M.Sc Thesis | |

M.Sc Student | Dolev Or |
---|---|

Subject | Priodic Euclidean CAT(0) Cube Complexes |

Department | Department of Mathematics |

Supervisor | PROF. Michah Sageev |

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, *E*^{n}, for
which there exists a group acting on *E*^{n} 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 *E*^{2}
or *E*^{3}. 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.