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.