A generating function for the Euler characteristic of out( F n)

Abstract

Let A be a group satisfying suitable homological finiteness conditions, and A' C A any torsion-free subgroup of finite index. Then the rational Euler characteristic %(A) of A is defined as x(A')/[A : A ' ] 9 where #(/4') is the usual alternating sum of the ranks of the homology groups H^A'.J.) (cf. [10]). The rational Euler characteristic has good multiplicative properties which often make it easier to compute and more appropriate for groups with torsion than the usual Euler characteristic, In addition, for many groups there is a strong connection between X(A) and number theory. Harder [5] has shown that the rational Euler characteristics of arithmetic groups are closely related to values of zeta functions. In particular, the rational Euler characteristics of the Z-points of algebraic groups can often be given in terms of Bernoulli numbers. Harer and Zagier [7] have shown that the rational Euler characteristics of mapping class groups can likewise be given as simple formulas involving Bernoulli numbers. In this paper, we concentrate on the group Out(Fn) of outer automorphims of a free group on n generators. In [3] it is shown that Out(/^) has homological finiteness properties in common with arithmetic groups and mapping class groups; specifically, it is of type VFL (cf. [1]). In general, however, very little is known about the homology of Out(/?). In this paper, we study the rational Euler characteristic %(Out(/^)), and in particular, we give an effective method for computing it. In an interesting departure from the arithmetic case, we do not know whether there is a formula for #(Out(/^)) in terms of Bernoulli numbers, though the denominators share some divisibility properties with the denominators of Bernoulli numbers. The authors would like to thank Ken Brown for many helpful comments and suggestions.