The Ore conjecture, posed in 1951, states that every element of every finite non-abelian
simple group is a commutator. Despite considerable effort, it remained open for various …

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 …

This paper contains a survey of recent developments in the investigation of word equations
in simple matrix groups and polynomial equations in simple (associative and Lie) matrix …

We establish new characterizations of primitive elements and free factors in free groups,
which are based on the distributions they induce on finite groups. For every finite group $ G …

In this paper we establish a congruence on the degree of the map from a component of a
Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves …

Let F k be the free group on k generators. A word w∈ F k is called primitive if it belongs to
some basis of F k. We investigate two criteria for primitivity, and consider more generally …

We provide new estimates on character values of symmetric groups which hold for all
characters and which are in some sense best possible. It follows from our general bound …

In recent years there has been much interest in word maps on groups, with various
motivations and applications. Substantial progress has been made and many fundamental …

The first part of the paper establishes results about products of commutators in ad-generator
finite group G, for example: if H⊲ G=< g 1,…, gr> then every element of the subgroup H, G is …

We present a survey of results on word equations in simple groups, as well as their
analogues and generalizations, which were obtained over the past decade using various …