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O. Bogopolski and E. Ventura (2008)

The mean Dehn functions of abelian groups

Journal of Group Theory, 11(4):569–586.

AbstractWhile 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 D mean(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) ˊ n 2 except in the 1-dimensional case), we show that the three variants D osmean(n), D smean(n) and D mean(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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