
TL;DR
This paper classifies isospectral but nonisometric manifolds, especially planar domains, using group theory and introduces properties that characterize these manifolds, revealing a surprising link to finite simple groups.
Contribution
It provides a classification of length equivalent manifolds satisfying certain properties, connecting their structure to finite simple groups and extending the Gassman-Sunada method.
Findings
Classifies length equivalent manifolds with properties FF, MAX, PAIR, INV.
Shows these manifolds arise from the Gassman-Sunada method.
Establishes a deep connection between these manifolds and finite simple groups.
Abstract
Virtually every known pair of isospectral but nonisometric manifolds - with as most famous members isospectral bounded -planar domains which makes one "not hear the shape of a drum" [13] - arise from the (group theoretical) Gassman-Sunada method. Moreover, all the known -planar examples (so counter examples to Kac's question) are constructed through a famous specialization of this method, called transplantation. We first describe a number of very general classes of length equivalent manifolds, with as particular cases isospectral manifolds, in each of the constructions starting from a given example that arises itself from the Gassman-Sunada method. The constructions include the examples arising from the transplantation technique (and thus in particular the planar examples). To that end, we introduce four properties - called FF, MAX, PAIR and INV - inspired by…
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