Spectral gaps for dense subgroups of compact simple Lie groups beyond algebraic generators: the almost-Diophantine conjecture, quantitative bounds, and the semisimple case

Bourgain and Gamburd (J. Eur. Math. Soc. 2012) proved that finitely many elements of the special unitary group $\mathrm{SU}(d)$ with algebraic entries that generate a dense subgroup yield an averaging (Hecke) operator on $L^2(\mathrm{SU}(d))$ with a spectral gap, i.e. the associated random walk equidistributes to Haar measure exponentially fast, and they expressed the expectation that the method extends to other continuous semisimple Lie groups. The group-generality strand was subsequently resolved for compact simple Lie groups by Benoist and de Saxcé (Invent. Math. 2016) and, in the appropriate local formulation, for non-compact simple Lie groups by Boutonnet, Ioana, and Salehi Golsefidy (Invent. Math. 2017). A precise open core of the source question survives. First, the algebraicity assumption is load-bearing in every known proof: whether every finite symmetric set generating a dense subgroup of a compact simple Lie group has a spectral gap — equivalently, whether every adapted probability measure is almost Diophantine — and even whether a Haar-generic generating tuple does (a question attributed to Sarnak), remain open; all known examples are, up to conjugation, algebraic-entry ones, and no counterexample is known either. Second, no explicit quantitative lower bound on the gap is known even in the algebraic case, and optimal (Ramanujan-type) constants are known only for special arithmetic generators. Third, no extension to compact semisimple groups that are not simple is available. The problem asks to settle these remaining strands of the Bourgain–Gamburd spectral gap program.

Background

A compact group $G$ carries a unique translation-invariant probability measure, its Haar measure. Given a finite symmetric set $S=\{s_1,\dots,s_k\}\subset G$ (symmetric meaning that $S=S^{-1}$), one forms the averaging (or Hecke) operator

$$T_S f(x)=\frac{1}{k}\sum_{j=1}^{k} f(s_j x),$$

a self-adjoint contraction of $L^2(G)$ whose largest eigenvalue, $1$, is attained exactly on the constant functions. Write $L^2_0(G)$ for the closed span of the mean-zero square-integrable functions. The set $S$, or the subgroup $\Gamma=\langle S\rangle$ it generates, has a spectral gap if the operator norm of $T_S$ restricted to $L^2_0(G)$ is strictly less than $1$. This is equivalent to exponential convergence of the simple random walk on $G$ with steps in $S$ to Haar measure, and it underlies strong ergodicity phenomena, for example uniqueness of finitely additive invariant means in the Banach–Ruziewicz problem.

If $G$ has an abelian quotient, spectral gaps genuinely fail: on the circle, whenever a finite set of rotations generates a dense subgroup, the eigenvalues of its averaging operator on the nontrivial characters accumulate at $1$ (for a single rotation of irrational angle $\theta$ they are the values $\cos(2\pi n\theta)$ with $n\neq 0$, which accumulate at $\pm 1$), so the operator norm on the mean-zero functions is exactly $1$. Quantitative rates on the circle are governed by Diophantine approximation: an angle is called Liouville if it is irrational yet approximable by rational angles faster than every power of the denominator, and for a Liouville angle the words in a rotation generator return to the identity faster than every exponential in the word length, so no exponential escape rate holds. For compact simple Lie groups — connected compact groups with no nontrivial connected normal subgroups, such as the special unitary group $\mathrm{SU}(d)$ (the $d\times d$ complex unitary matrices of determinant $1$), $\mathrm{SO}(n)$ (the rotations of $\mathbb{R}^n$, i.e. the real orthogonal matrices of determinant $1$), or $\mathrm{Sp}(n)$ (the compact symplectic group of unitary matrices preserving a quaternionic structure) — the situation is far more rigid. As recounted in the survey of Breuillard and Lubotzky cited below, Margulis and Sullivan (using projections of lattices with Kazhdan's property (T), the representation-theoretic rigidity property asserting that every unitary representation with almost-invariant vectors has an invariant vector) and Drinfeld (for $\mathrm{SO}(3)$, via automorphic-form estimates now standard in the theory of modular forms) showed that every compact simple Lie group contains some finite set generating a dense subgroup with a spectral gap, and Lubotzky, Phillips, and Sarnak later produced explicit arithmetic generators with optimal, so-called Ramanujan, gaps. The "worst case" question — deciding which generating sets of dense subgroups have a gap — is much harder, and it is the subject of this problem.

