Using an infinite number of Tietze transformations

Question

I have a group presentation $G\cong\langle R|S\rangle$ which I am willing to reduce to $G\cong\langle S'|R'\rangle$ by making use of Tietze transformations. In my case, I am only using the following transformation :

Removing a generator :

If a relation can be formed where one of the generators is a word in the other generators then that generator may be removed. In order to do this it is necessary to replace all occurrences of the removed generator with its equivalent word.

In my case, I can reduce my presentation to having $|S'|=2$. However, to achieve this goal, I am requiring to be able to use an infinite number of such transformations. My setting is that I have $R=\{t\}\coprod\{a_n,\;n\in\mathbb{Z}\}$ and I can express $a_n$ in terms of $t$ and $a_0$ using relations in $R$. Applying a finite number of Tietze transformations, I can reduce the presentation to any set of generators of the form : $$S''=\{t,a_0\}\coprod\{a_n,\;|n|\geqslant N\}$$ with arbitrarily large $N$. My question is therefore :

Is it allowed to make use of an infinite number of Tietze transformations to transform a presentation into another ?

I am expecting this result to be true in fact, since, admitting this is working just fine, the presentation I obtain in the end is exactly the one I am expected to give. Intuition tends to tell me that there's nothing wrong with this, but usually intuition is not working very well when messing with doing something "infinitely many times"...

I couldn't find anything anywhere about the proof of this fact. I even dug deep to try my luck with Tietze's 1908 paper, which was unfortunately never translated.

Answer 1

The definition in wikipedia is incorrect, essentially for situations like you describe. Wikipedia is really defining elementary Tietze transformations, which is fine for finite presentations but not in general.

The following definition is from Section 1.5 of the book Combinatorial group theory by Magnus, Karrass and Solitar. This is a solid textbook, and although I haven't compared it with the original 1908 paper, I see no reason to suspect it is incorrect. Theorem 1.5 of this book proves Tietze's result.

In 1908, H. Tietze showed that given a presentation \begin{align*} (7)&&\langle a, b, c, \ldots \mid P, Q, R\rangle \end{align*} for a group $G$, any other presentation for $G$ can be obtained by repeated application of the following transformations to (7):

(T1) If the words $S, T, \ldots$ are derivable from $P, Q, R, \ldots$, then add $S, T, \ldots$ to the defining relators in (7).

(T2) If some of the relators, say, $S, T, \ldots$, listed among the defining relators $P, Q, R, \ldots$ are derivable from the others, delete $S, T, \ldots$ from the defining relators in (7).

(T3) If $k, M, \ldots$ are words in $a, b, c, \ldots$, then adjoin the symbols $x, y, \ldots$ to the generating symbols in (7) and adjoin the relations $x=K, y=M, \ldots$ to the defining relators in (7).

(T4) If some of the defining relations in (7) take the form $p=V, q=W, \ldots$ where $p, q, \ldots$ are generators in (7) and $V, W, \ldots$ are words in the generators other than $p, q, \ldots$, then delete $p, q, \ldots$ from the generators, delete $p=V, q=W, \ldots$ from the defining relations, and replace $p, q, \ldots$ by $V, W, \ldots$ respectively, in the remaining defining relators in (7).

... a Tietze transformation is called elementary if it involves the insertion or deletion of one defining relator, or the insertion or deletion of one generator and the corresponding defining relation.

For example, by the above definition the group defined by $$\langle a, b, c, \ldots\mid, a=1, b=1,c=1, \ldots\rangle$$ is seen to be trivial, as a single application of (T4) gives the empty presentation. However, if we only allow the "elementary" moves, as in Wikipedia, we never obtain the empty presentation.