Surjectivity of certain word maps on PSL(2,C) and SL(2,C)
Tatiana Bandman, Yuri G. Zarhin
TL;DR
This paper investigates when certain word maps on PSL(2,C) and SL(2,C) are surjective, showing that most words induce surjective maps except those in the second derived subgroup.
Contribution
The paper characterizes the surjectivity of word maps on PSL(2,C) and SL(2,C), identifying specific algebraic conditions for surjectivity.
Findings
Most words in a free group induce surjective maps on PSL(2,C).
Certain words maps are surjective on SL(2,C) x SL(2,C).
Words in the second derived subgroup do not produce surjective maps.
Abstract
We show that an element w of a free group F on n generators defines a surjective word map of PSL(2,C)^n onto PSL(2,C) unless w belongs to the second derived subgroup of F. We also describe certain words maps that are surjective on SL(2,C) x SL(2,C). Here C is the field of complex numbers.
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Taxonomy
TopicsFinite Group Theory Research · Geometric and Algebraic Topology · Advanced Algebra and Geometry