Word maps on perfect algebraic groups

TL;DR

This paper extends Borel's theorem on word maps to perfect algebraic groups, explores surjectivity for specific words and groups, and surveys recent developments with new ideas and approaches.

Contribution

It generalizes Borel's theorem to perfect groups and words with constants, introducing new methods and connections in the study of word maps.

Findings

Extended Borel's theorem to perfect algebraic groups

Analyzed surjectivity for specific words and groups

Surveyed recent progress and introduced new approaches

Abstract

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.