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LOCALLY QUASICONVEX SMALL-CANCELLATION GROUPS JONATHAN P. MCCAMMOND 1 AND DANIEL T. WISE 2
 

Summary: LOCALLY QUASICONVEX SMALL-CANCELLATION GROUPS
JONATHAN P. MCCAMMOND 1 AND DANIEL T. WISE 2
Abstract. In this article we prove several results about the local quasicon-
vexity behavior of small cancellation groups. In addition to strengthening of
our previously obtained positive results, we also describe several families of
negative examples. Also, as the strength of the assumed small cancellation
conditions increases, the gap between our positive results and our counterex-
amples narrows. Finally, as an additional application of these techniques, we
include similar results and counterexamples for Coxeter groups.
It has been known for some time that the class of small cancellation groups
contains groups which are coherent, groups which are incoherent, groups which are
locally quasiconvex and groups which are not locally quasiconvex [2, 12, 13, 15].
However, there remains a large gap between the hypothesis necessary to obtain
positive results and available counterexamples. In this article, we begin closing this
gap by combining perimeter technique we introduced in [13] with the concept of a
fan we developed in [14]. On the positive side we derive a number of new results
based on the following theorem which is a combination of the main theorem of [14]
with one of the main theorems in [13].
Theorem 3.10. Let X be a compact weighted C(p)-T(q) complex, where p, q, and
k satisfy the Euclidean restrictions. If every minimal fan of type k in X is both

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics