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The space of non-extendable quasimorphisms

Morimichi Kawasaki, Mitsuaki Kimura, Shuhei Maruyama, Takahiro Matsushita, Masato Mimura

TL;DR

The paper develops a five-term exact sequence for the cohomology relative to the bounded subcomplex to study spaces of $G$-invariant homogeneous quasimorphisms on a normal subgroup $N$ that do not extend to the ambient group $G$, connecting these obstructions to ordinary cohomology. By introducing $N$-quasi-cocycles and a transgression map $\tau_{/b}$, the authors relate $\QQQ(N)^G / i^*\QQQ(G)$ and $\QQQ(N)^G /(\HHH^1(N)^G + i^*\QQQ(G))$ to $\HHH^2(\Gamma)$ and $\HHH^2(G)$, yielding finite-dimensionality under assumptions like $\Gamma=G/N$ boundedly $3$-acyclic or hyperbolicity of $G$. They apply the framework to flux kernels, IA-automorphism groups, and certain hyperbolic constructions (surface groups, hyperbolic mapping tori), obtaining concrete dimensions and nontrivial examples (e.g., nonzero Py-class contributions). The results yield sharp criteria for when stable commutator lengths $\mathrm{scl}_G$ and mixed scl $\mathrm{scl}_{G,N}$ are bi-Lipschitz equivalent and when they coincide, with consequences for mapping class groups, automorphism groups of free groups, and diffeomorphism groups, and they open new avenues for understanding extendability phenomena in geometric group theory.

Abstract

For a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the space of quasimorphisms and quasi-cocycles on $N$ non-extendable to $G$. To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As its application, we study the spaces associated with the kernel of the (volume) flux homomorphism, the IA-automorphism group of a free group, and certain normal subgroups of Gromov-hyperbolic groups. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the stable mixed commutator length for certain pairs of a group and its normal subgroup.

The space of non-extendable quasimorphisms

TL;DR

The paper develops a five-term exact sequence for the cohomology relative to the bounded subcomplex to study spaces of -invariant homogeneous quasimorphisms on a normal subgroup that do not extend to the ambient group , connecting these obstructions to ordinary cohomology. By introducing -quasi-cocycles and a transgression map , the authors relate and to and , yielding finite-dimensionality under assumptions like boundedly -acyclic or hyperbolicity of . They apply the framework to flux kernels, IA-automorphism groups, and certain hyperbolic constructions (surface groups, hyperbolic mapping tori), obtaining concrete dimensions and nontrivial examples (e.g., nonzero Py-class contributions). The results yield sharp criteria for when stable commutator lengths and mixed scl are bi-Lipschitz equivalent and when they coincide, with consequences for mapping class groups, automorphism groups of free groups, and diffeomorphism groups, and they open new avenues for understanding extendability phenomena in geometric group theory.

Abstract

For a pair of a group and its normal subgroup , we consider the space of quasimorphisms and quasi-cocycles on non-extendable to . To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As its application, we study the spaces associated with the kernel of the (volume) flux homomorphism, the IA-automorphism group of a free group, and certain normal subgroups of Gromov-hyperbolic groups. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the stable mixed commutator length for certain pairs of a group and its normal subgroup.

Paper Structure

This paper contains 32 sections, 82 theorems, 215 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main theorem
  4. Other applications of the main theorem
  5. On equivalences of $\mathop{\mathrm{\mathrm{scl}}}\nolimits_{G}$ and $\mathop{\mathrm{\mathrm{scl}}}\nolimits_{G,N}$
  6. The case of IA-automorphism groups of free groups
  7. Applications to volume flux homomorphisms
  8. Organization of the paper
  9. Preliminaries
  10. The spaces of non-extendable quasimorphisms
  11. Proofs of Theorems \ref{['main thm 2']} and \ref{['easy cor']}
  12. Proof of Theorem \ref{['thm surface group']}
  13. Proof of Theorem \ref{['mapping torus thm']} and a related example
  14. Other examples
  15. Cohomology classes induced by the flux homomorphism
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $l$ be an integer greater than $1$, $G = \pi_1(\Sigma_l)$ the surface group with genus $l$, and $N$ the commutator subgroup $[\pi_1(\Sigma_l),\pi_1(\Sigma_l)]$ of $\pi_1(\Sigma_l)$. Then

Figures (1)

  • Figure 1: $\alpha_1,\ldots,\alpha_l,\beta_1,\ldots,\beta_l\colon[0,1]\to \Sigma_l$

Theorems & Definitions (166)

  • Theorem 1.1: Non-zero finite dimensionality in surface groups
  • Theorem 1.2: Non-zero finite dimensionality in hyperbolic mapping tori
  • Theorem 1.3: KKMM2
  • Theorem 1.4: Five-term exact sequence of group cohomology
  • Theorem 1.5: Main Theorem
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 156 more