Abstract: Outer automorphisms of a free group are a fundamental example in geometric group theory and low dimensional topology. One approach to their study is by analogy with the mapping class groups of surfaces. This analogy is made concrete by the natural inclusions Mod(S) -> Out(F) that occur whenever S has free fundamental group. Outer automorphisms in the image of these inclusions are called

*geometric*. In 1992, Bestvina and Handel gave an algorithm for deciding when an irreducible outer automorphism is geometric, shortly followed by an algorithm due to Cohen and Lustig to decide when a Dehn twist is geometric. Aside from these two special cases, the problem has remained open. I will describe current joint work with Yulan Qing and Derrick Wigglesworth to give an algorithm to decide when a general outer automorphism is geometric, which recovers the other two algorithms as special cases.