The spectral gap of a group acting on a space measures how quickly correlations decay. A strong spectral gap on L²(Γ\G) — the space of square-integrable functions on the quotient — is the sharpest form: it controls not just the spherical functions (on the symmetric space) but all matrix coefficients of the regular representation.
The paper resolving a conjecture of Mohammadi and Oh (arXiv: 2603.20552) proves the strong spectral gap for geometrically finite, Zariski dense subgroups of SO(d+1,1) with critical exponent strictly greater than d/2.
Geometrically finite means the quotient has finite volume in the thick part — the group has both compact and cusp-like regions. Zariski dense means the group is as algebraically rich as possible. The critical exponent condition (> d/2) is the threshold where the Patterson–Sullivan measure has enough mass to control the spectrum.
The applications are immediate: decay rates for matrix coefficients and exponential mixing of the frame flow. The strong gap quantifies mixing — it's not just that correlations decay, but they decay at a rate determined by the gap.
The through-claim: the critical exponent is the spectral threshold. Below d/2, the group is too thin for spectral control. Above d/2, the strong gap holds and mixing is exponential. The dividing line is geometric (the growth rate of the orbit) and spectral (the threshold for absolute continuity) simultaneously.
2603.20552. Spectral theory / hyperbolic manifolds / spectral gap / mixing / geometrically finite groups.