Z ⁢ [ 1 / m ] such that, for g ∈ I ⁢ ( α ⁢ ( Z ⁢ [ 1 / m ] ) ) , k ≥ 1 , r ∈ Z ⁢ [ 1 / m ] ,

g k = α ⁢ ( r ) ⟹ r k ⁢ x 0 ∈ Z ⁢ [ 1 / m ] .

Write α ⁢ ( 1 ) = 1 n + X , with X ∈ N n ⁢ ( Z ⁢ [ 1 / m ] ) , as in Lemma 4.1. Note that X ≠ 0 as 𝛼 is injective. Then

g = α ~ ⁢ ( r k ) = ( 1 n + X ) r k = 1 n + r k ⁢ X + ⋯ .

For 1 ≤ i < n and an upper triangular matrix 𝑌 of size n × n , we denote by Y ( i ) the 𝑖-th parallel to the diagonal (moving upwards from the diagonal). Let 𝑖 be the smallest index such that X ( i ) ≠ 0 . Since ( X k ) ( i ) = 0 for k ≥ 2 , we have g ( i ) = r k ⁢ X ( i ) . Let x 0 be any non-zero coefficient of X ( i ) . Since g ∈ U n ⁢ ( Z ⁢ [ 1 / m ] ) , we have r k ⁢ x 0 ∈ Z ⁢ [ 1 / m ] as desired.

(3) Let π ⁢ ( m ) be the set of primes dividing 𝑚. An integer is a π ⁢ ( m ) -number if all its prime divisors are in π ⁢ ( m ) . Write x 0 = b t , with 𝑡 a π ⁢ ( m ) -number, and b ∈ Z is coprime to 𝑡. For g ∈ I ⁢ ( α ⁢ ( Z ⁢ [ 1 / m ] ) ) , with g k = α ⁢ ( r ) as above, write r k = a s ⁢ ℓ , where a , s , ℓ are pairwise coprime, 𝑠 is a π ⁢ ( m ) -number and ℓ is coprime with 𝑚. By the previous step, r k ⁢ x 0 = a ⁢ b s ⁢ t ⁢ ℓ ∈ Z ⁢ [ 1 / m ] . Since 𝑎 and ℓ are coprime, this may happen only if ℓ divides 𝑏. Finally,

g b = α ~ ⁢ ( r k ) b = α ~ ⁢ ( b ⁢ r k ) = α ~ ⁢ ( a ⁢ b s ⁢ ℓ ) .

But a ⁢ b s ⁢ ℓ ∈ Z ⁢ [ 1 / m ] as ℓ divides 𝑏. This implies that g b ∈ α ⁢ ( Z ⁢ [ 1 / m ] ) ; hence the exponent of I ⁢ ( α ⁢ ( Z ⁢ [ 1 / m ] ) ) / α ⁢ ( Z ⁢ [ 1 / m ] ) divides 𝑏. ∎

4.2 Special representations of BS ⁡ ( 1 , m )

For m ≥ 2 , set G m = BS ⁡ ( 1 , m ) = Z ⁢ [ 1 / m ] ⋊ Z . We will write A m for Z ⁢ [ 1 / m ] when viewed as a normal subgroup of G m .

Definition 4.4

A special representation of G m is an injective homomorphism ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] ) such that ρ ⁢ ( A m ) ⊂ U n ⁢ ( Z ⁢ [ 1 / m ] ) .

We note that the standard embedding G m → T 2 ⁢ ( Z ⁢ [ 1 / m ] ) is a special representation.

Lemma 4.5

The following statements hold.

If 𝜌 is a special representation, then ρ - 1 ⁢ ( U n ⁢ ( Z ⁢ [ 1 / m ] ) ) = A m .
If 𝑚 is even, then any injective homomorphism ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] ) is a special representation.

Proof

We work with the presentation G m = ⟨ a , t ∣ t ⁢ a ⁢ t - 1 = a m ⟩ , observing that the normal subgroup A m coincides with the normal subgroup generated by 𝑎.

