Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2023; 60(4): 823-833

Online first article June 13, 2023      Printed July 1, 2023

Copyright © The Korean Mathematical Society.

Cup product on relative bounded cohomology

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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