Atoroidal dynamics of subgroups of Out(F_N)

Abstract

We show that for any subgroup $H$ of Out($F_N$), either $H$ contains an atoroidal element or a finite index subgroup $H'$ of $H$ fixes a nontrivial conjugacy class in $F_N$. This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel-Mosher's subgroup theorem for Out($F_N$) in the setting of irreducible elements.