Chiral polytopes arising from almost-simple groups with socle PSL(2, q)
In terms of their symmetries, not many geometric objects have received more attention than polytopes. Since ancient times, polytopes have been considered with great interest, not only for aesthetic reasons, but also for scientific reasons. More recently, the theory of abstract polytopes, which generalises the concept of classical polytopes, has emerged as a powerful tool for studying symmetries. In the last decade, efforts to classify highly symmetric polytopes, including chiral and regular ones, have highly increased. Given a group ", a lot has been done in order to enumerate all polytopes of a certain type and which have " as their group of symmetries. Many results have been obtained for almost simple groups and atlases for small groups have been drawn up. However, not much is known about chiral polytopes, which are polytopes that have all possible rotational symmetries but no reflections. The purpose of this thesis is to investigate the almost simple groups with socle PSL(2, q), and to determine when they are automorphism groups of chiral polytopes. Furthermore, a comparison between regularity and chirality has emerged as a natural question. Accordingly, we also compare our results to those obtained for regularity.
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