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.

## Advisor/Supervisor/Committee Chair

Dr. David Benjamin Mcreynolds## Advisor/Supervisor/Committee co-chair

Dr. Freydoon Shahidi## Additional Committee Member 2

Dr. Sai Kee Yeung## Additional Committee Member 3

Dr. Lvhou Chen