B.1.1 The ring of m-adic integers
Let m∈Z∗. Recall that the ring ofm-adic integers Zm is the projective limit (in the category of topological rings) of the system
. . .→Z/mhZ→Z/mh−1Z→. . .→Z/m2Z→Z/mZ
where the arrows are the canonical (surjective) homomorphisms. This shows thatZm is compact. This topology is compatible with the ultrametric distance given, for ξ6=η, by
dm(ξ, η) = |m|−max{k∈N:ξ−η∈mkZm} .
We collect now (without proofs) some easy facts about m-adic integers which are useful in the following sections. Detailed proofs were given in the second-named author’s Ph.D. thesis [Sta05, Appendix C]1. Notice that Zm is the nul ring if |m|= 1.
Proposition B.1.1. Let m be an integer such that |m| > 2 and let m =
±pk11· · ·pkℓℓ be its decomposition in prime factors:
(a) One has an isomorphism of topological rings Zm ∼= Zp1 ⊕. . .⊕Zpℓ. In particular, if m is not a power of a prime number, the ring Zm has zero divisors.
(b) The group of invertible elements of Zm is given by Z×m =Zm\(p1Zm∪. . .∪pkZm).
(c) Any ideal of Zm is principal. Moreover any ideal of Zm containing a nonzero integer can be written pi11· · ·piℓℓZm with i1, . . . iℓ ∈N.
(d) For any i1, . . . iℓ ∈N, one has Z∩pi11· · ·piℓℓZm =pi11· · ·piℓℓZ.
Definition B.1.2. Let m be an integer such that |m| > 2 and let p1, . . . , pℓ
be its prime factors. If E is a subset of Zm containing a nonzero integer, the greatest common divisor (gcd) of the elements of E is the (unique) number pi11· · ·piℓℓ (withi1, . . . , iℓ ∈N) such that the ideal generated byE ispi11· · ·piℓℓZm.
If |m|= 1, we set by convention gcd(Zm) = 1.
Lemma B.1.3. Let m ∈ Z∗ and let m′ be a divisor of m. Let us write m′ = ±pj11· · ·pjℓℓ and m = ±pk11· · ·pkℓℓ their decomposition in prime factors (js 6 ks for all s = 1, . . . , ℓ). Let π : Zm → Zm′ the morphism induced by projections Z/mhZ→Z/(m′)hZ (for h>1). Then the following holds:
(a) One has π(n) = n for any integer n.
(b) For any d=±pi11· · ·piℓℓ with i1, . . . , iℓ ∈N, one has π−1(dZm′) =dZm. The ideal dZm is both open and closed in Zm. More generally :
Lemma B.1.4. Let m be in Z∗ and let m′ be an integer whose prime divisors are prime divisors of m.
There exists h = h(m′) such that for any ξ, η in Zm, the inequality dm(ξ, η) < |m|−h implies gcd(ξ, m′) = gcd(η, m′). In particular, the set of m-adic integers ξ such thatgcd(ξ, m′) =d is both open and closed in Zm. Proof. Consider h such that m′ divides mh. If dm(ξ, η) < |m|−h, one has ξ =k+mhµand η=k+mhν with k∈Z andµ, ν ∈Zm by Proposition B.1.1 (d). Hence gcd(ξ, m′) =gcd(k, m′) = gcd(η, m′).
1Note that the second part of Statement (a) was false there.
B.1.2 Marked groups and their topology
Introductory expositions of these topics can be found in [Cha00] or [CG05].
We only recall some basics and what we need in the following sections.
The free group on k generators will be denoted by Fk, or FS (with S = (s1, . . . , sk)) if we want to precise the names of (canonical) generating elements.
A marked group on k generators is a pair (G, S) where G is a group and S = (s1, . . . , sk)∈ Gk is a family which generates G. A marked group (G, S) is endowed with a canonical epimorphism φ : FS → G, which induces an isomorphism of marked groups between FS/kerφ and G. Hence a class of marked groups can always be represented by a quotient ofFS. In particular if a group is given by a presentation, this defines a marking on it. The nontrivial elements of R := kerφ are called relations of (G, S). Given w ∈ Fk we will often write ”w= 1 inG” or ”w=
G 1” to say that the image ofw inGis trivial.
Letw =xε11· · ·xεnn be a reduced word inFS (with xi ∈S and εi ∈ {±1}).
