The zero-norm subspace of bounded cohomology

Abstract

Let $\Sigma$ be a closed, orientable surface of genus > 1. In this paper, non-trivial elements $\alpha$ of the third bounded cohomology $H_b^3(\Sigma ;\mathbf{R})$ with $\Vert \alpha \Vert =0$ are given constructively by using both a hyperbolic metric and a singular euclidean metric on $\Sigma \times \mathbf{R}$ . Furthermore, it is shown that the dimension of the subspace $N^3(\Sigma )$ of $H_b^3(\Sigma ;\mathbf{R})$ consisting of zero-norm elements is the cardinality of the continuum.