(1) Suppose by contradiction that A m is strictly contained in ρ - 1 ⁢ ( U n ⁢ ( Z ⁢ [ 1 / m ] ) ) . Then there exists k > 0 such that ρ ⁢ ( t k ) ∈ U n ⁢ ( Z ⁢ [ 1 / m ] ) . Consider the subgroup 𝐻 of G m generated by A m ∪ { t k } so that ρ ⁢ ( H ) ⊂ U n ⁢ ( Z ⁢ [ 1 / m ] ) . Since 𝜌 is injective, we see that 𝐻 is nilpotent. As 𝐻 also has finite index in G m , we deduce that G m is virtually nilpotent, which is a contradiction.

(2) Suppose that 𝑚 is even and ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] ) is an injective homomorphism. It is enough to see that ρ ⁢ ( a ) belongs to U n ⁢ ( Z ⁢ [ 1 / m ] ) = ker ⁡ ( Δ ) . But we have

ρ ⁢ ( a m - 1 ) = ρ ⁢ ( [ a , t ] ) ∈ [ T n ⁢ ( Z ⁢ [ 1 / m ] ) , T n ⁢ ( Z ⁢ [ 1 / m ] ) ] ⊂ ker ⁡ ( Δ ) .

Now the image of Δ, i.e. D n ⁢ ( Z ⁢ [ 1 / m ] ) ≃ ( Z ⁢ [ 1 / m ] × ) n , contains only 2-torsion; since Δ ⁢ ( ρ ⁢ ( a ) ) m - 1 = 1 n and m - 1 is odd, we have Δ ⁢ ( ρ ⁢ ( a ) ) = 1 n as desired. ∎

We now head towards CSP for special representations of BS ⁡ ( 1 , m ) . The next lemma is proved exactly as [5, Lemma 4] by Formanek, using the same ingredient, namely a number-theoretical result by Chevalley [3].

Lemma 4.6

Let 𝑅 be a subring of a number field with R × finitely generated, and let 𝐺 be a subgroup of D n ⁢ ( R ) . There exists a function φ : N → N such that if g ∈ G satisfies g ≡ 1 n mod φ ⁢ ( r ) , then 𝑔 is an 𝑟-th power in 𝐺.∎

In [5, Theorem 5], Formanek proved that if 𝑅 is the ring of integers of a number field and 𝐺 is a subgroup of T n ⁢ ( R ) , then 𝐺 has CSP. The proof of the next result is inspired by Formanek’s proof.

Proposition 4.7

Let

ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] )

be a special representation of BS ⁡ ( 1 , m ) . Then ρ ⁢ ( G m ) has CSP.

Proof

Let N ◁ G m be a normal subgroup of finite index 𝑟. Denote by G m r the subgroup generated by 𝑟-th powers in G m . Then G m r ⊂ N by Lagrange’s theorem. Let 𝑏 be the exponent of I ⁢ ( ρ ⁢ ( A m ) ) / ρ ⁢ ( A m ) (which is finite by Lemma 4.3), and let 𝑒 be the exponent of the finite group T n ⁢ ( Z ⁢ [ 1 / m ] / ( b ⁢ r ) 2 ⁢ Z ⁢ [ 1 / m ] ) . Define then

M = ( b ⁢ r ) 2 ⁢ φ ⁢ ( r ⁢ e ) ,

where 𝜑 comes from Lemma 4.6 applied to Δ ⁢ ( ρ ⁢ ( G m ) ) . We will show that if ρ ⁢ ( g ) ≡ 1 n mod M , then g ∈ G n r so that ρ ⁢ ( N ) contains the congruence subgroup ρ ⁢ ( G m ) ⁢ ( M ) .

If ρ ⁢ ( g ) ≡ 1 n mod M , in particular Δ ⁢ ( ρ ⁢ ( g ) ) ≡ 1 n mod φ ⁢ ( e ⁢ r ) . By Lemma 4.6, the matrix Δ ⁢ ( ρ ⁢ ( g ) ) is an ( e ⁢ r ) -th power in Δ ⁢ ( ρ ⁢ ( G m ) ) , i.e. there exists z ∈ G m such that

Δ ⁢ ( ρ ⁢ ( g ) ) = Δ ⁢ ( ρ ⁢ ( z e ⁢ r ) ) = Δ ⁢ ( ρ ⁢ ( ( z e ) r ) ) .

