Finite Quotients of Free Groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:42:39Z http://mathoverflow.net/feeds/question/121374 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121374/finite-quotients-of-free-groups Finite Quotients of Free Groups Jibran D. U. D. E. 2013-02-10T05:43:18Z 2013-02-11T21:50:16Z <p>It is interesting FACT that given $l,m,n\geq 2$, there is (are) a <em>finite group</em> with elements $a,b$ such that $o(a)=l, o(b)=m$, and $o(ab)=n$ (see <a href="http://mathoverflow.net/questions/118035/order-of-elements" rel="nofollow">link</a> for a nice example by Derek Holt / B. Sury). </p> <p>Although, the group </p> <blockquote> <p>$G_{l,m,n}=\langle a,b\colon a^l,b^m,(ab)^n\rangle$</p> </blockquote> <p>contains elements with above property (?), it need not be finite. But for certain values of $l,m,n$, $G_{l,m,n}$ is finite (and these are interesting groups by their geometry):</p> <p>$G_{2,2,n}\cong D_{2n}$,</p> <p>$G_{2,3,3}\cong A_4$,</p> <p>$G_{2,3,4}\cong S_4$,</p> <p>$G_{2,3,5}\cong A_5$.</p> <p>A natural question I would like to ask is:</p> <blockquote> <p>For what values of $l,m,n$, the group $G_{l,m,n}$ is finite?</p> </blockquote> <hr> <p>I wondered by the above FACT, and very useful discussion on it in <strong>mathoverflow</strong>, with nice answer by Derek Holt/ B. Sury. I tried to solve the question in the <a href="http://mathoverflow.net/questions/118035/order-of-elements" rel="nofollow">link</a>, and for $(l,m,n)=(2,2,n)$, I quickly found that the group $D_{2n}$ is the best example for it. Then I came up with the natural question above. This question may have been discussed with different point of view; but I didn't know too much about it. </p> <p>Also, if possible, one may suggest(and edit) suitable title for this question. </p> http://mathoverflow.net/questions/121374/finite-quotients-of-free-groups/121534#121534 Answer by Misha for Finite Quotients of Free Groups Misha 2013-02-11T21:50:16Z 2013-02-11T21:50:16Z <p>These groups are called von Dyck groups, see e.g. here. Von Dyck group $D(l,m,n)$ is finite if and only if it is of <em>spherical type</em>: $$\chi=-1+ l^{-1} + m^{-1} + n^{-1}>0.$$ A side note: Von Dyck groups are fundamental groups of 2-dimensional oriented orbifolds. The number $\chi$ above is the orbifold Euler characteristic. This is explained nicely, for instance, in Peter Scott's paper "Geometries of 3-manifolds", Bull. London Math. Soc. 15 (1983), no. 5, 401–487. </p>