D ec 1 99 7 Solvable subgroups of Out ( F n ) are virtually abelian

Let Fn denote a free group of rank n. The group Out(Fn) contains mapping class groups of compact surfaces and maps onto GL(n,Z). It is perhaps not surprising that Out(Fn) behaves at times like a mapping class group and at times like a linear group. J. Birman, A. Lubotzky, and J. McCarthy [BLM83] showed that solvable subgroups of mapping class groups are finitely generated and virtually abelian. Of course, GL(3,Z) contains the Heisenberg group which is solvable but not virtually abelian. In this paper, we show that, with respect to the nature of solvable subgroups, Out(Fn) behaves more like mapping class groups. Theorem 1.1. Every solvable subgroup of Out(Fn) has a subgroup of index at most 3 2 that is finitely generated and free abelian.