A short proof of Handel and Mosher's alternative for subgroups of $\text{Out}(F_N)$

Abstract

We give a short proof of a theorem of Handel and Mosher stating that any finitely generated subgroup of $\text{Out}(F_N)$ either contains a fully irreducible automorphism, or virtually fixes the conjugacy class of a proper free factor of $F_N$, and we extend their result to non finitely generated subgroups of $\text{Out}(F_N)$.

arXiv comment10 pages

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• The Tits alternative for the automorphism group of a free product
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Ergodic decompositions for folding and unfolding paths in Outer space
• Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups