The Dimension of the Torelli Group for Out(F_n)
by
Kai-Uwe Bux
University of Virginia
Coauthors: Mladen Bestvina and Dan Margalit

Abelianizing induces a natural homomorphism from Out(F_n) to GL_n(Z). This homomorphism is surjective. In analogy to the situation in mapping class groups, we call its kernel the Torelli subgroup of Out(F_n), and we denote it by T_n. We show:

Theorem. For n >= 3, the following hold:

1. The Torelli subgroup T_n has an Eilenberg-Mac Lane complex of dimension 2n - 4.

2. Its integral homology in top dimension, H_2n-4(T_n;Z), is not finitely generated. In particular, T_n is not of type FP_2n-4.

Our approach is purely geometric: we construct an Eilenberg-Mac Lane space of dimension 2n-3 as a quotient of the spine of Outer Space. Then, we use combinatorial Morse theory to show that it is homotopy equivalent to a space of dimension 2n-4. Finally, we exhibit explicitly an infinite family of independent homology classes.

We note that our methods also yield a geometric proof of the classical fact (due to Magnus) that T_n is finitely generated.

We also note that a spectral sequence argument allows one to deduce similar statements for the Torelli subgroup of Aut(F_n).

Date received: April 11, 2006