The mean Dehn function of abelian groups
Abstract
While Dehn functions, D(n), of finitely presented groups are very well studied in the literature, mean Dehn functions are much less considered. M. Gromov introduced the notion of mean Dehn function of a group, $D_{mean}(n)$, suggesting that in many cases it should grow much more slowly than the Dehn function itself. Using only elementary counting methods, this paper presents some computations pointing into this direction. Particularizing them to the case of any finite presentation of a finitely generated abelian group (for which it is well known that $D(n)\sim n^2$ except in the 1dimensional case), we show that the three variations $D_{osmean}(n)$, $D_{smean}(n)$ and $D_{mean}(n)$ all are bounded above by $Kn(\ln n)^2$, where the constant $K$ depends only on the presentation (and the geodesic combing) chosen. This improves an earlier bound given by Kukina and Roman'kov.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606273
 Bibcode:
 2006math......6273B
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Combinatorics;
 20F65;
 20F05;
 20K99
 EPrint:
 View also http://www.epsem.upc.edu/~ventura/ and http://math.nsc.ru/~bogopolski/