Twisted conjugacy in $${\mathrm {GL}}_2$$ and $${\mathrm {SL}}_2$$ over polynomial algebras over finite fields

Abstract

Given an automorphism φ : G → G of an infinite group G, one has the twisted conjugation action of G on itself given by g.x = gxφ(g −1 ). The orbits of this action are the φ-twisted conjugacy classes. The Reidemeister number R(φ) is the number of φ-twisted conjugacy classes in G. One says that G has the R ∞ -property if R(φ) is infinite for every automorphism of G. We show that the groups2010 Mathematics Subject Classification. 20F28, 20G35.

Twisted conjugacy in classical groups over certain domains of Characteristic $p>0$

“…These isomorphisms of Chevalley groups can be read off from the corresponding isomorphisms in Lie algebras which are clear by looking at the corresponding Dynkin diagrams. The R ∞ -property of some rank 1 groups has been established in [11] where it is proved that the groups…”

Twisted Conjugacy in Linear Algebraic Groups

“…The reader may refer to [FT15] for an overview and more literature. Some recent work in this direction include [BDR20], [MS20], [GSW21], where the R ∞ -property has been studied for twisted Chevalley groups, for GL n (R), SL n (R) (where R = F q [t] or F q [t, t −1 ]), and for the fundamental groups of geometric 3-manifolds respectively.…”

Reidemeister classes, wreath products and solvability

“…On the other hand, in [11] we have detected an example of infinitely generated residually finite group which has neither TBFT nor TBFT f . 2) As an opposite case, to determine classes of groups for which any automorphism has infinite Reidemeister number (this property is called R ∞ ) -the list of results here is very extended, we mention only some expository or recent papers: [5,1,8,18,9,21,17,19]. 3) To study rationality and other properties of Reidemeister zeta function constructed from R(ϕ n ) (see e.g.…”