The space of finitely generated groups. (L’espace des groupes de type fini.) (French) Zbl 0959.20041

Champetier, Christophe

Topology 39, No. 4, 657-680 (2000).

A marked group is a group equipped with a fixed generating system. Let \(L_m\) be a marked free group with \(m\) generators, and let \({\mathcal G}_m\) be the set of quotient groups of \(L_m\). The set \({\mathcal G}_m\) is equipped with a natural topology (the Chabauty topology) for which it is compact and totally disconnected, and with the following equivalence relation: \(L_m/N_1\sim L_m/N_2\) if and only if these groups are isomorphic.
The author proves the following Theorem 1. For \(m\geq 2\), there is no injective measurable map from \({\mathcal G}_m/ \sim\) to the real numbers. In other words, there is no Borel map from \({\mathcal G}_m\) into the reals, which is constant on the equivalence classes and which separates these classes.
Let \({\mathcal H}_m\subset{\mathcal G}_m\) be the closure of the set of hyperbolic non-elementary groups. As the author shows, \({\mathcal H}_m\) is a Cantor set.
He obtains the following Theorem 2. The set \({\mathcal H}_m\) contains a \(G_\delta\)-dense set consisting of marked groups which are infinite and which are torsion groups (every element is of finite order).
Let \({\mathcal H}^{cc}_m\subset{\mathcal G}_m\) be now the closure of the set of hyperbolic non-elementary groups in which the centralizer of each element is cyclic, and let \({\mathcal H}^{st}_m\subset{\mathcal G}_m\) be the closure of the set of hyperbolic non-elementary groups without torsion elements.
The author proves Theorem 3. The set \({\mathcal H}^{cc}_m\) contains a \(G_\delta\)-dense set of marked groups, such that each group \(G\) in this set satisfies the following properties: (1) The equivalence class of \(G\) is dense in \({\mathcal H}^{cc}_m\) . (2) \(G\) contains every finite group with cyclic centralizers. (3) \(G\) satisfies Kazhdan’s property T. (4) \(G\) contains no free group on two generators. (5) \(G\) is perfect and generated by two elements. (6) \(G\) has no finite quotient. The author shows also that Theorem 3, except for Property (2), is also valid with \({\mathcal H}^{cc}_m\) replaced by \({\mathcal H}^{st}_m\).
Of course, the results in this paper are inspired by works of Gromov and of Ol’shanskij.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

20F67	Hyperbolic groups and nonpositively curved groups
20F65	Geometric group theory
20F05	Generators, relations, and presentations of groups
57M07	Topological methods in group theory

Keywords:

hyperbolic groups; finitely generated groups; generic properties of groups; Chabauty topology; torsion groups; marked groups