Twisted conjugacy in SLₙ and GLₙ over subrings of Fₚ̄(t)

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 F_p̄ . When n≥ 4 , we show that any subgroup H⊂ GLₙ(R) that contains SLₙ(R) also has the R∞ -property.