Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 2 / 3 / ii

The Messuage lemma for groups is the following effective collapse-versus-embedding statement: from a finite group presentation for $K$ and a word $w$, one can algorithmically write a finite presentation $K_w$ such that $w=1$ in $K$ makes $K_w$ trivial, whereas $w\ne 1$ makes the specified copy of $K$ embed in $K_w$. It can also be arranged that the image of $w$ normally generates $K_w$. We use this lemma as an assumed construction, as requested; the construction of its presentation does not require deciding which case applies.

Let $P$ be a finitely presented group with insoluble word problem for a group, whose existence is allowed. Set $K=P\ast ⟨z⟩$, with $z$ of infinite order. It is finitely presented, and the normal form theorem for a free product embeds both $P$ and the infinite cyclic factor. Thus an input word $w$ from $P$ is trivial in $K$ exactly when it is trivial in $P$, while $K$ is certainly infinite.

Apply the Messuage lemma for groups to this $K,w$. If $w=1$ in $P$, the constructed group $K_w$ is trivial and hence finite. If $w\ne 1$, it contains $K$ and is infinite. ConsequentlyA hypothetical algorithm deciding infinitude from arbitrary finite group presentations would therefore decide every input word in $P$, a contradiction. There is no such algorithm. The infinite cyclic free factor ensures the reduction does not need an additional assumption about the size of $P$.