The mean Dehn functions of abelian groups

O. Bogopolski 1  and E. Ventura 2
  • 1 O. Bogopolski, Institute of Mathematics, Koptjuga 4, Novosibirsk, 630090, Russia, and Universität Dortmund, Fachbereich Mathematik, Lehrstuhl VI (Algebra), Vogelpothsweg 87, 44221 Dortmund, Germany. E-mail:
  • 2 E. Ventura, Dept. Mat. Apl. III, Univ. Pol. Catalunya, Barcelona, Spain, and Centre de Recerca Matemàtica, Barcelona, Spain. E-mail:


While Dehn functions D(n) of finitely presented groups have been well studied in the literature, mean Dehn functions have received less attention. Gromov introduced the notion of the mean Dehn function Dmean(n) of a group, suggesting that in many cases it should grow more slowly than the Dehn function itself This paper presents computations pointing in this direction. In the case of any finite presentation of an infinite finitely generated abelian group (for which it is well known that D(n) ~ n2 except in the 1-dimensional case), we show that the three variants Dosmean(n), Dsmean(n) and Dmean(n) all are bounded above by Kn(In 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.

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The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.