Bourgain and Gamburd's breakthrough first treated $\mathrm{SU}(2)$ (Invent. Math. 2008) and then $\mathrm{SU}(d)$ (the source paper, J. Eur. Math. Soc. 2012): if $g_1,\dots,g_k\in\mathrm{SU}(d)$ have entries in the field $\overline{\mathbb{Q}}$ of algebraic numbers and generate a subgroup dense in $\mathrm{SU}(d)$ — equivalently, by their footnote, Zariski dense in $\mathrm{SL}_d(\mathbb{C})$, i.e. not contained in the zero set of any nonzero system of polynomial equations on $\mathrm{SL}_d(\mathbb{C})$ — then the Hecke operator has a spectral gap. The proof combines a non-commutative Balog–Szemerédi–Gowers reduction to approximate groups (finite symmetric subsets whose square is coverable by few translates of the subset itself) with a discretized product theorem and an $L^2$-flattening iteration. The algebraicity of the entries is used to work simultaneously at a non-archimedean place of a number field containing the entries (a completion with respect to a prime ideal, with ultrametric distances), where the classical theory of random matrix products, applied to exterior powers of the Lie algebra, overcomes the fact that unitary matrices are never expanding; and, through the effective Nullstellensatz (effective degree and height bounds for polynomial systems without common zeros), to make words escape from proper subspaces. The theorem is qualitative: no explicit lower bound on the gap $1-\lVert T_S\rVert$ is produced, and the paper states the expectation that the method extends to other continuous semisimple Lie groups.

The announced generalization was carried out for simple groups by others. Benoist and de Saxcé (Invent. Math. 2016) proved that on a compact connected simple Lie group $G$, an adapted probability measure $\mu$ (one whose support generates a dense subgroup) all of whose support elements act with algebraic matrices in the adjoint representation (the action of $G$ by conjugation on its own Lie algebra) is almost Diophantine — the mass that the $n$-fold convolution $\mu^{*n}$ places on the $e^{-c_1 n}$-neighborhood of any proper closed subgroup is at most $e^{-c_2 n}$ — and, more generally, that the almost-Diophantine property is equivalent to having a spectral gap. Boutonnet, Ioana, and Salehi Golsefidy (Invent. Math. 2017) established the corresponding local spectral gap (a version where the $L^2$ estimate is required only on each open subset, with constants independent of the subset) for the left-translation action of dense subgroups generated by algebraic elements of arbitrary connected simple, possibly non-compact, Lie groups, with applications to the Banach–Ruziewicz problem, orbit-equivalence rigidity, and expanders.

A principal application recorded already in the source paper comes from quantum computation: the Solovay–Kitaev algorithm approximates an arbitrary target gate by a short word in a fixed finite universal gate set, and the spectral gap of the gate set governs whether the required circuit depth grows logarithmically or worse in the desired precision — the distinction between "efficiently universal" and merely "computationally universal" gate sets.

Problem Statement

The source paper proved the spectral gap theorem for dense subgroups of $\mathrm{SU}(d)$ under an algebraicity hypothesis on the generators, expected the extension to other continuous semisimple Lie groups, and produced only a qualitative gap. After the generalization to all compact simple Lie groups (Benoist–de Saxcé) and to local spectral gaps for non-compact simple Lie groups (Boutonnet–Ioana–Salehi Golsefidy), the following strands of the source question remain open; they share one objective, namely completing the Bourgain–Gamburd spectral gap theorem for dense subgroups of compact semisimple Lie groups to its natural generality — arbitrary groups, arbitrary dense generating sets, and explicit constants.

(1) Removal of the algebraicity assumption (the worst-case problem). Let $G$ be a compact connected simple Lie group and let $S\subset G$ be a finite symmetric set (that is, $S=S^{-1}$) generating a dense subgroup of $G$. Does $T_S$ have a spectral gap on $L^2_0(G)$? Equivalently (Benoist–de Saxcé), is every adapted probability measure on $G$ almost Diophantine? A strictly weaker sub-case, attributed to Sarnak and open in both senses of genericity, asks whether a spectral gap holds for Haar-almost every (or Baire-generic) finite tuple of elements generating a dense subgroup; it is implied by the universal question above but does not imply it. Every known example with a spectral gap is, up to conjugation inside $G$, a generating set with algebraic entries; whether any dense generating set without a spectral gap exists is also unknown.

(2) Quantitative bounds. In the algebraic case, obtain an explicit, effectively computable lower bound for $1-\lVert T_S\rVert_{L^2_0(G)}$ in terms of effectively computable invariants of the generators (for instance the degree of the number field generated by their matrix entries and the heights of those entries, a standard measure of their arithmetic complexity); determine, at least asymptotically, the optimal constants, i.e. how close to the Ramanujan threshold $2\sqrt{k-1}$ (in the normalization where the trivial eigenvalue is $k$) the spectrum can be forced, a question resolved only for special arithmetic generators.

(3) Genuinely semisimple groups. In the algebraic case, extend the theorem to compact connected semisimple groups that are not simple (finite products of simple factors, such as $\mathrm{SO}(4)$), the literal wording of the source's expectation ("other continuous semi-simple Lie groups"), where the subgroup may project onto every factor yet not split as a product, the class of proper closed subgroups includes the simple factors and diagonal-type subgroups, and the quasirandomness mechanism (high multiplicity of irreducible representations) used in the simple case is unavailable.

A positive answer to the Haar-almost-every (or Baire-generic) sub-case of (1) is strictly weaker and does not by itself resolve (1).

References

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2. Jean Bourgain and Alex Gamburd, "On the spectral gap for finitely-generated subgroups of SU(2)", Inventiones mathematicae 171 (2008), no. 1, 83-121. DOI: 10.1007/s00222-007-0072-z, https://doi.org/10.1007/s00222-007-0072-z.
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