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} .

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

We compute the Bieri-Neumann-Strebel invariants Σ 1 for the generalized solvable Baumslag-Solitar groups Γ n and their finite index subgroups. Using Σ 1 , we show that certain finite index subgroups of Γ n cannot be isomorphic to Γ ñ for any ñ. In addition, we use the BNS-invariants to give a new proof of property R ∞ for the groups Γ n and their finite index subgroups.

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