Spheres and projections for Out(Fn).

The outer automorphism group $\mathrm {Out}(F_{2g})$ of a free group on $2g$ 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 $S^1\times S^2$ into the arc complex of $S_{g,1}$. Using this, we show that $\mathrm {Map} (S_{g,1})$ is a Lipschitz retract of $\mathrm {Out}(F_{2g})$. We use another ‘subsurface projection’ to give a simple proof of a result of Handel and Mosher [‘Lipschitz retraction and distortion for subgroups of $\mathrm {Out}(F_n)$’, 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 $\mathrm {Out}(F_n)$. [ABSTRACT FROM PUBLISHER]