Word maps and Waring type problems

Abstract

Waring’s classical problem deals with expressing every natural number as a sum of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th powers. Recently there has been considerable interest in similar questions for nonabelian groups and simple groups in particular. Here the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th power word is replaced by an arbitrary group word <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w not-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">w \ne 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the goal is to express group elements as short products of values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We give a best possible and somewhat surprising solution for this Waring type problem for various finite simple groups, showing that a product of length two suffices to express all elements. We also show that the set of values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is very large, improving various results obtained so far. Along the way we also obtain new results of independent interest on character values and class squares in symmetric groups. Our methods involve algebraic geometry, representation theory, probabilistic arguments, as well as results from analytic number theory, including three primes theorems (approximating Goldbach’s Conjecture).