Seminar

By foundational works of Elstrodt, Patterson, Sullivan, and Lax–Phillips, we know that the Laplace–Beltrami operator on the L² space of a (d + 1)-dimensional geometrically finite hyperbolic manifold has a spectral gap if and only if the critical exponent is strictly greater than d/2. In a representation theoretic language, this means that there exists a gap below the critical exponent for which the corresponding spherical complementary series do not occur in the L² space of the frame bundle. It is further known that if the critical exponent is greater than d - 1, then there exists such a gap which applies for the non-spherical complementary series as well, called strong spectral gap. This begs the question, and conjectured to hold by Mohammadi–Oh, whether the constraint on the critical exponent can be improved to the optimal one, d/2. Inspired by prior works on the relationship between spectral gap and dynamics, in a joint work with Dubi Kelmer and Osama Khalil, we use exponential mixing of the frame flow to answer the above question in the positive.