CUP PRODUCT ON RELATIVE BOUNDED COHOMOLOGY

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 G1 *A G2 are zero for every positive degree, assuming that free factors Gi 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 Z * Z and Zn*Zd Zm, where d is the greatest common divisor of n and m, are zero for every positive degree.