But ρ ⁢ ( z e ) ≡ 1 n mod ( b ⁢ r ) 2 by definition of 𝑒, so also ρ ⁢ ( z e ⁢ r ) ≡ 1 n mod ( b ⁢ r ) 2 . On the other hand, by definition of 𝑀, we have ρ ⁢ ( g ) ≡ 1 n mod ( b ⁢ r ) 2 , so

ρ ⁢ ( g - 1 ⁢ z e ⁢ r ) ≡ 1 n mod ( b ⁢ r ) 2 .

Since g - 1 ⁢ z e ⁢ r ∈ ker ⁡ ( Δ ) = U n ⁢ ( Z ⁢ [ 1 / m ] ) , [5, Lemma 1]^[2] applies to guarantee that g - 1 ⁢ z e ⁢ r is a ( b ⁢ r ) -th power in U n ⁢ ( Z ⁢ [ 1 / m ] ) , so we find h ∈ U n ⁢ ( Z ⁢ [ 1 / m ] ) such that

ρ ⁢ ( g - 1 ⁢ z e ⁢ r ) = h b ⁢ r .

Hence g - 1 ⁢ z e ⁢ r ∈ ρ - 1 ⁢ ( U n ⁢ ( Z ⁢ [ 1 / m ] ) ) = A m (the equality follows from the first part of Lemma 4.5). This means that h ∈ I ⁢ ( ρ ⁢ ( A m ) ) , so by definition of 𝑏, we have h b ∈ ρ ⁢ ( A m ) , say h b = ρ ⁢ ( y ) for y ∈ A m . Then

h b ⁢ r = ρ ⁢ ( y r ) and ρ ⁢ ( g - 1 ⁢ z e ⁢ r ) = ρ ⁢ ( y r ) .

As 𝜌 is injective, g - 1 ⁢ z e ⁢ r = y r , i.e. g = ( z e ) r ⁢ y - r , so g ∈ G m r . ∎

4.3 From special representations to all injective representations in T n ⁢ ( Z ⁢ [ 1 / m ] )

In this section, for any injective representation 𝜌 of G m into T n ⁢ ( Z ⁢ [ 1 / m ] ) , we show that the group ρ ⁢ ( G m ) has CSP. The main idea is to pass to subrepresentations which are special and extract information about 𝜌.

Lemma 4.8

Let 𝐺 be a subgroup of GL n ⁢ ( Z ⁢ [ 1 / m ] ) .

Let 𝐻 be a finite index subgroup of 𝐺, where 𝐻 has CSP. If 𝐻 contains a congruence subgroup of 𝐺, then 𝐺 also has CSP.
For 1 ≤ i ≤ k , let H i be a finite index subgroup of 𝐺. Suppose G = ⋃ i = 1 k H i ; then if all H i have CSP, 𝐺 also has CSP.

Proof

(1) Let 𝑁 be a finite index subgroup of 𝐺; then N ∩ H is a finite index subgroup of 𝐻 and thus contains some congruence subgroup of the form H ⁢ ( M ) . By assumption, there is some M ~ such that G ⁢ ( M ~ ) ⊂ H ; hence we have that

G ⁢ ( M ⁢ M ~ ) ⊂ G ⁢ ( M ) ∩ G ⁢ ( M ~ ) ⊂ H ⁢ ( M ) ⊂ N ∩ H ⊂ N .

(2) Suppose by contradiction that 𝐺 does not have CSP; then there is some finite index subgroup 𝑁 which does not contain any congruence subgroup. However, each N ∩ H i contains a congruence subgroup of the form H ⁢ ( M i ) . Setting M ¯ = ∏ i M i , there is some g ∈ G ⁢ ( M ¯ ) ∖ N . Note that this 𝑔 is necessarily in one of the H i ’s. So it is necessarily in H ⁢ ( M i ) since H ⁢ ( M ¯ ) ⊂ H ⁢ ( M i ) . However, it is not in N ∩ H i , a contradiction. ∎

Lemma 4.9

Let ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] ) be a monomorphism. If Δ ⁢ ( ρ ⁢ ( a ) ) ≠ 1 n and the diagonal of ρ ⁢ ( t ) is strictly positive, then ρ ⁢ ( G m ) has CSP.

Proof

Note that the subgroup B = ⟨ a 2 , t ⟩ is also a Baumslag–Solitar group isomorphic to BS ⁡ ( 1 , m ) . The restriction ρ | B is a special representation of BS ⁡ ( 1 , m ) ; indeed, by the proof of Lemma 4.5, we have Δ ⁢ ( ρ ⁢ ( a ) ) m - 1 = 1 n , so Δ ⁢ ( ρ ⁢ ( a ) ) has finite order in D n ⁢ ( Z ⁢ [ 1 / m ] ) , but the torsion subgroup of D m ⁢ ( Z ⁢ [ 1 / m ] ) is { ± 1 } n , so Δ ⁢ ( ρ ⁢ ( a 2 ) ) = 1 n . By Proposition 4.7, the group ρ ⁢ ( B ) has the CSP.

We show there is some congruence subgroup of ρ ⁢ ( G m ) that is fully contained inside ρ ⁢ ( B ) . Since Δ ⁢ ( ρ ⁢ ( a ) ) ≠ 1 n , there is some −1 on the diagonal; suppose it appears at the 𝑖-th position. Let 𝑠 be the element of ( Z ⁢ [ 1 / m ] ) × in the 𝑖-th position in 𝑡. By Lemma 2.10, we find M > 1 , coprime with 𝑚, such that 𝑠 has odd order in ( Z ⁢ [ 1 / m ] / M ⁢ Z ⁢ [ 1 / m ] ) × .

We note that if 𝜔 is a word with letters in { a , t } representing an element of G m ∖ B , then it contains an odd number of 𝑎’s. Hence the element on the 𝑖-th position on the diagonal of ρ ⁢ ( ω ) is of the form - s r for some r ∈ Z . If we suppose that ρ ⁢ ( ω ) ∈ ρ ⁢ ( G m ) ⁢ ( M ) , then s r = - 1 mod M . This is impossible since 𝑠 has odd order in ( Z ⁢ [ 1 / m ] / M ⁢ Z ⁢ [ 1 / m ] ) × and thus cannot have a power equal to −1. So ρ ⁢ ( ω ) cannot be in the corresponding congruence subgroup, i.e. ρ ⁢ ( G m ) ⁢ ( M ) ⊂ ρ ⁢ ( B ) . By Lemma 4.8, the result follows. ∎

We are ready to complement Proposition 4.7.

Proposition 4.10

For any injective homomorphism ρ : G m → T n ⁢ ( Z ⁢ [ 1 / m ] ) , the group ρ ⁢ ( G m ) has CSP.

Proof

Since we already know the result for 𝑚 even (by Proposition 4.7 and Lemma 4.5), we may and will assume that 𝑚 is odd.

The group G m has three subgroups: B 1 = ⟨ a 2 , t ⟩ , B 2 = ⟨ a 2 , a ⁢ t ⟩ , B 3 = ⟨ a , t 2 ⟩ , which are all isomorphic to BS ⁡ ( 1 , m ) or BS ⁡ ( 1 , m 2 ) . We show that they cover G m . Any element ω ∈ G m can be described by a word of the form t - k ⁢ a r ⁢ t ℓ , with k , ℓ ∈ N and r ∈ Z . Replacing 𝜔 by ω - 1 if necessary, we may assume k ≤ ℓ .

