Purdue University Graduate School
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Spectral Rigidity and Flexibility of Hyperbolic Manifolds

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posted on 2023-07-31, 15:48 authored by Justin E KatzJustin E Katz
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.


Degree Type

  • Doctor of Philosophy


  • Mathematics

Campus location

  • West Lafayette

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