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Abstract
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We show that the outer automorphism groups of graph products of finitely
generated abelian groups satisfy the Tits alternative, are residually finite,
their so-called Torelli subgroups are finitely generated, and they satisfy a
dichotomy between being virtually nilpotent and containing a nonabelian free
subgroup that is determined by a graphical condition on the underlying labeled
graph.
Graph products of finitely generated abelian groups simultaneously generalize
right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs),
providing a common framework for studying these groups. Our results extend a
number of known results for the outer automorphism groups of RAAGs and/or
RACGs by a variety of authors, including Caprace, Charney, Day, Ferov, Guirardel,
Horbez, Minasyan, Vogtmann, Wade, and the current authors.
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Keywords
graph product, automorphism group, Tits alternative,
residually finite
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Mathematical Subject Classification
Primary: 20F28, 20F65
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Milestones
Received: 5 September 2019
Revised: 10 June 2021
Accepted: 10 June 2021
Published: 15 October 2021
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