Craig J. Sutton

Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions

Abstract. We generalize Sunadas method to produce new examples of closed, locally non-isometric manifolds which are isospectral. In particular, we produce pairs of isospectral, simply-connected, locally non-isometric normal homogeneous spaces. These pairs also allow us to see that in general group actions with discrete spectra are not determined up to measurable conjugacy by their spectra. In particular, we show this for lattice actions.

Spectral isolation of naturally reductive metrics on simple Lie groups

We show that within the class of left-invariant naturally reductive metrics

Measures invariant under the geodesic flow and their projections

{\mathcal{M}_{{\rm Nat}}(G)}

Equivariant isospectrality and Sunada’s method

on a compact simple Lie group G, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.

Isospectral surfaces with distinct covering spectra via Cayley graphs

We demonstrate that every closed manifold of dimension at least two admits smooth metrics with respect to which the length spectrum is not a discrete subset of the real line. In contrast, we show that the length spectrum of any real analytic metric on a closed manifold is a discrete subset of the real line. In particular, the length spectrum of any closed locally homogeneous space forms a discrete set.

Spectral isolation of bi-invariant metrics on compact Lie groups

Let S n be the n-sphere of constant positive curvature. For n > 2, we will show that a measure on the unit tangent bundle of S 2n , which is even and invariant under the geodesic flow, is not uniquely determined by its projection to S 2n .