Twisted conjugacy in SL₍ and GL₍ over subrings of Fₚ (t)

Let G G be an automorphism of an infinite group G. One has an equivalence relation _ on G defined as x_ y if there exists a z G such that y=zx (z^-1). The equivalence classes are called -twisted conjugacy classes, and the set G/_ of equivalence classes is denoted by R (). The cardinality R () of R () is called the Reidemeister number of. We write R () = when R () is infinite. We say that G has the R_ -property if R () = for every automorphism of G. We show that the groups G=GL₍ (R), SL₍ (R) have the R_ -property for all n 3 when Ft R F (t), where F is a subfield of Fₚ. When n 4, we show that any subgroup H GL₍ (R) that contains SL₍ (R) also has the R_ -property.