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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian
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by Andrew Sale and Tim Susse PDF
Trans. Amer. Math. Soc. 372 (2019), 7785-7803 Request permission

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
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7785-7803
  • MSC (2010): Primary 20E36, 20F28, 20F55, 20F65
  • DOI: https://doi.org/10.1090/tran/7897
  • MathSciNet review: 4029681