Cubical Geometry

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Mathematics is the study of patterns. The most fundamental and elementary type of pattern is a symmetry. There are finite `groups' of symmetries, such as the group consisting of the 48 distinct symmetries of a cube, and there are also infinite groups of symmetries such as the group of symmetries of the tiling of the Euclidean plane by squares. While finite groups of symmetries can be understood quite well, some infinite groups are so complex that they are provably impossible to understand. Nevertheless, many important problems in geometry and topology can be reduced to understanding the infinite group of symmetries of some infinite geometric object. This research aims to understand certain infinite groups using geometric and topological reasoning. The work involves an interplay between algebra, geometry, and topology, and will shed light on all three disciplines. A primary aim of the proposed research is to examine applications of nonpositively curved cube complexes to our understanding of infinite groups and low-dimensional topology. A secondary aim is to understand the ubiquity of groups that are `locally quasiconvex' in the sense that all their finitely generated subgroups are quasiconvex.