Classical Projective Geometry and Modular Varieties

Abstract

This thesis uses projective geometry over both the real numbers and finite fields to explore the structure of locally symmetric spaces, following the 1989 work of MacPherson and McConnell. We detail fundamental projective geometry concepts such as cross ratios, harmonic quadruples, and Desargues' theorem before extending these ideas to the finite field setting. The heart of the exposition centers on the construction of a cell complex $W \subset X$, is a deformation retract of a modular variety $\Gamma(p) \backslash X$ associated with the arithmetic group $\Gamma(p)$.