Finite Generalized Tetrahedron Groups with a High-Power Relator

Abstract

An ‘ordinary’ tetrahedron group is a group with a presentati on 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 ei≥2 and fi≥2 for each i. Following Vinberg, we call groups defined by a presentation 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 Ri(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.