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On Dehn functions of free products of groups. (English) Zbl 0942.20023
Associated to a finite presentation of a group \(G\) is a Dehn function \(f\), where \(f(n)\) is the largest minimum number of conjugates of relators needed in a product that represents (in the free group on the generators) a word of length at most \(n\) that is trivial as an element of \(G\). Dehn functions for different finite presentations of \(G\) are equivalent in the sense that there is a positive integer \(C\) such that \(f(n)\leq Cg(Cn)+C\) and \(g(n)\leq Cf(Cn)+C\) for all \(n\). It is conjectured that every Dehn function is equivalent to a subnegative function, i.e., a function such that \(f(m)+f(n)\leq f(m+n)\) for all natural numbers \(m\) and \(n\). Any function \(f\) defined on the natural numbers has a subnegative closure \(\overline f\), defined by \(\overline f(n)=\max\{f(n_1)+\cdots+f(n_r)\}\) taken over all \(r\geq 1\) and all choices of \(n_j\) such that \(n_1+\cdots+n_r=n\). It is the smallest subnegative function with \(\overline f(n)\geq f(n)\) for all \(n\).
The main result of the paper is that if \(G_1\) and \(G_2\) are finitely presented nontrivial groups, then the Dehn function of \(G_1*G_2\) is equivalent to its subnegative closure. Using an inequality of S. G. Brick [Trans. Am. Math. Soc. 335, No. 1, 369-384 (1993; Zbl 0892.57001)], this implies that if \(f_i\) are Dehn functions for \(G_i\) for \(i=1,2\), then the Dehn function \(f\) of \(G_1*G_2\) is equivalent to \(\max\{\overline{f_1},\overline{f_2}\}\). The proof relies on careful examination and manipulation of diagrams.

MSC:
20F65 Geometric group theory
57M07 Topological methods in group theory
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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