If 𝑟 is even, then ω ∈ B 1 .
If 𝑘 and 𝑙 are both even or odd, then ω ∈ B 3 . Indeed,
ω = t - 2 ⁢ k ⁢ ( t k ⁢ a r ⁢ t - k ) ⁢ t k + ℓ = t - 2 ⁢ k ⁢ a r ⁢ m k ⁢ t k + ℓ ∈ B 3
as k + ℓ is even.
If 𝑟 is odd and k , ℓ do not have the same parity, then ω ∈ B 2 . Indeed,
t - k ⁢ a r ⁢ t ℓ = ( t - 1 ⁢ a - 1 ) k ⁢ a ∑ i = 0 k - 1 m i ⁢ a r ⁢ a - ∑ i = 0 ℓ - 1 m i ⁢ ( a ⁢ t ) l = ( a ⁢ t ) - k ⁢ a r - ∑ i = k ℓ - 1 m i ⁢ ( a ⁢ t ) ℓ .
Since r , ℓ - k and 𝑚 are odd, the exponent of 𝑎 in the latter expression is even, so ω ∈ B 2 .

Note that if Δ ⁢ ( ρ ⁢ ( a ) ) = 1 n , then it is a special representation and Proposition 4.7 applies. If it is not, then the restrictions of 𝜌 to B 1 and B 2 are special representations, and the restriction of 𝜌 to B 3 satisfies the assumptions of Lemma 4.9. Hence the ρ ⁢ ( B i ) ’s all have CSP. Now simply applying Lemma 4.8 finishes the argument. ∎

4.4 An alternative approach to CSP for BS ⁡ ( 1 , p ℓ ) (𝑝 prime)

We use the language of ℚ-algebraic groups. Denoting by G a (resp. G m ) the additive group (resp. the multiplicative group), we set G = G a ⋊ G m , the affine group viewed as a subgroup of GL 2 via the standard embedding.

Let p 1 , … , p r be distinct primes, and let S = { p 1 , … , p r , ∞ } be viewed as a set of places of ℚ. Then the ring O S of 𝑆-integers in ℚ is precisely

O S = Z ⁢ [ 1 / p 1 , … , 1 / p r ]

so that O S × = { ± p 1 k 1 ⁢ ⋯ ⁢ p r k r : k 1 , … , k r ∈ Z } and

G ⁢ ( O S ) = { ( ± p 1 k 1 ⁢ ⋯ ⁢ p r k r a 0 1 ) : k 1 , … , k r ∈ Z , a ∈ Z ⁢ [ 1 / p 1 , … , 1 / p r ] } .

It is known that G ⁢ ( O S ) satisfies CSP; see [12, formula (∗∗), p. 108].

Note that, taking r = 1 , i.e. S = { p . ∞ } and O S = Z ⁢ [ 1 / p ] , makes BS ⁡ ( 1 , p ℓ ) appear as a finite index subgroup in G ⁢ ( O S ) , namely [ G ( O S ) : BS ( 1 , p ℓ ) ] = 2 ℓ .

Proposition 4.11

Let H ⊂ GL n be a ℚ-subgroup, and let ρ : G → H be a ℚ-isomorphism. Then ρ ⁢ ( BS ⁡ ( 1 , p ℓ ) ) satisfies the CSP.

Proof

Let 𝐾 be a finite index subgroup in ρ ⁢ ( BS ⁡ ( 1 , p ℓ ) ) . Since G ⁢ ( O S ) satisfies the CSP, we find N > 0 such that G ⁢ ( O S ) ⁢ ( N ) ⊂ ρ - 1 ⁢ ( K ) . By [9, Lemma 3.1.1 (ii), Chapter I], the subgroup ρ ⁢ ( G ⁢ ( O S ) ⁢ ( N ) ) is an 𝑆-congruence subgroup in H ⁢ ( O S ) , i.e. it contains H ⁢ ( O S ) ⁢ ( M ) for some M > 1 . Then

( ρ ⁢ ( BS ⁡ ( 1 , p ℓ ) ) ) ⁢ ( M ) ⊂ H ⁢ ( O S ) ⁢ ( M ) ⊂ ρ ⁢ ( G ⁢ ( O S ) ⁢ ( N ) ) ⊂ K . ∎

Funding source: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Award Identifier / Grant number: 200021_188578

Funding statement: L. Hayez was supported by grant 200021_188578 of the Swiss National Fund for Scientific Research.

Communicated by: Dessislava Kochloukova

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Received: 2021-04-22

Revised: 2022-08-30

Published Online: 2022-11-15

Published in Print: 2023-05-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2021-0069