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I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup of the trace formula, which is given by equation (5.3) of the above reference. In the particular case of a compact hyperbolic surface (or finite area surface with cusps), the trace formula specializes to act as a bridge between spectral data (of the Laplace-Beltrami operator) and the length of geodesics (see Theorem 5.6 of the above reference for a precise example of such a formulation).

My question is, is there some higher dimensional analogue of the latter formulation? In other words, is there a trace formula linking spectral (eigenvalues of the Laplacian) and geometric data for higher dimensional manifolds (for example, arithmetic hyperbolic manifolds)? The question is a bit vague, but I am just looking for some general references.

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    $\begingroup$ Have you googled something like "trace formula" and "hyperbolic manifolds"? Anyway, try the book by Elstrodt, Grunewald and Mennicke. That does 3-folds, and probably has some references for higher rank. $\endgroup$
    – Kimball
    May 17, 2022 at 11:33

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