Bradley Forrest

Ph.D. (2009) Cornell University

First Position

Assistant professor at the Richard Stockton College of New Jersey

Dissertation

Degree Subcomplexes of Auter Space and Ribbon Graph Complexes

Advisor:

Karen Vogtmann

Research Area:

Topology and geometric group theory

Abstract: The group Aut(F_n) of automorphisms of a finitely generated free group acts properly and cocompactly on a simply-connected simplicial complex known as the degree 2 subcomplex of the spine of Auter space. In the first part of this thesis, we show that the degree 2 subcomplex contains a proper, invariant, simply-connected subcomplex K, and use K to simplify a finite presentation of Aut(F_n) given by Armstrong, Forrest, and Vogtmann. Further, we prove that K is contained in every Aut(F_n)-invariant simply-connected subcomplex of the degree 2 subcomplex.

The mapping class group of an orientable, basepointed, punctured surface (Σ,u) acts properly and cocompactly on a simplicial complex known as R_(Σ,u) the ribbon graph complex of (Σ,u). We define a filtration J₀ ⊂ J₁ ⊂ J₂ … on R_(Σ,u) and prove that J_i is i-dimensional and (i – 1)-connected.