Measuring the tameness of almost convex groups

Hermiller, Susan; Meier, John

doi:10.1090/S0002-9947-00-02717-3

Measuring the tameness of almost convex groups
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by Susan Hermiller and John Meier

Trans. Amer. Math. Soc. 353 (2001), 943-962

DOI: https://doi.org/10.1090/S0002-9947-00-02717-3

Published electronically: October 11, 2000

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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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