# Limits of conjugacy classes under iterates of hyperbolic elements of Out($F$)

### Pritam Ghosh

Ashoka University, Sonipat, India

## Abstract

For a free group $F$ of finite rank such that rank $(F)≥3$, we prove that the set of weak limits of a conjugacy class in $F$ under iterates of some hyperbolic $ϕ∈Out(F)$ is equal to the collection of generic leaves and lines with endpoints in attracting fixed points of $ϕ$.

As an application we describe the ending lamination set for a hyperbolic extension of $F$ by a hyperbolic element of Out($F$) in a new way and use it to prove results about Cannon–Thurston maps for such extensions. We also use it to derive conditions for quasiconvexity of finitely generated, infinite index subgroups of $F$ in the extension group. These results generalize similar results obtained in [23] and [19] and use different techniques.

## Cite this article

Pritam Ghosh, Limits of conjugacy classes under iterates of hyperbolic elements of Out($F$). Groups Geom. Dyn. 14 (2020), no. 1, pp. 177–211

DOI 10.4171/GGD/540