Strong spectral gap for geometrically finite subgroups of SO(n, 1)

Vortrag von Dr. Pratyush Sarkar

Datum: 05.11.25  Zeit: 13.30 - 14.30  Raum: Y27H28

Let Γ be a geometrically finite subgroup of G := SO(d + 1, 1) for d ≥ 1. By foundational works of Elstrodt, Patterson, Sullivan, and Lax–Phillips, we know that if the critical exponent of Γ is greater than d/2, then the Laplace⁠–Beltrami operator on the L² space of the associated hyperbolic manifold has a spectral gap. 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 L²(G/Γ). It is further known that if the critical exponent is greater than d - 1, then there exists such a gap which also applies for the non-spherical complementary series, called strong spectral gap. This begs the question 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.