Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups

Abstract

In this two part work we prove that for every finitely generated subgroup $\Gamma <\text{Out}(F_n)$, either $\Gamma$ is virtually abelian or $H_b^2(\Gamma ;\mathbb{R})$ contains an embedding of ${\ell }^1$. The method uses actions on hyperbolic spaces, for purposes of constructing quasimorphisms. Here in Part I, after presenting the general theory, we focus on the case of infinite lamination subgroups $\Gamma$ - those for which the set of all attracting laminations of all elements of $\Gamma$ is infinite - using actions on free splitting complexes of free groups.

73 pages. The latest version corrects errors involving axial actions, and has been shortened by splitting off some material now found in our preprint "Second bounded cohomology and WWPD" (arXiv:TBA)

References in corpus (4)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Subgroup classification in Out(F_n)
• Relative free splitting and free factor complexes I: Hyperbolicity
• A short proof of Handel and Mosher's alternative for subgroups of $\text{Out}(F_N)$

Cited by in corpus (1)

• Second bounded cohomology and WWPD