The Waring problem for Lie groups and Chevalley groups

TL;DR

This paper investigates how elements of Lie and Chevalley groups over infinite fields can be expressed as products of values of a fixed non-trivial word, extending classical number theory problems to group theory.

Contribution

It establishes that for large rank classical Lie groups, every element can be written as a product of two word values, and similarly for Chevalley groups over various fields, including p-adic fields.

Findings

In large rank compact Lie groups, w(G)^2=G.

In Chevalley groups over R or p-adic fields, every element is a product of two values of w.

Every element in Chevalley groups over infinite fields is a product of two squares.

Abstract

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given non-trivial word w. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed non-trivial word w and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)^2=G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over R or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.