The R _∞ and S _∞ properties for linear algebraic groups

Abstract

In this paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear algebraic groups over some fields of zero characteristic possess the R ∞ and S ∞ properties.

Reidemeister spectrum of special and general linear groups over some fields contains 1

“…The author studied conditions which imply the R ∞ -property for different linear groups over rings [12,14] and fields [6,13]. In particular, it was proved that if F is an algebraically closed field of zero characteristic which has finite transcendence degree over Q, then every reductive linear algebraic group G with nontrivial quotient G/R(G), where R(G) denotes the solvable radical of G, possesses the R ∞ -property [6]. Also it was proved that if F is a field of zero characteristic which has periodic group of automorphisms, then every Chevalley group (of normal type) over the field F possesses the R ∞ -property [13].…”

The 𝑅∞-property for nilpotent quotients of Baumslag–Solitar groups

“…Several families of groups have been studied by many authors. A non-exhaustive list of references is [1,2,3,4,5,6,7,8,9,10,14].Of particular interest for this paper is the fact that in [5] it was proved that the Baumslag-Solitar groups BS(m, n) have the R ∞ -property except for m = n = 1 (or m = n = −1 which is the same group). Recently in [3], motivated by the results of [1], new examples of groups which have the R ∞ -property were obtained by looking at quotients of a group which has the R ∞ -property by the terms of the lower central series as well the derived central series.…”

“…Several families of groups have been studied by many authors. A non-exhaustive list of references is [1,2,3,4,5,6,7,8,9,10,14].…”

Twisted conjugacy classes in unitriangular groups

“…The study of this problem has been quite an active research topic in recent years. We refer to the paper [5] for an overview of the families of groups which have been studied in this context until 2016. More recent results can be found in [2,13,19,20].…”

“…More recent results can be found in [2,13,19,20]. The author studied twisted conjugacy classes and the R ∞ -property for classical linear groups [5,[14][15][16][17]. For the immediate consequences of the R ∞ -property for topological fixed point theory see [9].…”