On Small Gaps in the Length Spectrum

TL;DR

This paper investigates the size of gaps in the length spectrum of negatively curved manifolds, establishing bounds and generic properties, with implications for understanding geometric and spectral characteristics.

Contribution

It provides new bounds for gaps in the length spectrum and shows that small gaps are topologically generic in certain settings.

Findings

Existence of exponential lower bounds for gaps with algebraic fundamental groups

Small gaps are topologically generic for surfaces and negatively curved metrics

Potential for gaps not being too small for most metrics

Abstract

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 (Theorem 3.1), and for the space of negatively curved metrics (Theorem 4.1). While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. One result in this direction is presented in Section 5.