M.Sc Thesis


M.Sc StudentDolev Or
SubjectPriodic Euclidean CAT(0) Cube Complexes
DepartmentDepartment of Mathematics
Supervisor PROF. Michah Sageev


Abstract

The goal of this thesis is to study and classify Periodic Euclidean CAT(0) cube complexes. Those are sub-complexes of the standard cubulation of the Euclidean space, En, for which there exists a group acting on En leaving the subcomplex invariant such that this action is essential and cofinite on the hyperplanes. Such complexes arise naturally when studying skewer complexes, which have drawn interest in recent years, due to the fact that any skewer complex can be embedded into some Euclidean space satisfying the above properties.

In this thesis we focus on classifying all Periodic Euclidean complexes which embed into E2 or E3. In the two-dimensional complex we use the fact that hyperplanes are connected sets and embed into R to deduce they are either lines, rays or intervals. In addition, we prove that all hyperplanes of a two-dimensional Periodic Euclidean complex have the same type from the above three and use that in order to provide a full description of all such complexes.

We then generalize this concept into n-dimensional Periodic Euclidean complexes by looking at hyperlines which are intersections of n-1 hyperplanes. Those hyperlines are connected and embed into R and thus again can be classified into one of three types. We prove that parallel hyperlines must be of the same types and thus provide a classification the complexes into finitely many types. We then split our analysis based on this classification to provide a full description of all three-dimensional Periodic Euclidean complexes and also obtain some information about Periodic Euclidean complexes of higher dimensions.