Word maps and Waring type problems

By Michael Larsen, Aner Shalev. 2007

110 citations recorded by the source corpus.

Waring's problem asks whether every natural number is a sum of g(k) fcth powers (where g is a suitable function). This was solved affirmatively by Hubert in 1909. Optimizing g(k) has been a central problem in additive number theory ever since (see [Na] for more detail and background). Recently there has been considerable interest in group theoretic analogues of this phenomenon, where the aim is to present group elements as short products of certain "special" elements. These special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group. The motivation for this is sometimes topological. Let G be a (topologically) finitely generated pro-p group. Using expressions of elements of the commutator subgroup G1 as bounded products of commutators, Serre showed that G' is closed and deduced that every finite index subgroup of G is open. A recent deep result of Nikolov and Segal [NSI, NS2] shows that this holds for every finitely generated profinite group. Again the core of the proof is presenting group elements as short products of values of certain words.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5d58caa5e39a648754928755a721aabba48411d5

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Citations in the corpus

Commutator maps, measure preservation, and T-systems (2007, background): “Word maps on algebraic groups and on finite simple groups have been the subject of active investigations in recent years, see [Bo], [LiSh2], [La], [Sh] and [LaSh].”
Characters of symmetric groups: sharp bounds and applications (2008, methodology): “This improves a technical result from [LaSh], where it is shown that cyc(σ) ≤ n implies |χ(σ)| ≤ χ(1).”
Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture (2020, unknown): “The latest result of Larsen–Shalev–Tiep [72] (see also [71, 73, 74]), is as follows:”