Summary: THE LENGTH SPECTRUM OF A COMPACT
CONSTANT CURVATURE COMPLEX IS DISCRETE
NOEL BRADY 1 AND JON MCCAMMOND 2
In this short note we prove that the length spectrum of a compact constant
curvature complex is discrete. After recalling the relevant definitions and reducing
to a relatively simple situation, the result follows easily from a foundational result
about real semialgebraic sets and the MorseSard theorem. We conclude with a
conjecture which remains open, a few remarks and an easy application.
Definition 1 (Constant curvature complexes). A constant curvature complex with
curvature # is a polyhedral cell complex where each cell has been assigned a metric
of curvature # with the requirement that the induced metrics agree on the overlaps.
In the positive curvature case, there is an additional restriction that each closed
polyhedral cell must lie in some open hemisphere. The generic term is M # complex
where # is the curvature which all of the cells have in common. In practice, con
stant curvature complexes can be rescaled so that # is 1, 0 or -1 and subdivided
into simplices. Thus, we only need to consider piecewise spherical, piecewise Eu
clidean and piecewise hyperbolic simplicial complexes. See  for a comprehensive
discussion of these types of complexes.
It is an early foundational result of Bridson that compact M # complexes are
geodesic metric spaces. In other words, every pair of points in a compact M #