Finite Generalized Tetrahedron Groups with a High-Power Relator期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Finite Generalized Tetrahedron Groups with a High-Power Relator

Authors:

Martin Edjvet James Howie Gerhard Rosenberger Richard M. Thomas

Affiliation:

(1) School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, England;(2) Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, Scotland;(3) Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany;(4) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, England

Abstract:

ordinary

tetrahedron group is a group with a presentati on of the form

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

where e_i

2 and f_i

2 for each i. Following Vinberg, we call groups defined by a presentation of the form

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

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.

Keywords:

essential representation generalized tetrahedron group generalized triangle group spelling theorem trace polynomial triangle of groups

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