M.Sc Student | Kassis Ameer |
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Subject | Equidistribution Through Thickening and Arithmetic Extension |

Department | Department of Mathematics |

Supervisor | Professor Uri Shapira |

Full Thesis text |

The main object of this thesis is to
display two applications of the thickening trick around compact orbits in
homogeneous spaces. In particular, we use it in order to prove two results concerning
equidistribution of some families of the horocycle and geodesic orbits in the space
of the unimodular lattices in the two dimensional Euclidean space. We start by
introducing lattices in the n-dimensional real Euclidean space R^{n},
mainly as a motivation for the more general case, which is lattices in locally
compact groups. A well known fact is that locally compact groups carry
invariant measures with respect to translations. This establishes the existence
of invariant measures on their homogeneous spaces, with respect to the group
actions. After introducing the building blocks, we characterize and study
periodic horocycle lattices, and prove that they are equidistributed. Along the
way, we prove a special case of the Howe-Moore theorem, and present the notion
of tubes which is the basic object of the thickening trick. Using tools from
dynamical analysis on p-arithmetic homogeneous spaces, we establish a similar
result regarding geodesics.