Boundedness of bundle diffeomorphism groups over a circle

TL;DR

This paper investigates the boundedness and perfectness of bundle diffeomorphism groups over a circle, introducing a key integer parameter and constructing a function to analyze their algebraic properties.

Contribution

It introduces a new integer invariant for bundle diffeomorphism groups over a circle and characterizes their boundedness and perfectness based on this invariant.

Findings

When k ≥ 1, the group is uniformly perfect with bounded commutator length.

When k = 0, the group admits an unbounded quasimorphism and is not uniformly perfect.

Explicit examples of bounded and unbounded groups are provided.

Abstract

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