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Second bounded cohomology and WWPD

Michael Handel, Lee Mosher

TL;DR

This work develops a robust WWPD framework as a weaker alternative to WPD to study the second bounded cohomology $H^2_b(\Gamma;\mathbb{R})$, culminating in the Global WWPD Theorem. The authors introduce multiple equivalent WWPD formulations, prove their equivalence, and construct quasimorphisms from WWPD elements to obtain embedded copies of $\ell^1$ in $H^2_b(\Gamma;\mathbb{R})$ under a global WWPD hypothesis. The Global WWPD Theorem is proved via a Kaloujnin–Krasner embedding that replaces wreath-product hypotheses, and in the special case $N=\Gamma$ reduces to obtaining an independent pair of loxodromic WWPD elements yielding an $\ell^1$-embedding. The results have applications to subgroups of $\mathrm{Out}(F_n)$ and related areas, providing structural insights into when $H^2_b(\Gamma;\mathbb{R})$ has large dimension and linking dynamical group actions on hyperbolic spaces to bounded cohomology.

Abstract

Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded cohomology using a 'really' weak property discontinuity property known as WWPD that was introduced by Bestvina, Bromberg, and Fujiwara.

Second bounded cohomology and WWPD

TL;DR

This work develops a robust WWPD framework as a weaker alternative to WPD to study the second bounded cohomology , culminating in the Global WWPD Theorem. The authors introduce multiple equivalent WWPD formulations, prove their equivalence, and construct quasimorphisms from WWPD elements to obtain embedded copies of in under a global WWPD hypothesis. The Global WWPD Theorem is proved via a Kaloujnin–Krasner embedding that replaces wreath-product hypotheses, and in the special case reduces to obtaining an independent pair of loxodromic WWPD elements yielding an -embedding. The results have applications to subgroups of and related areas, providing structural insights into when has large dimension and linking dynamical group actions on hyperbolic spaces to bounded cohomology.

Abstract

Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded cohomology using a 'really' weak property discontinuity property known as WWPD that was introduced by Bestvina, Bromberg, and Fujiwara.

Paper Structure

This paper contains 14 sections, 6 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Weakening weak proper discontinuity
  3. Basic concepts.
  4. Hyperbolic actions.
  5. $2^{\text{nd}}$ bounded cohomology.
  6. Variant definitions of WWPD
  7. WWPD (\ref{['ItemWWPDAcyl']}): $h$ is equivariantly acylindrical:
  8. WWPD (\ref{['ItemEqDiscYes']}): $h$ has equivariantly discrete endpoint pairs:
  9. WWPD (\ref{['ItemWWPDQuasiAxes']}): $h$ has equivariantly bounded projections
  10. WWPD (\ref{['ItemBadRayNo']}): $h$ is equivariantly acylindrical (variant):
  11. Equivalence $g \sim h$ in a hyperbolic group action
  12. Remark.
  13. The Global WWPD Theorem
  14. Some notations:

Key Result

Corollary 2.4

A loxodromic $h \in \Gamma$ satisfies the individual WPD property if and only if $h$ satisfies WWPD and the subgroup $\mathop{\mathrm{\mathsf{Stab}}}\nolimits(\partial_\pm h) < \Gamma$ is virtually cyclic.

Theorems & Definitions (22)

  • Definition 2.1: Osin:AcylHyp
  • Definition 2.2: BestvinaFujiwara:bounded
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • proof : Proof of Proposition \ref{['PropDefWWPD']}
  • Definition 2.7
  • ...and 12 more