Twisted conjugacy in $\mathrm{SL}_{n}$ and $\mathrm{GL}_{n}$ over subrings of $\overline{\mathbb{F_p}}(t)$

Oorna Mitra¹  ; Parameswaran Sankaran² 

¹Chennai Mathematical Institute, Kelambakkam, India; Indian Statistical Institute, Bengaluru, India
²Chennai Mathematical Institute, Kelambakkam, India

Groups, geometry, and dynamics, Tome 18 (2024) no. 3, pp. 939-962

Résumé

Let φ:G→G be an automorphism of an infinite group G. One has an equivalence relation ∼φ​ on G defined as x∼φ​y if there exists a z∈G such that y=zxφ(z−1). The equivalence classes are called φ-twisted conjugacy classes, and the set G/∼φ​ of equivalence classes is denoted by R(φ). The cardinality R(φ) of R(φ) is called the Reidemeister number of φ. We write R(φ)=∞ when R(φ) is infinite. We say that G has the R∞​-property if R(φ)=∞ for every automorphism φ of G. We show that the groups G=GLn​(R),SLn​(R) have the R∞​-property for all n≥3 when F[t]⊂R⊊F(t), where F is a subfield of Fp​​. When n≥4, we show that any subgroup H⊂GLn​(R) that contains SLn​(R) also has the R∞​-property.

Détail

DOI : 10.4171/ggd/758

Classification : 20F28, 20G35
Mots-clés : twisted conjugacy, general linear group, special linear group, automorphism group, Reidemeister number