anabelian

English

Etymology

From an- (“not”) +‎ abelian (“commutative”); coined to imply a nuancedly stronger condition than "merely" noncommutative (see an-).

Adjective

anabelian (not comparable)

1. (mathematics, algebraic geometry, arithmetic geometry) Of or relating to anabelian geometry, a proposed theory describing the way the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian group, in a sense to be made more precise.
   • 2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 522:

     We end up this section with the following remarks, both for the sake of completeness, but mostly because of the relevance of the facts we want to mention for the anabelian phenomena.

   • 2012, Florian Pop, “Lectures on anabelian phenomena in geometry and arithmetic”, in John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page 5:

     We will formulate Grothendieck's anabelian conjectures in a more general context later, after having presented the basic facts about ́etale fundamental groups.

   • 2013, Anna Cadoret, “Galois Categories”, in Pierre Dèbes, Michel Emsalem, Matthieu Romagny, A. Muhammed Uludağ, editors, Arithmetic and Geometry Around Galois Theory, Springer (Birkhäuser), page 172:

     But the interplay between those two parts remains mysterious and is at the source of some of the most standard conjectures about fundamental groups such as anabelian conjectures or the section conjecture.

Usage notes

• The term reflects the notion that the further G is from being commutative, the more information it contains about V, and therefore the greater relevance it has to the theory.

Derived terms

• anabelioid