Owen Baker
Owen Baker

Ph.D. (2011) Cornell University

First Position
Dissertation
Advisor:
Research Area:
Abstract: The outer automorphism group Out(F_{n}) of a finite rank free group admits a proper, cocompact action on a contractible space X_{n} known as Outer space. Likewise, the outer automorphism group GL(n,Z) of a finite rank free abelian group admits such an action on the homogeneous space Y_{n} of positive definite quadratic forms. Both spaces have been used to study the homology of the respective groups. In this thesis, a Jacobian map J from X_{n} to Y_{n} compatible with the two actions is investigated. The image of J is discussed. It is shown that the preimage of Soulé’s wellrounded retract of Y_{3} is a deformation retract of X_{3}. The quotient K of this preimage by the kernel of the natural map from Out(F_{n}) to GL(n,Z) is an EilenbergMacLane space for this kernel. A 2dimensional subspace Ã of K is found whose integral homology groups surject onto those of the kernel. The kernel of the map H_{2}(Ã;Z) to H_{2}(K;Z) is shown to be generated by two elements as a GL_{3}(Z)module.