On the automorphisms of a graph product of abelian groups

Abstract

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 .

A Generating Set for the Automorphism Group of a Graph Product of Abelian Groups

“…We write (X, F ) = X Γ or simply X . In that case, by [10,Lemma 2.5], the inclusion i defined in Sec. 1 is a retraction.…”

“…Theorem 3.1 of [10] shows that Aut G is the semi-direct product Aut 0 G Aut 1 G. (2) Theorem 4.1 of [12] (see also [13,Theorem 2.2]) states that Aut 0 G is generated by the partial conjugations (set (4) in Sec. 1).…”

“…By the results of [10], it suffices to show that G 1 generates Aut 1 G. Suppose η ∈ Aut 1 G; then, for some γ ∈ Aut(Γ, o) we have that θ = γη ∈ Aut 1 G satisfies (4.1) for all v ∈ Γ. Let ∆, Θ ∈ MCS(Γ) as in Definition 5.1 and suppose v ∈ ∆.…”

Group metrics for graph products of cyclic groups

“…(5) We say that the word w is reduced if there is no word with fewer syllables which spells the same element of G. (6) We say that the consecutive syllables a αi i and a αi+1 i+1 are adjacent if a i E Γ a i+1 . (7) We say that the word w is a normal form for g if it spells g and it is reduced. (1) If the word a α1 1 · · · a α k k spelling the element g ∈ G is not reduced, then there exist 1 p < q k such that a p = a q and a p is adjacent to each vertex a p+1 , a p+2 , ..., a q−1 .…”

Normal Forms for Automorphisms of Universal Coxeter Groups and Palindromic Automorphisms of Free Groups

“…It is shown in [7] that Out 0 W is isomorphic to the subgroup of Aut 0 W generated by the following set of partial conjugation:…”