Boundedness of diffeomorphism groups of manifold pairs -- Circle case --

In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group $\mathcal{D}\equiv \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M,N{)}_0$ of a closed manifold pair $(M,N)$ with $\dim N\ge 1$, first we clarify the relation among the fragmentation norm, the conjugation generated norm, the commutator length $cl$ and the commutator length with support in balls $clb$ and show that $\mathcal{D}$ is weakly simple relative to a union of some normal subgroups of $\mathcal{D}$. For the boundedness of these norms, this paper focuses on the case where $N$ is a union of $m$ circles. In this case, the rotation angle on $N$ induces a quasimorphism $\nu :\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{t}(M,N{)}_0\to {\mathbb{R}}^m$, which determines a subgroup $A$ of ${\mathbb{Z}}^m$ and a function $\widehat{\nu }:\mathcal{D}\to {\mathbb{R}}^m\mathrm{/}A$. If $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}A=m$, these data leads to an upper bound of $clb$ on $\mathcal{D}$ modulo the normal subgroup $\mathcal{G}\cong {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_c(M-N{)}_0$. Then, some upper bounds of $cl$ and $clb$ on $\mathcal{D}$ are obtained from those on $\mathcal{G}$. As a consequence, the group $\mathcal{D}$ is uniformly weakly simple and bounded when $\dim M\mathrm{\ne }2,4$. On the other hand, if $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}A<m$, then the group $\mathcal{D}$ admits a surjective quasimorphism, so it is unbounded and not uniformly perfect. We examine the group $A$ in some explicit examples.