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user6976
user6976

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group (${\mathbb Z}$) with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).

Update Recall that Markov proved undecidability of every Markov property for semigroups (see http://iopscience.iop.org/0036-0279/19/3/M05/pdf/0036-0279_19_3_M05.pdf), and later Adyan and Rabin proved it for groups (see Lyndon and Schupp, Combinatorial group theory, Theorem 4.1, Chapter 4).

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group (${\mathbb Z}$) with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group (${\mathbb Z}$) with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).

Update Recall that Markov proved undecidability of every Markov property for semigroups (see http://iopscience.iop.org/0036-0279/19/3/M05/pdf/0036-0279_19_3_M05.pdf), and later Adyan and Rabin proved it for groups (see Lyndon and Schupp, Combinatorial group theory, Theorem 4.1, Chapter 4).

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user6976
user6976

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group ($\mathbb Z}$${\mathbb Z}$) with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group ($\mathbb Z}$ with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group (${\mathbb Z}$) with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).

Source Link
user6976
user6976

There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group ($\mathbb Z}$ with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).