CSA-groups and separated free constructions

1 3 M ay 1 99 6 CSA groups and separated free constructions

On superstable CSA-groups

We prove that a non-abelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of non-abelian CSAgroup of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of E. Mustafin and B. Poiza...

Existentially closed CSA-groups

We study existentially closed CSA-groups. We prove that existentially closed CSA-groups without involutions are simple and divisible, and that their maximal abelian subgroups are conjugate. We also prove that every countable CSA-group without involutions embeds into a finitely generated one having the same maximal abelian subgroups, except maybe the infinite cyclic ones. We deduce from this tha...

Poly-free Constructions for Right-angled Artin Groups

We show that every right-angled Artin group AΓ defined by a graph Γ of finite chromatic number is poly-free with poly-free length bounded between the clique number and the chromatic number of Γ. Further, a characterization of all right-angled Artin groups of poly-free length 2 is given, namely the group AΓ has poly-free length 2 if and only if there exists an independent set of vertices D in Γ ...

Stability of Universal Equivalence of Groups under Free Constructions

In his important paper in [3] J. Stallings introduced a generalisation of amalgamated products of groups – called a pregroup, which is a particular kind of a partial group. He then defined the universal group U(P ) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P ). The universal group turned out to be a ...