Random generation of the special linear group

<p>It is well known that the proportion of pairs of elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper S upper L left-parenthesis n comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>SL</mml:mi> <mml:mo>⁡<! -- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SL}(n,q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which generate the group tends to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="q Superscript n Baseline right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">→<! -- → --></mml:mo> <mml:mi mathvariant="normal">∞<! -- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q^n\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. </p> <p>An essential step in our proof is an estimate for the average of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="left-parenthesis o r d g right-parenthesis Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ord</mml:mi> <mml:mo>⁡<! -- ⁡ --></mml:mo> <mml:mi>g</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<! -- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\operatorname {ord} g)^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ranges over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper G upper L left-parenthesis n comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>⁡<! -- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {GL} (n,q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which may be of independent interest. We prove that this average is <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="exp left-parenthesis minus left-parenthesis 2 minus o left-parenthesis 1 right-parenthesis right-parenthesis StartRoot n log n log q EndRoot right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡<! -- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<! -- − --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>−<! -- − --></mml:mo> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:msqrt> <mml:mi>n</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡<! -- ⁡ --></mml:mo> <mml:mi>n</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡<! -- ⁡ --></mml:mo> <mml:mi>q</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> <mml:mo>. </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp (-(2-o(1)) \sqrt {n \log n \log q}). </mml:annotation> </mml:semantics> </mml:math> \] </disp-formula></p>