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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian


Authors: Andrew Sale and Tim Susse
Journal: Trans. Amer. Math. Soc. 372 (2019), 7785-7803
MSC (2010): Primary 20E36, 20F28, 20F55, 20F65
DOI: https://doi.org/10.1090/tran/7897
Published electronically: September 10, 2019
MathSciNet review: 4029681
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Abstract: We generalize the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph $\Gamma$ of a right-angled Coxeter group $W_\Gamma$ so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group $F_2$ as a quotient. When $\operatorname {Out}(W_\Gamma )$ is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite or equivalently $\Gamma$ contains no SIL.


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Additional Information

Andrew Sale
Affiliation: Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall (Keller Hall 401A), Honolulu, Hawaii 96822
MR Author ID: 1075109

Tim Susse
Affiliation: Department of Mathematics, Bard College at Simon’s Rock, Great Barrington, Massachusetts 01230
MR Author ID: 1126468

Received by editor(s): September 19, 2017
Received by editor(s) in revised form: February 8, 2019
Published electronically: September 10, 2019
Article copyright: © Copyright 2019 American Mathematical Society