We discuss upper and lower bounds for the size of gaps in the
length spectrum of negatively curved manifolds. For manifolds with
algebraic generators for the fundamental group, we establish the existence
of exponential lower bounds for the gaps. On the other hand, we show that
the existence of arbitrary small gaps is topologically generic: this is
established both for surfaces of constant negative curvature, and for the
space of negatively curved metrics. While arbitrary small gaps are
topologically generic, it is plausible that the gaps are not too small for
almost every metric; we discuss one result
This is joint work with Dima Dolgopyat.