The R ∞ R_{\infty} property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups

Abstract

Abstract We say a group 𝐺 has property R ∞ R_{\infty} if the number R ⁢ ( φ ) R(\varphi) of twisted conjugacy classes is infinite for every automorphism 𝜑 of 𝐺. For such groups, the R ∞ R_{\infty} -nilpotency degree is the least integer 𝑐 such that G / γ c + 1 ⁢ ( G ) G/\gamma_{c+1}(G) has property R ∞ R_{\infty} . In this work, we compute the R ∞ R_{\infty} -nilpotency degree of all Generalized Solvable Baumslag–Solitar groups Γ n \Gamma_{n} . Moreover, we compute the lower central series of Γ n \Gamma_{n} , write the nilpotent quotients Γ n , c = Γ n / γ c + 1 ⁢ ( Γ n ) \Gamma_{n,c}=\Gamma_{n}/\gamma_{c+1}(\Gamma_{n}) as semidirect products of finitely generated abelian groups and classify which invertible integer matrices can be extended to automorphisms of Γ n , c \Gamma_{n,c} .