2019/12/21 by Oorna Mitra, Mitra, Oorna, Parameswaran Sankaran +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1912.10184
openalex publication_date 2019/12/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let ϕ:G→ G be an automorphism of an infinite group G. One has an equivalence relation ∼ϕ on G defined as x∼ϕy 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 \mathcal R(ϕ). The cardinality R(ϕ) of \mathcal R(ϕ) is called the Reidemeister number of ϕ. We write R(ϕ)=∞ when \mathcal R(ϕ) is infinite. We say that G has the R_∞-\it 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\subsetneq F(t) where F is a subfield of \mathbb Fp. When n≥ 4, we show that any subgroup H⊂ GLn(R) that contains SLn(R) also has the R_∞-property.