My dissertation focuses on the existence of metrics of non-positive curvature for the simplicial complexes constructed recently by Tom Brady and Daan Krammer for the braid groups and other Artin groups of finite type. In particular, for each Artin group of finite type, there is a recently constructed finite simplicial Eilenberg-Mac Lane space known as its Brady-Krammer complex. The Brady-Krammer complexes are highly symmetric objects. Prior work on the relationship between the Brady-Krammer complexes and the theory of CAT(0)spaces has produced some positive results in low-dimensions. More specifically, the Brady-Krammer complexes of dimension at most 3 have been shown to support piecewise Euclidean metrics of non-positive curvature. Similarly, the 4dimensional Brady-Krammer complexes of type A4 and type B4 also support such metrics. In every instance, the metrics assigned respect all of the symmetries alluded to above. The main results of my dissertation show that this pattern does not extend to the Brady-Krammer complexes of type F4 and D4. These are the first negative results known about the curvature of these Brady-Krammer complexes. The proofs of my main theorems involve a combination of combinatorial results and computer calculations. These negative results are particularly striking since Ruth Charney, John Meier and Kim Whittlesey have shown that a particular complex closely related to each Brady-Krammer complex admits an asymmetric metric satisfying a weak version of non-positive curvature. Thus, one corollary of my results is that the weak asymmetric version of a CAT(0) metric (initially defined by Mladen Bestvina) is strictly weaker than the traditional version.

My dissertation focuses on the existence of metrics of non-positive curvature for the simplicial complexes constructed recently by Tom Brady and Daan Krammer for the braid groups and other Artin groups of finite type. In particular, for each Artin group of finite type, there is a recently constructed finite simplicial Eilenberg-Mac Lane space known as its Brady-Krammer complex. The Brady-Krammer complexes are highly symmetric objects. Prior work on the relationship between the Brady-Krammer complexes and the theory of CAT(0)spaces has produced some positive results in low-dimensions. More specifically, the Brady-Krammer complexes of dimension at most 3 have been shown to support piecewise Euclidean metrics of non-positive curvature. Similarly, the 4dimensional Brady-Krammer complexes of type A4 and type B4 also support such metrics. In every instance, the metrics assigned respect all of the symmetries alluded to above. The main results of my dissertation show that this pattern does not extend to the Brady-Krammer complexes of type F4 and D4. These are the first negative results known about the curvature of these Brady-Krammer complexes. The proofs of my main theorems involve a combination of combinatorial results and computer calculations. These negative results are particularly striking since Ruth Charney, John Meier and Kim Whittlesey have shown that a particular complex closely related to each Brady-Krammer complex admits an asymmetric metric satisfying a weak version of non-positive curvature. Thus, one corollary of my results is that the weak asymmetric version of a CAT(0) metric (initially defined by Mladen Bestvina) is strictly weaker than the traditional version.