Final Report Summary - WORDS (Words and Waring type problems)
Our ERC project focuses on words, word maps on groups, related Waring type problems, and on various applications. This is a highly active field of research in recent years, which arises in different contexts, such as Burnside type problems, profinite groups, as well as the theory of finite simple groups. The origins of this topic date back to Number Theory and Waring problem on representing numbers as short sums of powers.
Recall that a word w is an element of a free group, and it defines a word map on any group G (by substituting elements of G in the variables of w) whose image is denoted by w(G). Borel showed that word maps are dominant on simple algebraic groups. Our ERC research proposal aimed at showing that, in various situations, w(G) is large in various senses, and many of its detailed objectives were achieved. In a paper by Larsen, Tiep and myself, which appeared in the prestigious Annals of Mathematics, we show that for any non-trivial word w and any sufficiently large (nonabelian) finite simple group G, we have w(G)^2=G, namely every element is a product of two w-values. The proof is long and complicated, combining representation theory with geometry, and it yields further applications, such as an approximation to a famous conjecture of Thompson.
In a subsequent work by the same authors we study similar problems for finite quasi-simple groups. While w(G)^2 need not equal to G here (indeed we find infinitely many examples showing this) we prove that w(G)^3=G (for G large). In another work on quasi-simple groups, by Liebeck, O'Brien, Tiep and myself, we prove Ore conjecture in such groups, with given (finitely many) exceptions.
Some word maps turn out to be surjective on ALL finite simple groups (not only large ones). We prove (with Liebeck, O'Brien and Tiep) that this is the case for the word x^2y^2, namely, every element of a finite simple group is a product of two squares. This can be regarded as a non-commutative analogue of Lagrange four squares theorem in Number Theory. Further results on surjective and non-surjective words were recently obtained by us, by Guralnick and Malle, and other authors.
The size of fibers of word maps on finite simple groups were studied by Larsen and myself, with interesting applications to subgroup growth and representation varieties. Papers of Liebeck, Nikolov and myself deal with covering finite simple groups by short products of conjugates of a given subgroup or a subset H, using several tools including the recent deep theory of approximate groups.
Puder and Parzanchevski characterized primitive words as words which are uniform on all finite groups (extending a previous similar result of Puder on words in two variables). This is a deep result proved by research students supported by my ERC grant. Another research student, Schul, is studying class expansion, and (with Parzanchevski) Fourier expansion of the distribution of word maps, and already published a paper on this topic.
Recently we obtained interesting results on the behavior of word maps on certain infinite groups. p-adic groups are dealt with in a paper of Avni, Glenader, Kassabov and myself. We show for these groups results of the form w(G)^3=G. In a subsequent paper with my post-doc Hui and Larsen we study word maps on compact connected classical real Lie groups. We prove strong results of type w(G)^2=G, and also obtain results for infinite Chevalley groups. Various additional results were obtained which we cannot describe in this short outline.
We regard our ERC project as fruitful and inter-disciplinary, yielding a variety of new results and methods. It is also a productive project, giving rise to over 20 papers by myself and my ERC team, and had a big impact on the mathematical community.
Recall that a word w is an element of a free group, and it defines a word map on any group G (by substituting elements of G in the variables of w) whose image is denoted by w(G). Borel showed that word maps are dominant on simple algebraic groups. Our ERC research proposal aimed at showing that, in various situations, w(G) is large in various senses, and many of its detailed objectives were achieved. In a paper by Larsen, Tiep and myself, which appeared in the prestigious Annals of Mathematics, we show that for any non-trivial word w and any sufficiently large (nonabelian) finite simple group G, we have w(G)^2=G, namely every element is a product of two w-values. The proof is long and complicated, combining representation theory with geometry, and it yields further applications, such as an approximation to a famous conjecture of Thompson.
In a subsequent work by the same authors we study similar problems for finite quasi-simple groups. While w(G)^2 need not equal to G here (indeed we find infinitely many examples showing this) we prove that w(G)^3=G (for G large). In another work on quasi-simple groups, by Liebeck, O'Brien, Tiep and myself, we prove Ore conjecture in such groups, with given (finitely many) exceptions.
Some word maps turn out to be surjective on ALL finite simple groups (not only large ones). We prove (with Liebeck, O'Brien and Tiep) that this is the case for the word x^2y^2, namely, every element of a finite simple group is a product of two squares. This can be regarded as a non-commutative analogue of Lagrange four squares theorem in Number Theory. Further results on surjective and non-surjective words were recently obtained by us, by Guralnick and Malle, and other authors.
The size of fibers of word maps on finite simple groups were studied by Larsen and myself, with interesting applications to subgroup growth and representation varieties. Papers of Liebeck, Nikolov and myself deal with covering finite simple groups by short products of conjugates of a given subgroup or a subset H, using several tools including the recent deep theory of approximate groups.
Puder and Parzanchevski characterized primitive words as words which are uniform on all finite groups (extending a previous similar result of Puder on words in two variables). This is a deep result proved by research students supported by my ERC grant. Another research student, Schul, is studying class expansion, and (with Parzanchevski) Fourier expansion of the distribution of word maps, and already published a paper on this topic.
Recently we obtained interesting results on the behavior of word maps on certain infinite groups. p-adic groups are dealt with in a paper of Avni, Glenader, Kassabov and myself. We show for these groups results of the form w(G)^3=G. In a subsequent paper with my post-doc Hui and Larsen we study word maps on compact connected classical real Lie groups. We prove strong results of type w(G)^2=G, and also obtain results for infinite Chevalley groups. Various additional results were obtained which we cannot describe in this short outline.
We regard our ERC project as fruitful and inter-disciplinary, yielding a variety of new results and methods. It is also a productive project, giving rise to over 20 papers by myself and my ERC team, and had a big impact on the mathematical community.