On the spectral gap for finitely-generated subgroups of \(\text{SU}(2)\)
Let \(g_1,\dots ,g_k\), \(k\geq 2\), be elements of the group \(G=\text{SU}(2)\). Consider the averaging operator \[ z=z_{g_1,\dots ,g_k}=\sum_{j=1}^k (f(g_jx)+f(g_j^{-1}x)). \] This operator is selfadjoint on \(L_2(G)\), the constant function is its eigenfunction with the eigenvalue \(\lambda_0=2k\). Denote by \(\lambda_1\) the supremum of the eigenvalues of \(z\) on the orthogonal complement of the constant functions. The element \(z\) is said to have a spectral gap if \(\lambda_1<2k\). The authors prove the spectral gap property for elements satisfying a noncommutative diophantine condition. In particular, if \(g_1,\dots ,g_k\) generate a subgroup containing elements with algebraic entries, which generate a non-Abelian free group, then \(z\) has a spectral gap. The authors describe connections of the problem and their techniques with the problem of uniqueness of the finitely additive rotation invariant measure, the Katz-Tao discretized ring conjecture (proved by Bourgain), etc. Open problems are formulated.
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