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The Waring problem for finite simple groups. (English) Zbl 1283.20008
Summary: The classical Waring problem deals with expressing every natural number as a sum of $$g(k)$$ $$k$$-th powers. Recently there has been considerable interest in similar questions for non-Abelian groups, and simple groups in particular. Here the $$k$$-th power word can be replaced by an arbitrary group word $$w\neq 1$$, and the goal is to express group elements as short products of values of $$w$$.
We give a best possible and somewhat surprising solution for this Waring type problem for (non-Abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements.
Along the way we also obtain new results, possibly of independent interest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes.
Our methods involve algebraic geometry and representation theory, especially Lusztig’s theory of representations of groups of Lie type.

##### MSC:
 20D05 Finite simple groups and their classification 11P05 Waring’s problem and variants 20C33 Representations of finite groups of Lie type 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F05 Generators, relations, and presentations of groups 20E45 Conjugacy classes for groups 20G40 Linear algebraic groups over finite fields 20E05 Free nonabelian groups
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