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Spectral gap conjecture

Conjecture in ergodic theory

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Source updated July 6, 2026

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 $S^{2}$ .

01Statement

Any matrix $U\in SU(2)$ defines an isometry of the sphere $S^{2}$ , which in turn defines an operator $\phi _{U}$ on the Hilbert space $L^{2}(SU(2))$ . The spectral gap conjecture states that for any integer $n>2$ , if $n$ isometries $U_{1},\dots ,U_{n}$ are chosen uniformly at random, then the operator $\phi _{U_{1}}+\phi _{U_{1}}^{-1}+\cdots +\phi _{U_{n}}+\phi _{U_{n}}^{-1}$ has a nontrivial spectral gap with probability 1.

02Progress

In 2007, Jean Bourgain and Alex Gamburd proved that when the matrices $U_{i}$ have entries which are all algebraic numbers up to simultaneous conjugation, the resulting operator has a spectral gap. This result was later generalized to the case of $SU(d)$ . It is known that either there is a nontrivial spectral gap with probability 1 or that the spectral gap is trivial with probability 1. If true, the statement would have applications to quantum computing and the design of universal quantum gate sets.

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Sources and credits

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