[Paper Review] A spectral gap theorem in simple Lie groups

Yves Benoist, Nicolas de Saxcé|arXiv (Cornell University)|May 8, 2014

Advanced Algebra and Geometry8 references4 citations

TL;DR

This paper establishes a spectral gap theorem for dense subgroups generated by algebraic elements in compact simple Lie groups, proving that such measures have a spectral gap if and only if they are almost Diophantine. Using random matrix products over local fields and effective Nullstellensatz techniques, the authors generalize Bourgain and Gamburd's results from SU(2) to arbitrary compact simple Lie groups, providing a sharp characterization of spectral gap via Diophantine properties.

ABSTRACT

We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.

Motivation & Objective

• To establish a spectral gap criterion for Borel probability measures on compact simple Lie groups.
• To generalize Bourgain and Gamburd's spectral gap result for SU(2) to arbitrary compact simple Lie groups.
• To prove that a measure has a spectral gap if and only if it is almost Diophantine, linking spectral properties to Diophantine approximation.
• To use the theory of random matrix products over local fields to analyze the behavior of measures supported on algebraic elements.
• To establish a quantitative connection between spectral gap and the decay of measure mass near proper closed subgroups.

Proposed method

• Define the spectral gap via the spectral radius of the averaging operator on $ L^2_0(G) $, showing it is less than one.
• Introduce the notion of 'almost Diophantine' measures, quantifying the decay of $ u^{*n}(H^{(e^{-C_1n})}) $ for proper closed subgroups $ H $.
• Use the discretized Product Theorem from [14] and Bourgain-Gamburd's method to relate spectral gap to the Diophantine property.
• Apply the effective Nullstellensatz to show that if no common zero exists for a system of polynomials, a contradiction arises from size bounds on coefficients.
• Construct polynomial maps $ P_{I_0,g} $ encoding the action of group elements on wedge powers of the Lie algebra, with coefficients in a number field.
• Bound the size of these polynomials using the ring of integers of the number field generated by the algebraic entries of $ \operatorname{Ad}g $.

Experimental results

Research questions

• RQ1 Does every adapted probability measure on a compact simple Lie group with algebraic entries have a spectral gap?
• RQ2 Is the spectral gap property equivalent to the almost Diophantine condition for such measures?
• RQ3 Can the spectral gap be characterized uniformly across all compact simple Lie groups using Diophantine approximation?
• RQ4 What is the quantitative decay rate of measure mass near proper closed subgroups for algebraic measures?
• RQ5 How do random matrix products over local fields help in proving spectral gap for algebraic measures?

Key findings

• A Borel probability measure $ \mu $ on a compact simple Lie group $ G $ has a spectral gap if and only if it is almost Diophantine.
• For any compact simple Lie group $ G $, if $ \mu $ is adapted and supported on elements with algebraic $ \operatorname{Ad}g $ entries, then $ \mu $ is almost Diophantine and thus has a spectral gap.
• The spectral gap is quantitatively controlled by the Diophantine exponent: $ \mu^{*n}(H^{(e^{-C_1n})}) \leq e^{-c_2n} $ for some $ c_2 > 0 $.
• The proof relies on effective bounds in the Nullstellensatz, showing that if a system of polynomials has no common zero, then a non-zero algebraic integer of bounded size must exist.
• The size of the polynomials $ P_{I_0,g} $ is controlled by $ \|q^n P_{I_0,g}\| \leq q^{2n} $, where $ q $ depends only on the generating set.
• The contradiction in the proof arises when $ C_1 $ is chosen large enough so that $ q^{-Mn} \leq C q^{Cn} e^{-C_1n} $ fails, implying a common zero must exist.