On small gaps in the length spectrum

Abstract

We discuss upper and lower bounds for the size of gaps in thelength spectrum of negatively curved manifolds. For manifolds with algebraicgenerators for the fundamental group, we establish the existence of exponentiallower bounds for the gaps. On the other hand, we show that the existenceof arbitrarily small gaps is topologically generic: this is established both forsurfaces of constant negative curvature (Theorem 3.1) and for the space ofnegatively curved metrics (Theorem 4.1). While arbitrarily small gaps aretopologically generic, it is plausible that the gaps are not too small for almostevery metric. One result in this direction is presented in Section 5.

Locations

• arXiv (Cornell University) - View - PDF
• Journal of Modern Dynamics - View - PDF