Numerical study of length spectra and low-lying eigenvalue spectra of compact hyperbolic 3-manifolds (Q2715855)

The author studies the length spectra and the low eigenvalues of the Laplace-Beltrami operator for a large number of compact hyperbolic 3 manifolds. The periodic orbit sum method is used to study the first non-zero eigenvalues and to compare them with various geometric quantities: the volume, diameter, and length of the shortest periodic geodesic. The computer program `SnaPea' by Weeks is used in the investigation. NEWLINENEWLINENEWLINEThe author studies the fluctuating property of the length spectra and checks the accuracy of the low eigenvalues using the periodic orbit sum method of the trace formula. The deviation of the low lying eigenvalue spectra from the asymptotic distribution is measured using the \(\zeta\) function and the spectral distance for manifolds that have a region similar to a neighborhood of a cusped point.