Spheres and projections for Out(Fn)

Abstract

The outer automorphism group Out ( F 2 g ) of a free group on 2 g generators naturally contains the mapping class group of a punctured genus g surface S g , 1 as a subgroup. We define a ‘subsurface projection’ of the sphere complex of the connected sum of n copies of S 1 × S 2 into the arc complex of S g , 1 . Using this, we show that Map ( S g , 1 ) is a Lipschitz retract of Out ( F 2 g ) . We use another ‘subsurface projection’ to give a simple proof of a result of Handel and Mosher [‘Lipschitz retraction and distortion for subgroups of Out ( F n ) ’, Geom. Topol. 17 (2013) 1535–1580] stating that stabilizers of conjugacy classes of free splittings and corank 1 free factors in a free group F n are Lipschitz retracts of Out ( F n ) .