Let K be a compact space. By induction on ordinals, define I0(K) as the set of isolated points of K, and for α > 0, define Iα(K) as the set of isolated points in K −S
β<αIβ(K); call Iα(K) the set of α-isolated points of K. Let Cond(K) denote the condensation points, that is the complement of the union S
αIα(K).
IfK is metrizable, then the sequence (Iα(K)) breaks off after a countable number of steps. In the case of the space of finitely generated groups (in which case we simply write Iα and Cond), what is this number?
It might be interesting to studyIα for small values of α. For instance, it is easy to check that Zn∈In for all n.
The study of Cond is also of interest. It is characterized by: a finitely generated group G is in Cond if and only if every neighbourhood of G is uncountable. By the results of Champetier [Cha00] Cond contains all non-elementary hyperbolic groups. It can be showed that it also contains the wreath product Z≀Z and the first Grigorchuk group Γ.
A related variant of these definitions is the following: consider a group G (not necessarily finitely generated). We say that G is in the class IIα if G∈Iα(G(G)). This holds for at most oneα; otherwise say thatG belongs to the class ICond. The new I stands for “Inner” or “Intrinsic”. Note that, if we restrict to finitely generated groups, we have ICond⊂ Cond, and every group in Iα belongs to IIβ for some β ≤ α. Note also that the two classes coincide in restriction to finitely presented groups, but for instance the first Grigorchuk group Γ belongs to II1.
Limits of Baumslag-Solitar groups
We give a parametrization by m-adic integers of the groups obtained as limit of Baumslag-Solitar groups marked with a canonical set of generators. We first study the injectivity and the continuity of this map. Second, we construct, for each limit a tree on which it acts transitively. We deduce then informations on the structure and defining relations of these limit groups.
Introduction
The set of marked groups on k generators (see Section B.1 for definitions) has a natural topology in which it is a compact totally disconnected space.
This topology received several names: “topology on marked groups”, “Cayley topology”, “Grigorchuk topology”. . . Two marked groups are close if there are large balls of their Cayley graphs which are isomorphic as (pointed) labelled graphs.
This topology has been used for several purposes. Let us cite the following examples :
• Stepin [Ste84] used it to prove the existence of amenable but non ele-mentary amenable groups.
• To prove that every finitely generated Kazhdan group is a quotient of some finitely presented Kazhdan group, Shalom proved in [Sha00] that Kazhdan’s property (T) defines an open subset of the space of marked groups.
• In [CG05], Champetier et Guirardel gave a characterization of limit groups of Sela in terms of the topology on marked groups.
There also exists several papers about questions whose formulation involves the topology on marked groups language. Let us cite the following ones :
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• In [Cha00], Champetier showed that the quotient of the space of marked groups on k generators by the group isomorphism relation is not a stan-dard Borel space. He also studied the closure of non elementary hyper-bolic groups.
• In [Sta06a], the second author gave an almost complete characterization of convergent sequences among Baumslag-Solitar groups.
We are interested in the closure of Baumslag-Solitar groups and its ele-ments, which we study for their own right. Theorem 6 of [Sta06a] allows us to define the following elements of the closure (Definition B.1.7):
BS(m, ξ) = lim
n→∞BS(m, ξn)
where m ∈ Z∗, ξ ∈Zm, (ξn)n is any sequence of integers such that ξn →ξ in Zm and |ξn| → ∞ (for n→ ∞).
Nota Bene. We denote by Nthe set {0,1,2, . . .}of non negative integers and by N∗ and Z∗ the set of positive and non zero integers. If A is any ring, then A× is the set of invertible elements of A. For instance,Z× ={−1,1} whereas Z∗ =Z\ {0}.
Outline of the paper and description of results. Section B.1contains the material we want to recall and the definitions of the main groups appearing in the article.
Section B.2 describes carefully the “parametrization” BSm : Zm → G2 given by ξ7→BS(m, ξ). More specifically, we study the lack of injectivity and the continuity. Precise statements are as follows:
Theorem B.0.1 (Theorem B.2.1 and Corollary B.2.9). Let m ∈ Z∗ and let ξ, η ∈ Zm. The equality of marked groups BS(m, ξ) = BS(m, η) holds if and only if there is some d ∈ Z∗ such that gcd(ξ, m) = gcd(η, m) = d and the images of ξ/d and η/d in Zm/d are equal. In particular, the map BSm is injective on the set of invertible m-adic intergers.
Theorem B.0.2 (Corollary B.2.11). The map BSm is continuous.
These theorems allow us to describe the set of groups BS(m, ξ), with ξ invertible, as a boundary of Baumslag-Solitar groups (Corollary B.2.13).
It is well-known that a Baumslag-Solitar group acts on its Bass-Serre tree by automorphisms and onQby affine transformations. The first action is faithful whereas thesecond is not in general. Section B.3 is devoted to actions and structure of the groups BS(m, ξ). First BS(p, ξ) acts affinely on Qp when pis a prime (see Theorem B.3.1). Second, we construct an action of BS(m, ξ) by automorphisms on a tree which is in some sense a “limit” of Bass-Serre trees (see Theorem B.3.4). This action allows us to prove the following results:
Theorem B.0.3 (Theorem B.3.12). For any m∈Z∗ andξ ∈Zm, there exists an exact sequence 1→F2 →BS(m, ξ)→ Z≀Z→1, where F2 is a free group.
Corollary B.0.4 (Corollary B.3.14). The limits BS(m, ξ)have the Haagerup property and are residually solvable.
Let us quote here for comparison some straightforward properties of the groups BS(m, ξ) as they must share any property of Baumslag-Solitar groups which defines a closed set in the space of marked groups: they are torsion free and centerless, their subgroup generated by a and bab−1 is a non abelian free group for |m| ≥ 2, and they are non Kazhdan (the last one could also be deduced from Corollary B.0.4).
Section B.4 is devoted to presentations of the limits BS(m, ξ)’s. First, we discuss finite presentability. It turns out that limits of free groups and our limits have very different properties.
Theorem B.0.5 (see Theorem B.4.1). For any m ∈Z∗ and ξ ∈ Zm \mZm, the group BS(m, ξ) is not finitely presented.
However, the proof uses crucially the definition as limit of Baumslag-Solitar groups. Moreover, we exhibit explicit presentations of our limits, which are again related to the action by tree automorphisms (see Theorem B.4.4).
Finally, we collect results about classical sequences of groups inAppendix B.5. In the spirit of [Sta06a], we treat the problem of convergence of Torus knots groups. The solution happens to be simpler than for Baumslag-Solitar groups; indeed, the torus knot groups depend on two parameters p, q, but fixing pand letting q go to∞, one only gets convergent sequences (see Propo-sition B.5.1). We also discuss the cases of Baumslag-Solitar groups with other markings (see Theorem B.5.4) and other Baumslag’s one-relator groups (see Proposition B.5.7).
Acknowledgements. We would like to thank Laurent Bartholdi and Thierry Coulbois for having pointed out Theorem B.5.4 (a) and Remark B.3.3 to us. We would also thank our advisors, Goulnara Arzhantseva and Alain Valette, for their valuable comments on previous versions of this article.