Lecture #2: Quantitative questions in spectral geometry

In 1992, Reid posed the question of whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to Reid's question, Futer and Millichap have recently constructed innitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same rst n geodesic lengths. In the present lecture, we show that this phenomenon is surprisingly common in the arithmetic setting. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.

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