On the number of outer automorphisms of the automorphism group of a right-angled Artin group

Fullarton, Neil J.

doi:10.4310/mrl.2016.v23.n1.a8

Cited by 5 publications

(10 citation statements)

References 20 publications

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“…The description of the automorphism group of a direct product of RAAGs is due to N. Fullarton [10], and G. Giovanni and N. Wahl [11], whose results we present in this section, together with its implications for the R ∞ -property for RAAGs. Recall that a direct product on group theoretical level corresponds to a simplicial join on graph theoretical level.…”

Section: Automorphism Group Of Direct Product Of Raagsmentioning

confidence: 94%

The R∞-property for right-angled Artin groups

Dekimpe¹,

Senden²

2021

Topology and its Applications

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Given a group G and an automorphism ϕ of G, two elements x, y ∈ G are said to be ϕ-conjugate if x = gyϕ(g) −1 for some g ∈ G. The number of equivalence classes is the Reidemeister number R(ϕ) of ϕ, and if R(ϕ) = ∞ for all automorphisms of G, then G is said to have the R ∞ -property.A finite simple graph Γ gives rise to the right-angled Artin group A Γ , which has as generators the vertices of Γ and as relations vw = wv if and only if v and w are joined by an edge in Γ. We conjecture that all non-abelian right-angled Artin groups have the R ∞ -property and prove this conjecture for several subclasses of right-angled Artin groups.have showed that twin groups, a subfamily of the right-angled Coxeter groups, all possess the R ∞ -property [21].We suspect that, amongst all RAAGs, only the free abelian ones do not possess the R ∞property:Conjecture. Let Γ(V, E) be a finite non-complete graph, i.e. V is finite and there are two (distinct) vertices not joined by an edge. Then A Γ has the R ∞ -property.We first reduce the conjecture to graphs belonging to three specific classes, after which we prove the conjecture for one of these classes and for several subclasses of the other two.We start by recalling two ways of proving that an automorphism has infinite Reidemeister number. Definition 1.1.1. Let G, H be two groups. If H ∼ = G/N for some characteristic subgroup N of G, we call H a characteristic quotient of G. The following result is well-known, see e.g. [20, Lemma 2.1].

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Section: Automorphism Group Of Direct Product Of Raagsmentioning

confidence: 94%

The R∞-property for right-angled Artin groups

Dekimpe¹,

Senden²

2021

Topology and its Applications

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“…The above results indicate that both Out(Aut(A Γ )) and Out(Out(A Γ )) are either small or trivial for A Γ = Z n and A Γ = F n , independent of n. For more general RAAGs, the second author has shown in [5] that this behavior does not hold. More precisely, he proves that for any n > 0 there exist graphs Γ 1 , Γ 2 so that | Out(Aut(A Γ 1 ))| > n, and | Out(Out(A Γ 2 ))| > n. Theorem A of this paper substantially strengthens the second author's result in the case of Out(Out(A Γ )).…”

Section: Introductionmentioning

confidence: 93%

“…The second author [5] also introduced the notion of an austere graph. If Γ is austere, then Out(A Γ ) is, in some sense, as simple as possible.…”

Section: Introductionmentioning

confidence: 99%

“…If Γ is austere, then Out(A Γ ) is, in some sense, as simple as possible. The second author previously conjectured that for austere graphs Γ, the group Aut(A Γ ) is complete (see the remarks after Proposition 5.1 in [5]). However the following theorem establishes that the order of Out(Aut(A Γ )) in the austere case is at least exponential in n.…”

Section: Introductionmentioning

confidence: 99%

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Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group

Bregman¹,

Fullarton²

2017

Michigan Math. J.

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We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A Γ ), for certain right-angled Artin groups A Γ . This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A Γ )). This demonstrates a marked departure from the known behavior of Out(Out(A Γ )) when A Γ is free or free abelian, as in these cases Out(Out(A Γ )) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A Γ )).

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“…(1) (graph automorphisms) automorphisms of the graph Γ via a permutation of its set of vertices V , The next proposition builds on the work of Fullarton [5] to show how automorphisms of RAAGs interact with the direct product decomposition of a RAAG.…”

Section: Corollary 32mentioning

confidence: 99%