# Non-escape of mass for automorphic forms in hyperbolic 4-manifolds

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Abstract: The arithmetic quantum unique ergodicity (AQUE) conjecture predicts that the L^2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”, and it can be a delicate matter to rule out such a possibility. This was achieved by Soundararajan for arithmetic surfaces, which when combined with celebrated work of Lindenstrauss completed the proof of AQUE for surfaces.

We establish non-escape of mass for Hecke-Maass cusp forms on a congruence quotient of hyperbolic 4-space. Unlike in the setting of hyperbolic 2- or 3-manifolds (for which AQUE has been proved), the number of terms in the Hecke relations is unbounded, which prevents us from applying Cauchy-Schwarz. We instead view the isometry group as a group of quaternionic matrices and rely on non-commutative unique factorization, along with certain crucial structural features of the Hecke relations. Joint work with Zvi Shem-Tov.