We study the automorphisms of a graph product of finitely-generated abelian groups W . More precisely, we study a natural subgroup Aut * W of Aut W , with Aut * W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition Aut * W = (Inn W ⋊ Out 0 W ) ⋊ Aut 1 W . We also give a number of applications, some of which are geometric in nature.complete subgroups are a set of representatives for the conjugacy classes of maximal finite subgroups of W [9, Lemma 4.5] and each automorphism of W maps each maximal complete subgroup to a conjugate of some maximal complete subgroup. This is not true in an arbitrary graph product of directlyindecomposable cyclic groups, but we may pretend that it is by restricting our attention to a natural subgroup of Aut W .Definition 1.1. Write Aut * W for the subgroup of Aut W consisting of those automorphisms which map each maximal complete subgroup to a conjugate of a maximal complete subgroup.The following lemma is immediate from the discussion above and the main result of [18]. For each 1 ≤ i ≤ N, we write L i (resp, S i ) for the link (resp. star ) of v i . Lemma 1.2. If W is a graph product of directly-indecomposable cyclic groups, then Aut * W = Aut W in each of the following cases:1. W is a graph product of primary cyclic groups; 2. W is a right-angled Artin group and L i ⊆ L j for each pair of distinct non-adjacent vertices v i , v j ∈ V ;3. W is a right-angled Artin group and Γ contains no vertices of valence less than two and no circuits of length less than 5.Remark 1.3. Case (2) can be substantially generalized to groups that are not right-angled Artin groups.We now report the main results of the present article. They concern the structure of Aut * W and shall make reference to the subgroups and quotients of Aut W defined in Figure 1. In writing Aut 0 W for the subgroup of 'conjugating automorphisms', we follow Tits [26]. Mühlherr [24] writes Spe(W ) for the same subgroup. Charney, Crisp and Vogtmann [5] use the notation Aut 0 W and Out 0 W for different subgroups of the automorphism group of a right-angled Artin group than described here.Tits [26] proved that if W is a right-angled Coxeter group, then Aut W = Aut 0 W ⋊ Aut 1 W . Our first main result is a generalization of Tits' splitting.Theorem 1.4 (cf. [26]). If W is a graph product of directly-indecomposable cyclic groups, then Aut * W = Aut 0 W ⋊ Aut 1 W .

We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.

The present paper records more details of the relationship between primitive elements and palindromes in F 2 , the free group of rank two. We characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F 2 . Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes.

We show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T 0 property. Our results build on results by Droms, Laurence and Radcliffe.2000 Mathematics subject classification: primary 20E34, 20E06.

We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.