Boundedness of bundle diffeomorphism groups over a circle

Abstract

In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle $π: M \to S^1$ with fiber $N$ and structure group $Γ$ and $r \in {\Bbb Z}_{\geq 0} \cup \{ \infty \}$ we distinguish an integer $k = k(π, r) \in {\Bbb Z}_{\geq 0}$ and construct a function $\widehatν : {\rm Diff}_π(M)_0 \to {\Bbb R}_k$. When $k \geq 1$, it is shown that the bundle diffeomorphism group ${\rm Diff}_π(M)_0$ is uniformly perfect and $clb_π\,{\rm Diff}^r_π(M)_0 \leq k+3$, if ${\rm Diff}_{ρ, c}(E)_0$ is perfect for the trivial fiber bundle $ρ: E \to {\Bbb R}$ with fiber $N$ and structure group $Γ$. On the other hand, when $k = 0$, it is shown that $\widehatν$ is a unbounded quasimorphism, so that ${\rm Diff}_π(M)_0$ is unbounded and not uniformly perfect. We also describe the integer $k$ in term of the attaching map $ϕ$ for a mapping torus $π: M_ϕ\to S^1$ and give some explicit examples of (un)bounded groups.