Automorphisms of partially commutative groups III: Inversions and transvections

The automorphism group of a partially commutative group [Formula: see text] is generated by four types of automorphism: graph automorphisms, inversions, transvections and vertex conjugating automorphisms. We find a characterisation of the subgroup [Formula: see text] generated by inversions and transvections, in terms of stabilisers of subgroups of [Formula: see text] generated by the so-called “admissible” subsets of [Formula: see text]. This is used to give a decomposition of [Formula: see text] as a chain of semi-direct products of (mostly) tractable and recognisable subgroups; which in turn gives rise to a presentation of [Formula: see text].

Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups

The first main result of the paper is a criterion for a partially commutative group 𝔾 to be a domain. It allows us to reduce the study of algebraic sets over 𝔾 to the study of irreducible algebraic sets, and reduce the elementary theory of 𝔾 (of a coordinate group over 𝔾) to the elementary theories of the direct factors of 𝔾 (to the elementary theory of coordinate groups of irreducible algebraic sets).Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group ℍ. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of ℍ has quantifier elimination and that arbitrary first-order formulas lift from ℍ to ℍ * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

CENTRALISER DIMENSION OF FREE PARTIALLY COMMUTATIVE NILPOTENT GROUPS OF CLASS 2

In an effort to extend the theory of algebraic geometry over groups beyond free groups, Duncan, Kazatchkov and Remeslennikov have studied the notion of centraliser dimension for free partially commutative groups. In this paper we consider the centraliser dimension of free partially commutative nilpotent groups of class 2, showing that a free partially commutative nilpotent group of class 2 with non-commutation graph Γ has the same centraliser dimension as the free partially commutative group represented by the non-commutation graph Γ.

PARTIALLY COMMUTATIVE MAGNUS TRANSFORMATIONS

Grâce à l’introduction de la notion de forme exponentielle réduite, nous montrons l’injectivité de la transformation de Magnus associée au groupe partiellement commutatif libre F(A, ϑ). Nous en déduisons la séparation de la topologie p-adique de F(A, ϑ), l’existence de plus petites racines dans F(A, ϑ) et enfin la structure des centralisateurs dans F(A, ϑ). We introduce the notion of reduced exponential form in the free partially commutative group F(A, ϑ). This allows us to prove that the Magnus transformation associated with F(A, ϑ) is injective. We deduce from this result that the p-adic topology on F(A, ϑ) is Hausdorff, that there exist least roots in F(A, ϑ) and finally the exact structure of centralizers in F(A, ϑ).