The R-infinity property for pure Artin braid groups

Author(s)

Dekimpe, Karel ^[1] ; Goncalves, Daciberg Lima ^[2] ; Ocampo, Oscar ^[3]

Affiliation

^[1] Katholieke Univ Leuven, Campus Kulak Kortrijk, Etienne Sabbelaan 53, B-8500 Kortrijk - Belgium

^[2] IME USP, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[3] Univ Fed Bahia, Dept Matemat IME, Av Adhemar de Barros S-N, BR-40170110 Salvador, BA - Brazil

• Document type: Journal article
• Source: MONATSHEFTE FUR MATHEMATIK; v. 195, n. 1 FEB 2021.
• Web of Science Citations: 0

Abstract

In this paper we prove that all pure Artin braid groups P-n (n >= 3) have the R-infinity property. In order to obtain this result, we analyse the naturally induced morphism Aut(P-n) -> Aut(Gamma(2)(P-n)/Gamma(3)(P-n)) which turns out to factor through a representation rho : Sn+1 -> Aut(Gamma(2)(P-n)/Gamma(3)(P-n)). We can then use representation theory of the symmetric groups to show that any automorphism alpha of P-n acts on the free abelian group Gamma(2)(P-n)/Gamma(3)(P-n) via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number R(alpha) of alpha is infinity. (AU)

• FAPESP's process: 16/24707-4 - Algebraic, geometric and differential topology
• Grantee: Daciberg Lima Gonçalves
• Support Opportunities: Research Projects - Thematic Grants