Presentations of the unity group

Question

I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for it but didn't seem to find anything relevant. So is this article available online? If so, can you point me to it? In any case, can you give me a couple of examples of presentations of that group?

Answer 1

One such example is $\langle a,b,c \mid a^{-1}ba=b^2, b^{-1}cb=c^2, c^{-1}ac=a^2 \rangle$.

You can use this to construct a sequence of examples of increasing complexity. The example above is the first in the sequence and has total relator length $15$. The second group in the sequence is

$$\langle a,b,c \mid A^{-1}BA=B^2, B^{-1}CB=C^2, C^{-1}AC=A^2 \rangle,$$ where $A=a^{-1}bab^{-2}$, $B=b^{-1}cbc^{-2}$, $C=c^{-1}aca^{-2}$, so the total relator length is $75$. You can then repeat this idea to get further more complicated examples.

If you believe that the first presentation defines the trivial group, then it is not hard to prove that the second one does too. The first group is easily proved trivial by coset enumeration programs, but the second one is much harder.