## Test your intuition 24: Which of the following three groups is trivial

Martin Bridson

We have three finitely presented groups

A is generated by two generators a and b and one relation  $a^{-1} \cdot b\cdot a = b^2$

B is generated by three generators a, b, c and three relations $a^{-1} \cdot b\cdot a = b^2$,  $b^{-1}\cdot c\cdot b = c^2,$  $c^{-1}\cdot a\cdot c = a^2$.

C is generated by four generators a, b, c, d and four relations $a^{-1} \cdot b\cdot a = b^2$,  $b^{-1}\cdot c\cdot b = c^2,$  $c^{-1}\cdot d\cdot c = d^2$, and $d^{-1}\cdot a \cdot d = a^2$.

Test your intuition: which of the groups A, B or C is trivial

Update: I did not know the answer (and I feel now better about it).

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### 10 Responses to Test your intuition 24: Which of the following three groups is trivial

1. Paco Santos says:

One of them is easy to discard, since its abelianization is not trivial. (Gil, feel free to delete this if you think it is too much of a clue)

• domotorp says:

Moreover, there are two generators and only one equation.

• Gabor Pete says:

Moreover, one can discard that without any group-theoretical knowledge: if that was trivial, the other two would also be trivial.

Btw, the task is a bit ill-posed: it is to test our intuition, so we need to think enough to start having an intuition, but need to stop thinking before starting to get the actual answer?

• Gil Kalai says:

No, no you can think enough to have an intuition and then when you get the actual answer you can test your own intuition. (Thinking about it more, and even working it out is most welcome. Of course, one can look in the literature or wait for the answer.)

• Gil Kalai says:

Dear Paco, domotorp, and Gabor. You are all correct, of course. Indeed Martin tested us for just two out of the three. (Gabor, Initially I sillingly misunderstood your comment to be: “If its trivial to answer for one group, than it must be trivial to answer for the other groups” 🙂 .)

2. Yiftach Barnea says:

Gil, I think there is a typo. Shouldn’t it be $c^{-1}dc=d^2$ in group C?

GK: yes yes it was a typo. corrected now, thanks Yiftach

3. Yiftach Barnea says:

Thinking about it, don’t you want also the relation $b^-1ab=a^2$ in A?

• Gil Kalai says:

Dear Yiftach, what difference does it make?

• Yiftach Barnea says:

I believed you are following some pattern trying to show that even though the there is a pattern in relations, there isn’t one in the answers. But maybe I was looking for a pattern where there isn’t one.

4. Please can someone explain the answer, or tell me where I can find such an explanation?