Spectral gap conjecture
In ergodic theory, a branch of mathematics, the spectral gap conjecture of Alexander Lubotzky, Ralph S. Phillips, and Peter Sarnak is a statement on the spectral gaps of certain actions of a free group on the sphere S 2 {\displaystyle S^{2}} . == Statement == Any matrix U ∈ S U ( 2 ) {\displaystyle U\in SU(2)} defines an isometry of the sphere S 2 {\displaystyle S^{2}} , which in turn defines an operator ϕ U {\displaystyle \phi _{U}} on the Hilbert space L 2 ( S U ( 2 ) ) {\displaystyle L^{2}(SU(2))} .