Spheres and Projections for $\mathrm{Out}(F_n)$

TL;DR

This paper introduces a subsurface projection technique for the sphere complex in free groups, demonstrating that certain subgroups are Lipschitz retracts of Out(F_n), thereby providing new geometric insights into the structure of Out(F_n).

Contribution

It defines a novel subsurface projection for the sphere complex and proves that specific subgroups are Lipschitz retracts of Out(F_n), extending geometric understanding of free group automorphisms.

Findings

Subsurface projection maps the sphere complex into the arc complex.

Certain subgroups are shown to be Lipschitz retracts of Out(F_n).

Provides a simplified proof of known subgroup retraction results.

Abstract

The outer automorphism group Out(F_2g) of a free group on 2g generators naturally contains the mapping class group of a punctured surface as a subgroup. We define a subsurface projection of the sphere complex of the connected sum of n copies of S^1 x S^2 into the arc complex of the surface and use this to show that this subgroup is a Lipschitz retract of Out(F_2g). We also use subsurface projections to give a simple proof of a result of Handel and Mosher [HM10] stating that stabilizers of conjugacy classes of free splittings and corank 1 free factors in a free group Fn are Lipschitz retracts of Out(F_n).