A Quasi-Isometry Invariant Loop Shortening Property for Groups
We first introduce a loop shortening property for metric spaces, generalizing the property considered by M. Elder on Cayley graphs of finitely generated groups. Then using this metric property, we define a very broad loop shortening property for finitely generated groups. Our definition includes Elder's groups, and unlike his definition, our property is obviously a quasi-isometry invariant of the group. Furthermore, all finitely generated groups satisfying this general loop shortening property are also finitely presented and satisfy a quadratic isoperimetric inequality. Every CAT(0) cubical group is shown to have this general loop shortening property.
International Journal of Algebra and Computation
Brick, S. G.,
Corson, J. M.,
(2008). A Quasi-Isometry Invariant Loop Shortening Property for Groups. International Journal of Algebra and Computation, 18(8), 1243-1257.
Available at: http://aquila.usm.edu/fac_pubs/1519