J. Korean Math. Soc. 2023; 60(4): 823-833
Online first article June 13, 2023 Printed July 1, 2023
https://doi.org/10.4134/JKMS.j220345
Copyright © The Korean Mathematical Society.
HeeSook Park
Sunchon National University
In this paper, we define cup product on relative bounded cohomology, and study its basic properties. Then, by extending it to a more generalized formula, we prove that all cup products of bounded cohomology classes of an amalgamated free product \( G_{1}\ast_{A}G_{2} \) are zero for every positive degree, assuming that free factors \( G_i \) are amenable and amalgamated subgroup \( A \) is normal in both of them. As its consequences, we show that all cup products of bounded cohomology classes of the groups \( \mathbb{Z} \ast \mathbb{Z} \) and \( \mathbb{Z}_{n} \ast_{\mathbb{Z}_{d}}\mathbb{Z}_m \), where \( d \) is the greatest common divisor of \( n \) and \( m \), are zero for every positive degree.
Keywords: Relative bounded cohomology, cup product, amalgamated free product
MSC numbers: Primary 55N99; Secondary 18G99
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