Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups

Details Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups

2011-06-12

arxiv.org/abs/1106.2331v2

Mathematics

Group Theory

7 figures, 63 pages. Notation and definitions clarified and typos corrected. 2 new figures added. Appendix containing details

Scientific paper

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show how Aut(G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decompostion of Aut(G) into a subgroup of locally conjugating automorphisms by the subgroup stabilising a certain lattice of "admissible subsets" of the vertices of C. We then characterise those graphs for which Aut(G) is a product (not necessarily semi-direct) of two such subgroups.

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