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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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