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

Nikolai Gordeev; Boris Kunyavskiĭ; Eugene Plotkin

doi:10.1016/j.jalgebra.2017.03.016

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

Nikolai Gordeev, Boris Kunyavskiĭ, Eugene Plotkin

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We consider word maps and word maps with constants on a simple algebraic group G. 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 the map induced by w∈F_m and the structure of the representation variety R(Γ_w,G) of the group Γ_w=F_m/〈w〉.

Original language	English
Pages (from-to)	390-424
Number of pages	35
Journal	Journal of Algebra
Volume	500
DOIs	https://doi.org/10.1016/j.jalgebra.2017.03.016
State	Published - 15 Apr 2018

Bibliographical note

Funding Information:
The research of the first author was supported by the Ministry of Education and Science of the Russian Federation and RFBR grant 14-01-00820 . The research of the second and third authors was supported by ISF grants 1207/12 , 1623/16 and the Emmy Noether Research Institute for Mathematics .

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

Algebraic groups
Representation varieties
Word maps

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10.1016/j.jalgebra.2017.03.016

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abstract = "We consider word maps and word maps with constants on a simple algebraic group G. 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 the map induced by w∈Fm and the structure of the representation variety R(Γw,G) of the group Γw=Fm/〈w〉.",

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AU - Plotkin, Eugene

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AB - We consider word maps and word maps with constants on a simple algebraic group G. 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 the map induced by w∈Fm and the structure of the representation variety R(Γw,G) of the group Γw=Fm/〈w〉.

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JF - Journal of Algebra

ER -

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

Abstract

Bibliographical note

Keywords

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