### M Edjvet, J Howie, G Rosenberger and R M Thomas

#### Abstract

A "ordinary" tetrahedron group is a group with a presentation of the form

$$\langle x,y,z:x^{e_1}=y^{e_2}=z^{e_3}=(xy^{-1})^{f_1}=(yz^{-1})^{f_2}=(zx^{-1})^{f_3}=1\rangle$$

where $e_i\ge 2$ and $f_i\ge 2$ for each $i$. Following Vinberg, we call groups defined by a presentations of the form

$$\langle x,y,z:x^{e_1}=y^{e_2}=z^{e_3}=R_1(x,y)^{f_1}=R_2(y,z)^{f_2}=R_3(z,x)^{f_3}=1\rangle,$$

where each $R_i(a,b)$ is a cyclically reduced word involving both $a$ and $b$, generalized tetrahedron groups}. These groups appear in many contexts, not least as subgroups of generalized triangle groups. In this paper, we build on previous work to start on a complete classification as to which generalized tetrahedron groups are finite; here we treat the case where at least one of the $f_i$ is greater than three.