Word maps, word maps with constants and representation varieties of one-relator groups

TL;DR

This paper investigates the behavior of word maps with constants on simple algebraic groups, establishing dominance results and linking unipotent elements in images to the structure of representation varieties.

Contribution

It proves a dominance theorem for word maps with constants and relates unipotent elements in images to the structure of representation varieties.

Findings

General word maps with constants are dominant.

Existence of unipotents in images relates to representation variety structure.

Analogues of Borel's theorem for word maps with constants.

Abstract

We consider word maps and word maps with constants on a simple algebraic group. We present results on the images of such maps, in particular, we prove a theorem on the dominance of general word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the corresponding representation variety.