| Spectral theory is the subfield of differential geometry which provided the solution to Kac's famous question, “Can you hear the shape of a drum?” That is, can we use the Laplace spectrum of a manifold to draw conclusions about its geometry or topology? Here we ask this question in the context of Riemannian orbifolds. Two results concerning the topological similarity of isospectral orbifolds are obtained. First it is proven that an isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many possible isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Peterson, as well as comparison geometry techniques developed by Grove and Petersen.; It is also shown that a family of isospectral orbifolds with only isolated singularities, satisfying a lower sectional curvature bound; contains orbifolds with underlying spaces of only finitely many homotopy types. Except for the spectral hypotheses, this is an alternative proof of a fact first shown by Perel'man. |
