User mike - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:21:49Z http://mathoverflow.net/feeds/user/10780 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45798/presentations-of-the-trivial-group presentations of the trivial group Mike 2010-11-12T06:53:02Z 2010-11-12T14:55:56Z <p>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 $S^2$.</p>