Ph.D. Dissertation

Title: The existence of metrics of nonpositive curvature on the Brady-Krammer complexes for finite-type Artin groups

The general area of my research is the field of geometric group theory. More specifically, my dissertation focuses on the existence of metrics of nonpositive 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 I show that the complexes for the Artin groups corresponding to the $D_4$ and $F_4$ Dynkin diagrams do not support such metrics. These are the first negative results known about these complexes.

The proof involves a combination of combinatorial results and computer calculations. More specifically, I programmed the computational algebra software GAP (Groups, Algorithms, Programming) to calculate the system of linear inequalities that would need to be satisfied by a hypothetical metric of nonpositive curvature. These inequalities are derived from the graphs which represent the codimension 2 links in the Brady-Krammer complex. In both the $D_4$ and $F_4$ cases, analyzing the linear system shows that each system has  a unique solution. This solution assigns to each simplex a metric commonly used in the study of Coxeter groups. Since it was already shown by Jon McCammond and Tom Brady that this particular metric fails to be nonpositively curved for these complexes, no such metric exists. Ruth Charney, John Meier and Kim Whittlesey have shown that Mladen Bestvina's construction also applies these complexes. In combination with my results this demonstrates that there is a distinction between weak nonpositively curved metrics (in the sense of Bestvina) and nonpositively curved metrics in the strict sense.

I have presented these results at the SUNY-Albany conference on topology and group theory in October 2003 and in the topology seminar at the University of Oklahoma in November 2003. 

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