Deduce unsolvability of $\operatorname{IP}(G_0)$ from the Adian–Rabin Theorem

Question

$\operatorname{IP}(G_0)$: the special isomorphism problem for $G_0$, i.e., given $G_0$, determine if $G$ is isomorphic to $G_0$. My question is that how can we deduce from the Adian–Rabin theorem that $\operatorname{IP}(G_0)$ is unsolvable for any finitely presented $G_0$? I have found the following paper:

Stillwell - The word problem and the isomorphism problem for groups

Is it true to say that Theorem 5, page 53 is derived from the Adian–Rabin Theorem? And if no, do you know a reference including the deduction of unsolvability of $\operatorname{IP}(G_0)$ from the Adian–Rabin Theorem?

Answer 1

The triviality problem (input: a finite group presentation of a group $G$, output: yes/no according to whether $G$ is a trivial group) is not solvable by an algorithm. Indeed just apply the Adian-Rabin theorem to the property "being trivial".

Now we deduce the general case of your question as follows:

Suppose by contradiction that you have an algorithm $A$ whose input is a finite presentation of a group $G$ and output is yes/no according to whether $G$ is isomorphic to the given finitely presented group $G_0$.

Then $A$ solves the triviality problem. Namely input $G$, and apply $A$ to the concatenation of presentations of $G$ and $G_0$, which is a presentation of the free product $G\ast G_0$. So the answer is yes/no according to whether $G\ast G_0$ is isomorphic to $G_0$. And by Grushko's theorem (see next paragraph), the latter holds if and only $G$ is a trivial group. So we get a contradiction.

Grushko's theorem says that the generating rank (minimal number of generators) of a free product $B\ast C$ equals the sum of the generating ranks of $B$ and $C$. (This is false for direct products: if $B$ and $C$ are cyclic of order $2$ and $3$, then the generating rank of $B$, $C$, $B\times C$ are all equal to $1$.) In particular, for every finitely generated group $B$ and every group $C$, the free product $B\ast C$ is isomorphic to $B$ if and only if $C$ is a trivial group.