Word maps on perfect algebraic groups

We extend Borel’s theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel’s theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections.

The Algebraic Independence of Certain Exponential Functions

In 1897 E. Borel proved a general theorem which implied as a special case the following result equivalent to his celebrated generalization of Picard's theorem [2]: If f1 … ,fm are entire functions such that for each, C then the functions exp f1, … , exp f/m are linearly independent over C. In 1929 R. Nevanlinna [6] extended Borel's theorem to consider arbitrary C-linearly independent meromorphic functions < ϕi, … , < ϕm satisfying < ϕ1 + … + ϕm = 1.