Coherence, Locally Quasiconvexity, and the Perimeter of 2-complexes

A group is called coherent when all of its finitely generated subgroups are finitely presented. Although many examples of coherent groups are known, (most notably polycyclic groups and 3-manifold groups) few tests are known for positively determining the coherence of an arbitrary group presentation. In this article we provide such a test. The test is based upon a definition of the perimeter of a map between two finite 2-complexes which is introduced here. As an application we prove that one-relator groups with sufficient torsion are coherent. We also prove the coherence of several small cancellation groups and Coxeter groups. 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 most of these groups we can show in addition that they are locally quasiconvex. A subspace Y of a geodesic metric space X is called quasiconvex if there is an ǫ neighborhood of Y which contains all of the geodesics in X which start and end in Y . In group theory, a subgroup H of a group G generated by A is called quasiconvex if the vertices corresponding to H form a quasiconvex subspace of the Cayley graph Γ(G,A). If all of its finitely generated subgroups are quasiconvex, then G (generated by A) is called locally quasiconvex. Finally, our results give an alternative proof of the coherence and local quasiconvexity of many 3-manifold groups.