On certain finite generalized tetrahedron groups (Q2702045)

A generalized tetrahedron group has a presentation of the form NEWLINE\[NEWLINE\langle x,y,z\mid x^{e_1}=y^{e_2}=z^{e_3}=R_1(x,y)^{f_1}=R_2(y,z)^{f_2}=R_3(z,x)^{f_3}=1\rangleNEWLINE\]NEWLINE where \(R(a,b)\) is a reduced word involving both \(a\) and \(b\). In the present paper the question is considered when such a group is infinite (this question has been solved before for generalized triangle groups, obtained from the presentation above by deleting one generator and all relations containing it). The paper surveys some known results and gives some new conditions for the groups to be infinite. For example, a generalized tetrahedron group is infinite if \(1/{f_1}+1/{f_2}+1/{f_3}\leq 1\) (noting that a generalized tetrahedron group is a ``triangle'' of three generalized triangle groups, the proof uses van Kampen diagrams to show that a non-spherical triangle of groups is infinite).NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].