A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$

Abstract

Abstract Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature $$-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> . For each $$n\in {\mathbf {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math> , let $$X_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> be a random degree- n cover of X sampled uniformly from all degree- n Riemannian covering spaces of X . An eigenvalue of X or $$X_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> is an eigenvalue of the associated Laplacian operator $$\Delta _{X}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:math> or $$\Delta _{X_{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:math> . We say that an eigenvalue of $$X_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> is new if it occurs with greater multiplicity than in X . We prove that for any $$\varepsilon &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ε</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> , with probability tending to 1 as $$n\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> , there are no new eigenvalues of $$X_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> below $$\frac{3}{16}-\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mfrac><mml:mn>3</mml:mn><mml:mn>16</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:math> . We conjecture that the same result holds with $$\frac{3}{16}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mn>3</mml:mn><mml:mn>16</mml:mn></mml:mfrac></mml:math> replaced by $$\frac{1}{4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac></mml:math> .

Topics & Concepts