# Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf{Out}(F_n)$. Part II: Finite lamination subgroups

## TL;DR

This paper investigates the second bounded cohomology of finitely generated subgroups of Out(F_n), focusing on finite lamination subgroups, and constructs hyperbolic actions to analyze their algebraic properties.

## Contribution

It extends the theory to finite lamination subgroups by constructing hyperbolic actions, enabling the application of the general framework from Part I.

## Key findings

- Finite lamination subgroups have hyperbolic actions compatible with the theory.
- Such subgroups are either virtually abelian or have rich second bounded cohomology.
- The work advances understanding of subgroup dynamics in Out(F_n).

## Abstract

This is the second part of a two part work in which we prove that for every finitely generated subgroup $\Gamma < \mathsf{Out}(F_n)$, either $\Gamma$ is virtually abelian or its second bounded cohomology $H^2_b(\Gamma;\mathbb{R})$ contains an embedding of $\ell^1$. Here in Part II we focus on finite lamination subgroups $\Gamma$ --- meaning that the set of all attracting laminations of elements of $\Gamma$ is finite --- and on the construction of hyperbolic actions of those subgroups to which the general theory of Part I is applicable.