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Finite Generalized Tetrahedron Groups with a High-Power Relator

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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 e i ≥2 and f i ≥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 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.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, England

    Martin Edjvet

  2. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, Scotland

    James Howie

  3. Fachbereich Mathematik, Universität Dortmund, 44221, Dortmund, Germany

    Gerhard Rosenberger

  4. Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, England

    Richard M. Thomas

Authors
  1. Martin Edjvet
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  2. James Howie
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  3. Gerhard Rosenberger
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  4. Richard M. Thomas
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Edjvet, M., Howie, J., Rosenberger, G. et al. Finite Generalized Tetrahedron Groups with a High-Power Relator. Geometriae Dedicata 94, 111–139 (2002). https://doi.org/10.1023/A:1020912827484

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