Coherence, local quasiconvexity, and the perimeter of 2-complexes

2002/12/31 by Jonathan P. McCammond, McCammond, Jonathan P., Daniel T. Wise +1 · 2 citations
Mathematics · #20F06 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F06

paper · pdf · doi:10.48550/arxiv.math/0212381

60 pages

arxiv created 2002/12/31 · arxiv updated 2009/11/30

Abstract

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that one-relator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3-manifold groups. The main application is to establish the coherence and local quasiconvexity of many small cancellation groups.

Cited by

• Quasi-convex surface subgroups in some one-relator groups with torsion
• The geometry of subgroups of mapping tori of free groups

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