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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Conditions on the monodromy for a surface group extension to be CAT(0)
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by Kejia Zhu
Proc. Amer. Math. Soc. 151 (2023), 4643-4651
DOI: https://doi.org/10.1090/proc/16518
Published electronically: July 28, 2023

Abstract:

In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py [Math. Ann. 380 (2021), pp. 449–485] give a necessary condition: given a surface-by-surface group $G$ with infinite monodromy, if $G$ is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let $G$ be a group with a normal surface subgroup $R$. Assume $G/R$ satisfies the property that for every infinite normal subgroup $\Lambda$ of $G/R$, there is an infinite finitely generated subgroup $\Lambda _0<\Lambda$ so that the centralizer $C_{G/R}(\Lambda _0)$ is finite. We then prove that if $G$ is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. This applies in particular if $G/R$ is acylindrically hyperbolic.
References
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Bibliographic Information
  • Kejia Zhu
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92507
  • MR Author ID: 1499581
  • Email: kzhumath@gmail.com
  • Received by editor(s): July 1, 2022
  • Received by editor(s) in revised form: December 21, 2022, December 22, 2022, and March 27, 2023
  • Published electronically: July 28, 2023
  • Communicated by: Genevieve S. Walsh
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 4643-4651
  • MSC (2020): Primary 20F65, 51H30, 20A05
  • DOI: https://doi.org/10.1090/proc/16518
  • MathSciNet review: 4634870