Automorphisms of partially commutative groups III: Inversions and transvections.

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