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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Measuring the tameness of almost convex groups


Authors: Susan Hermiller and John Meier
Journal: Trans. Amer. Math. Soc. 353 (2001), 943-962
MSC (2000): Primary 20F65; Secondary 20F69, 57M07
Published electronically: October 11, 2000
MathSciNet review: 1804409
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Abstract:

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points $x$ in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions.


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Additional Information

Susan Hermiller
Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: smh@math.unl.edu

John Meier
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
Email: meierj@lafayette.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02717-3
PII: S 0002-9947(00)02717-3
Keywords: Tame combings, almost convex groups, covering conjecture, rewriting systems, isoperimetric inequalities
Received by editor(s): June 21, 1999
Published electronically: October 11, 2000
Additional Notes: Susan Hermiller acknowledges support from NSF grant DMS-9623088
John Meier acknowledges support from NSF RUI grant DMS-9704417
Article copyright: © Copyright 2000 American Mathematical Society