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ERIC Number: ED526024
Record Type: Non-Journal
Publication Date: 2009
Pages: 109
Abstractor: As Provided
Reference Count: 0
ISBN: ISBN-978-1-1095-8384-7
ISSN: N/A
On State Complexes and Special Cube Complexes
Peterson, Valerie J.
ProQuest LLC, Ph.D. Dissertation, University of Illinois at Urbana-Champaign
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. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]
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Publication Type: Dissertations/Theses - Doctoral Dissertations
Education Level: N/A
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A