Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces

TL;DR

This paper demonstrates that the collection of totally geodesic subspaces uniquely identifies the commensurability class of certain arithmetic hyperbolic orbifolds, extending to related locally symmetric spaces.

Contribution

It establishes that totally geodesic spectra determine the commensurability class for arithmetic hyperbolic orbifolds and related spaces, using algebraic and quadratic form techniques.

Findings

Totally geodesic subspaces determine commensurability classes in hyperbolic orbifolds.

Results extend to locally symmetric spaces of types B_n and D_n.

Techniques combine algebraic group theory and quadratic forms.

Abstract

In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to prove several results about these spaces.