Doubles, finiteness properties of groups, and quadratic isoperimetric inequalities (Q1291085)

Let \(G\) be a group and \(H\) a subgroup of \(G\). Then the double of \(G\) along \(H\), which the author denotes by \(\Delta_2(G;H)\), is the amalgamated free product of two copies of \(G\) with the two copies of \(H\) identified by the identity map. More generally, the author denotes by \(\Delta_m(G;H)\) the amalgamated free product of \(m\) copies of \(G\) along \(H\). We recall that the Dehn function \(f_G\) of \(G\) describes the optimal isoperimetric equality satisfied by \(G\). The author proves the following Theorem. Let \(G_1\) and \(G_2\) be finitely presented groups with epimorphisms to \(\mathbb{Z}\) (the group of integers), let \(\phi\colon G_1\times G_2\to\mathbb{Z}\) be the induced map, and let \(K=\ker\phi\). For every \(m\geq 2\), the group \(\Delta_m(G_1\times G_2;K)\) is finitely presented and its Dehn function is \(\simeq\max\{f_{G_1},f_{G_2}\}+n^2\). Some well-known groups studied by \textit{J. Stallings} [in Am. J. Math. 85, 541-543 (1963; Zbl 0122.27301)] (as the first examples of finitely generated groups which are not of type \(F_3\)) and reinterpreted by \textit{R. Bieri} [in Homological dimension of discrete groups (Queen Mary College Lecture Notes, 1976; Zbl 0357.20027)], can be described in terms of such doublings. Using this description, the author shows that these groups satisfy a quadratic isoperimetric inequality.