Subfactor projections

TL;DR

This paper introduces a new projection method for free factors in free groups, demonstrating properties similar to subsurface projections, and uses it to analyze the action of Out(F_n) on hyperbolic spaces, with applications to hyperbolicity and growth.

Contribution

It defines subfactor projections in free groups, establishes their properties, and applies them to construct hyperbolic actions of Out(F_n), extending the understanding of free group automorphisms.

Findings

Projections satisfy properties analogous to subsurface projections.

Constructs an action of Out(F_n) on a product of hyperbolic spaces.

Proves a version of the Bounded geodesic image theorem.

Abstract

When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor projections to construct an action of Out(F_n) on a finite product of hyperbolic spaces where every automorphism with exponential growth acts with positive translation length. We also prove a version of the Bounded geodesic image theorem. In the appendix, we give a sketch of the proof of the Handel-Mosher hyperbolicity theorem for the splitting complex using (liberal) folding paths.