The integer n is called thelength of w and denoted|w|. If (G, S) is a marked group on k generators and g ∈G, the length of g is
|g|G := min{n :g =s1· · ·sn with si ∈S⊔S−1}
= min{|w|:w∈FS, φ(w) = g}.
LetGk be the set of marked groups onk generators (up to marked isomor-phism). Let us recall that the topology on Gk comes from the following ultra-metric distance: for (G1, S1) 6= (G2, S2) ∈ Gk we set d (G1, S1),(G2, S2)
:=
e−λ where λ is the length of a shortest element of Fk which vanishes in one group and not in the other one. But what the reader has to keep in mind is the following characterization of convergent sequences.
Lemma B.1.5. [Sta06a, Proposition 1] Let (Gn)n>0 be a sequence of marked groups in Gk. The sequence (Gn)n>0 converges if and only if for any w ∈ Fk, we have either w = 1 in Gn for n large enough, or w 6= 1 in Gn for n large enough.
The reader could remark that the latter condition characterizes exactly Cauchy sequences. Another useful statement we will use in the paper is the following:
Lemma B.1.6. [CG05, Lemma 2.3] If a sequence(Gn)n>0 in Gk converges to a marked group G ∈ Gk which is given by a finite presentation, then, for n large enough, Gn is a marked quotient of G.
We address the reader to the given references for proofs.
B.1.3 Notation and conventions
We give now some notation and conventions which hold in the whole paper.
First, recall that we define the Baumslag-Solitar groups by BS(m, n) =
a, b abma−1 =bn
(m, n∈Z∗).
Then, we define a new family of groups in the following way:
Definition B.1.7. For m ∈ Z∗ and ξ ∈ Zm, one defines a marked group on two generators BS(m, ξ) by the formula
BS(m, ξ) = lim
n→∞BS(m, ξn)
where (ξn)n is any sequence of integers such that ξn→ξ in Zm and |ξn| → ∞ (for n→ ∞).
Notice that BS(m, ξ) is well defined for any ξ ∈ Zm by Theorem 6 of [Sta06a]. Note also that for any n ∈ Z∗, one has BS(m, n) 6= BS(m, n).
Indeed, the word abma−1b−n represents the identity element in BS(m, n), but not in BS(m, n).
When considered as marked groups, the free groupF2 =F(a, b), Baumslag-Solitar groups, and groups BS(m, ξ) are all (unless stated otherwise) marked by the pair (a, b).
Another group which plays an important role in this article is Z≀Z=Z ⋉tZ[t, t−1]∼=Z ⋉s
M
Z
Z
where the generator of the first copy of Z acts on Z[t, t−1] by multiplication by t or, equivalently, on L
ZZ by shifting the indices. This group is assumed (unless specified otherwise) to be marked by the generating pair consisting of elements (1,0) and (0, t0).
The last groups we introduce here are Γ(m, n) =Z ⋉mn Z[gcd(m,n)lcm(m,n)] (m, n∈ Z∗) where the generator of the first copy ofZacts onZ[gcd(m,n)lcm(m,n)] by multiplica-tion by mn. This group is assumed (unless specified otherwise) to be marked by the generating pair consisting of elements (1,0) and (0,1). The latter elements are the images of (1,0) and (0, t0) by the homomorphismZ≀Z→Γ(m, n) given by the evaluation t = mn; they are also the images of the elements a and b of BS(m, n) by the homomorphism defined by a 7→ (1,0) and b 7→ (0,1). Ob-serve that the group Z ⋉ Z[gcd(m,n)lcm(m,n)] acts affinely on Q(orR) by (1,0)·x= mnx and (0, y)·x=x+y.
We introduce the homomorphism σa : F2 → Z defined by σa(a) = 1 and σa(b) = 0. It factors through all groupsBS(m, n),BS(m, ξ),Z≀Zand Γ(m, n).
The induced morphisms are also denoted by σa. To end this section we de-fine the homomorphism ¯ : F2 → F2 given by ¯a = a and ¯b = b−1. Note that it is compatible with quotient maps and also defines homomorphisms
¯ : BS(m, n) → BS(m, n) for m, n ∈ Z∗ and ¯ : BS(m, ξ) → BS(m, ξ) for m ∈Z∗, ξ∈Zm.