thesis.pdf (464.5 kB)

# Spectral Rigidity and Flexibility of Hyperbolic Manifolds

In the first part of this thesis we show that, for a given non-arithmetic closed hyperbolic$n$manifold$M$, there exist for each positive integer$j$, a set$M_1,...,M_j$of pairwise nonisometric, strongly isospectral, finite covers of$M$, and such that for each$i,i'$one has isomorphisms of cohomology groups$H^*(M_i,\Zbb)=H^*(M_{i'},\Zbb)$which are compatible with respect to the natural maps induced by the cover. In the second part, we prove that hyperbolic$2$- and$3$-manifolds which arise from principal congruence subgroups of a maixmal order in a quaternion algebra having type number$1$are absolutely spectrally rigid. One consequence of this is a partial answer to an outstanding question of Alan Reid, concerning the spectral rigidity of Hurwitz surfaces.

## History

## Degree Type

- Doctor of Philosophy

## Department

- Mathematics

## Campus location

- West Lafayette