MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I just came across this statement in Bowditch's notes on geometric group theory that $\langle a,b\ |\ aba^{-1}b^{-2},a^{-2}b^{-1}ab \rangle$ is a presentation of the trivial group. Does anyone know if all presentations of the form $\langle a,b\ |\ a^{i_1}b^{j_1}\cdots a^{i_n}b^{j_n},a^{j_1}b^{i_1}\cdots a^{j_n}b^{i_n} \rangle$ generally present the trivial group? We can realize the presented group as the fundamental group of $$\text{glue two disks to $S^1\vee S^1$ along the relations}$$ and it seems like this construction is homotopy equivalent to $S^2$.

share|cite|improve this question
If $n=1$ and $i_1=j_1$, the group is not trivial. – Mariano Suárez-Alvarez Nov 12 '10 at 7:03
Put $I=i_1+\dotsb+i_n$ and $J=j_1+\dotsb+j_n$. Then the abelianisation has order $|I^2-J^2|$ (or is infinite if $I^2=J^2$). – Neil Strickland Nov 12 '10 at 7:50

If $n$ is big enough comparing to $i$'s and $j$'s, and $i$'s and $j$'s are sufficiently different, then this group satisfies the small cancellation condition $C'(\lambda)$ with $\lambda\le 1/6$ (that is these words do not contain common subwords of length $\gt 1/6$ of their length). This implies that the group is infinite, hyperbolic, and not virtually cyclic. This means that "generically" this group is infinite. By the way, the group in Bowditch's notes is a particular case of complexes of Baumslag-Solitar groups studied here.

share|cite|improve this answer

For any $n$, setting all $i$'s and $j$'s to $k$ gives a group which certainly surjects onto $C_k\times C_k$, for any $k$, which means it's definitely non-trivial.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.