$\operatorname {Aut}$-invariant quasimorphisms on groups
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- by Francesco Fournier-Facio and Richard D. Wade
- Trans. Amer. Math. Soc. 376 (2023), 7307-7327
- DOI: https://doi.org/10.1090/tran/8980
- Published electronically: June 21, 2023
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Abstract:
For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian.
This was known for $F_2$ by a result of Brandenbursky and Marcinkowski [Comment. Math. Helv. 94 (2019), pp. 661–687], but is new even for free groups of higher rank, settling a question of Miklós Abért. The case of graph products of finitely generated abelian groups settles a question of Michał Marcinkowski. As a consequence, we deduce that a variety of $Aut$-invariant norms on such groups are unbounded.
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Bibliographic Information
- Francesco Fournier-Facio
- Affiliation: Department of Mathematics, ETH Zürich, Switzerland
- MR Author ID: 1532652
- ORCID: 0000-0003-0386-2071
- Email: francesco.fournier@math.ethz.ch
- Richard D. Wade
- Affiliation: Mathematical Institute, Oxford University, United Kingdom
- MR Author ID: 951412
- ORCID: 0000-0001-9274-3474
- Email: wade@maths.ox.ac.uk
- Received by editor(s): February 18, 2023
- Received by editor(s) in revised form: April 28, 2023, and May 4, 2023
- Published electronically: June 21, 2023
- Additional Notes: The second author was funded by The Royal Society through a University Research Fellowship.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 7307-7327
- MSC (2020): Primary 20F65, 20E36; Secondary 20F67, 20J05
- DOI: https://doi.org/10.1090/tran/8980
- MathSciNet review: 4636691