# von dyck groups and solvability

A von Dyck group is a group with presentation $< a,b | a^m=b^n=(ab)^p=1 >$ with m,n,p natural numbers. Is it known which of these groups are solvable and which are not? Is there a reference for this? Thanks.

• The groups with $\frac{1}{m} + \frac{1}{n} + \frac{1}{p} > 1$ are all finite, and all solvable except $\Delta(2,3,5) \cong A_5$. The groups with $\frac{1}{m} + \frac{1}{n} + \frac{1}{p} = 1$ are all infinite and solvable: their commutator subgroups are isomorphic to $\mathbb{Z}^2$. The groups with $\frac{1}{m} + \frac{1}{n} + \frac{1}{p} < 1$ are all infinite and nonsolvable: indeed, with only finitely many exceptions, each of these groups has a simple group $PSL_2(\mathbb{F}_q)$ as a quotient. Jun 21, 2010 at 2:27
• Thank you for the comment. Do you know where I can find a proof for the case of 1/m + 1/n + 1/p = 1?
– dave
Jun 21, 2010 at 3:41
• These days, these groups are usually called triangle groups. You might like to look at the answers to the following question, which was very similar. mathoverflow.net/questions/22459/x-y-xp-yp-xyp-1/22463#22463
– HJRW
Jun 22, 2010 at 21:27