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Star Graphs, Projective Planes and Free Subgroups in Small Cancellation Groups

Star Graphs, Projective Planes and Free Subgroups in Small Cancellation Groups STAR GRAPHS, PROJECTIVE PLANES AND FREE SUBGROUPS IN SMALL CANCELLATION GROUPS MARTIN EDJVET and JAMES HOWIE [Received 15 March 1987] 1. Introduction Small cancellation groups, like one-relator groups, are generalizations of surface groups, and satisfy many of the nice properties associated with those groups. The present paper concerns the property of admitting a free subgroup of rank 2. A result of Collins [5] asserts that, with a few obvious exceptions, a group given by a presentation satisfying C(4) and T(4) contains a free subgroup of rank 2. The analogous result for C(6) groups was proved by Al-Janabi [1] (see also [2]). An alternative proof of Collins's result, using star graphs, is given by El-Mosalamy [7]. In this paper we also use star graphs, and prove an analogous theorem for groups satisfying the third of the standard sets of (non-metric) small-cancellation conditions C(3) and T(6) (see §2 for a definition). MAIN THEOREM. Let G be a group admitting a presentation which satisfies C(3) and T(6). Then either G contains a free subgroup of rank 2, or G is isomorphic to one of the following: (i)Z (/i^l) ; (ii) Z; (iii) Z * Z ; 2 2 (iv) ZxZ ; (v) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Proceedings of the London Mathematical Society Wiley

Star Graphs, Projective Planes and Free Subgroups in Small Cancellation Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6115
eISSN
1460-244X
DOI
10.1112/plms/s3-57.2.301
Publisher site
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Abstract

STAR GRAPHS, PROJECTIVE PLANES AND FREE SUBGROUPS IN SMALL CANCELLATION GROUPS MARTIN EDJVET and JAMES HOWIE [Received 15 March 1987] 1. Introduction Small cancellation groups, like one-relator groups, are generalizations of surface groups, and satisfy many of the nice properties associated with those groups. The present paper concerns the property of admitting a free subgroup of rank 2. A result of Collins [5] asserts that, with a few obvious exceptions, a group given by a presentation satisfying C(4) and T(4) contains a free subgroup of rank 2. The analogous result for C(6) groups was proved by Al-Janabi [1] (see also [2]). An alternative proof of Collins's result, using star graphs, is given by El-Mosalamy [7]. In this paper we also use star graphs, and prove an analogous theorem for groups satisfying the third of the standard sets of (non-metric) small-cancellation conditions C(3) and T(6) (see §2 for a definition). MAIN THEOREM. Let G be a group admitting a presentation which satisfies C(3) and T(6). Then either G contains a free subgroup of rank 2, or G is isomorphic to one of the following: (i)Z (/i^l) ; (ii) Z; (iii) Z * Z ; 2 2 (iv) ZxZ ; (v)

Journal

Proceedings of the London Mathematical Society – Wiley

Published: Sep 1, 1988

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