Spectral Rigidity and Flexibility of Hyperbolic Manifolds

<pre>In the first part of this thesis we show that, for a given non-arithmetic closed hyperbolic <i>$</i><i>n</i><i>$</i> manifold <i>$</i><i>M</i><i>$</i>, there exist for each positive integer <i>$</i><i>j</i><i>$</i>, a set <i>$</i><i>M_</i><i>1</i><i>,...,M_j</i><i>$</i> of pairwise nonisometric, strongly isospectral, finite covers of <i>$</i><i>M</i><i>$</i>, and such that for each <i>$</i><i>i,i'</i><i>$</i> one has isomorphisms of cohomology groups <i>$</i><i>H^*(M_i,</i><i>\Zbb</i><i>)=H^*(M_{i'},</i><i>\Zbb</i><i>)</i><i>$</i> which are compatible with respect to the natural maps induced by the cover. In the second part, we prove that hyperbolic <i>$</i><i>2</i><i>$</i>- and <i>$</i><i>3</i><i>$</i>-manifolds which arise from principal congruence subgroups of a maixmal order in a quaternion algebra having type number <i>$</i><i>1</i><i>$</i> 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.</pre>