This thesis presents the first steps toward a classification of non-positively curved cube complexes called state complexes. A state complex is a configuration space for a reconfigurable system, i.e., an abstract system in which local movements occur in some discrete manner. Reconfigurable systems can be used to describe, for example, situations arising in chemistry (digital microfluidics), biology (protein folding), robotics (motion-planning algorithms), and psychology (media theory). We consider state complexes in relation to a similar class of objects, called special cube complexes, which in turn leads to a discussion of clean VH-complexes and right-angled Artin groups. Group theoretic, topological, and geometric methods are used throughout.
We find that state complexes are non-positively curved metric spaces, aspherical topological spaces, and have fundamental groups that are subgroups of right-angled Artin groups. In addition, we find that the property state is preserved when taking finite products, as is true in the case of special. Unlike special, however, state is not inherited by convex subcomplexes or finite covers; the latter fact is somewhat surprising to experts. Several other classification results are presented, along with methods for realization; we conclude with directions for further study. Numerous examples and figures supplement the text.
|Advisor:||Ghrist, Robert W.|
|School:||University of Illinois at Urbana-Champaign|
|School Location:||United States -- Illinois|
|Source:||DAI-B 71/01, Dissertation Abstracts International|
|Keywords:||Cube complexes, Curved cube complexes, State complexes|
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