On the Totally Geodesic Commensurability Spectrum of Arithmetic Locally Symmetric Spaces.

Abstract

Mark Kac famously posited in 1966, “can you hear the shape of a drum?” This question simply and elegantly summarizes our quest in spectral geometry to find collections of topological or geometric data, in this case the Laplace spectrum, which captures a Riemannian manifold's “geometric class.” The work of Milnor, Vigneras, Sunada, and others by the 1980’s showed that there are (infinitely many) examples of isospectral nonisometric spaces, however their methods always produce commensurable spaces. Since then much work has been done determining to what extent the rational length spectrum determines a commensurability class. For example, Reid, Chinburg, Hamilton, and Long showed that if two 2- or 3-dimensional arithmetic hyperbolic manifolds are length commensurable then they are commensurable. However in 2009, Prasad and Rapinchuk produced examples of noncommensurable, length commensurable hyperbolic manifolds.

In this thesis, we consider a different spectrum, the set of proper nonflat finite volume totally geodesic subspaces, and use this to prove new results that distinguish between commensurability classes of arithmetic locally symmetric spaces. We show that this spectrum of totally geodesic subspaces determines the commensurability class of arithmetic locally symmetric spaces coming from quadratic forms. In particular, this collection of spaces includes all even dimensional arithmetic locally symmetric spaces and “half” of all odd dimensional arithmetic hyperbolic spaces. In order to prove our results, we establish a correspondence between the local invariants of a quadratic form and the Tits index of its isometry group. These techniques enabled us to prove several new results as well as give an alternate proof of Maclachlan’s parametrization of even dimensional arithmetic hyperbolic manifolds.