Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces

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.Comment: 34 Pages. Corrected typos. Added references. Improved expositio

Periodicity and Growth in a Lattice Gas with Dynamical Geometry

We study a one-dimensional lattice gas "dynamical geometry model" in which local reversible interactions of counter-rotating groups of particles on a ring can create or destroy lattice sites. We exhibit many periodic orbits and and show that all other solutions have asymptotically growing lattice length in both directions of time. We explain why the length grows as $\sqrt{t}$ in all cases examined. We completely solve the dynamics for small numbers of particles with arbitrary initial conditions.Comment: 18 pages, LaTe