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

Kazuhiko Fukui, Tatsuhiko Yagasaki

TL;DR

The paper develops a comprehensive framework to study boundedness of conjugation-invariant norms on Diffiffeomorphism groups of manifold pairs, emphasizing the circle-case. It connects fragmentation, commutator-lengths, and conjugation-generated norms, and introduces vector- and lattice-valued quasimorphisms obtained from rotation angles on circles. A key dichotomy hinges on the rank of the lattice $A$ associated with these rotations: if ${\rm rank}\,A = m$, upper bounds for $clb$ (and hence boundedness) arise; if ${\rm rank}\,A < m$, unboundedness and lack of uniform perfectness follow due to a surjective quasimorphism. The article proves uniform weak simplicity and boundedness for most dimensions ($\dim M \neq 2,4$) and provides explicit examples and factorization techniques to realize these bounds, with broad implications for foliations and Seifert-fibered settings.

Abstract

In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group ${\mathcal D} \equiv {\rm Diff}(M,N)_0$ of a closed manifold pair $(M, N)$ with $\dim N \geq 1$, 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 $ν: {\rm Isot}(M, N)_0 \to {\Bbb R}^m$, which determines a subgroup $A$ of ${\Bbb Z}^m$ and a function $\widehatν : {\mathcal D} \to {\Bbb R}^m/A$. If ${\rm rank}\,A = m$, these data leads to an upper bound of $clb$ on ${\mathcal D}$ modulo the normal subgroup ${\mathcal G} \cong {\rm Diff}_c(M - N)_0$. 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 $\dim M \neq 2,4$. On the other hand, if ${\rm rank}\,A < m$, 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.

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

TL;DR

The paper develops a comprehensive framework to study boundedness of conjugation-invariant norms on Diffiffeomorphism groups of manifold pairs, emphasizing the circle-case. It connects fragmentation, commutator-lengths, and conjugation-generated norms, and introduces vector- and lattice-valued quasimorphisms obtained from rotation angles on circles. A key dichotomy hinges on the rank of the lattice associated with these rotations: if , upper bounds for (and hence boundedness) arise; if , unboundedness and lack of uniform perfectness follow due to a surjective quasimorphism. The article proves uniform weak simplicity and boundedness for most dimensions () and provides explicit examples and factorization techniques to realize these bounds, with broad implications for foliations and Seifert-fibered settings.

Abstract

In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group of a closed manifold pair with , first we clarify the relation among the fragmentation norm, the conjugation generated norm, the commutator length and the commutator length with support in balls and show that is weakly simple relative to a union of some normal subgroups of . For the boundedness of these norms, this paper focuses on the case where is a union of circles. In this case, the rotation angle on induces a quasimorphism , which determines a subgroup of and a function . If , these data leads to an upper bound of on modulo the normal subgroup . Then, some upper bounds of and on are obtained from those on . As a consequence, the group is uniformly weakly simple and bounded when . On the other hand, if , then the group admits a surjective quasimorphism, so it is unbounded and not uniformly perfect. We examine the group in some explicit examples.
Paper Structure (31 sections, 29 theorems)

This paper contains 31 sections, 29 theorems.

Table of Contents

  1. Introduction and statement of results
  2. Conjugation invariant norms and quasimorphisms
  3. Conjugation invariant norms --- Weak simplicity
  4. Conjugation invariant norms modulo a subgroup
  5. Vector-valued quasimorphisms
  6. A lattice-valued map induced from a quasimorphism
  7. The case $\ell < m$
  8. The case $\ell = m$
  9. The case $m = 1$
  10. Restriction and Reduction of Diagarm $(\sharp)$ in Fact \ref{['fact_setting']}
  11. Diffeomorphism groups of manifold pairs
  12. Basic notions
  13. Groups of diffeomorphisms and isotopies of manifold pairs
  14. Isotopy extension theorem
  15. Fragmentation Lemma and the norms $\eta$, $\zeta$, $clb$ on ${\rm Diff}^r_c(M,N)_0$
  16. ...and 16 more sections

Key Result

Theorem 1.1

Suppose $M$ is a connected $C^\infty$ manifold without boundary and $N$ is a proper $C^\infty$ submanifold of $M$ with $\dim N \geq 1$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.1
  • Theorem 1.3
  • Example 1.1
  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • ...and 65 more