An asymptotic Freiheitssatz for finitely generated groups
The authors consider a generalization of \textit{W. Magnus}' well-known Freiheitssatz [Math. Ann. 106, 295-307 (1932; Zbl 0004.09709)] in a certain statistical sense: in almost every group presented with finitely many relations and with \(k\) generators a subgroup generated by \(k-1\) generators is a free group. Namely, let \(G=\langle X\mid R\rangle=\langle X\mid r_1,\dots,r_l\rangle\) be a group presentation with a generating system \(X\) of \(k\geq 2\) elements and with \(l\geq 1\) (cyclically reduced) relations \(r_1=1,\dots,r_l=1\) of lengths \(|r_i|=n_i\); \(i=1,\dots,l\). Denote by \(Pr(X,n_1,\dots,r_l)\) the set of group presentations of the type mentioned. It is clear that \(|Pr(X,n_1,\dots,r_l)|<\infty\) holds. A given property \(P\) of group presentations is said to be generic in the sense of Gromov if \[ \lim_{n\to\infty}{{|\{\langle X\mid R\rangle\in Pr(X,n_1,\dots,r_l)\mid\langle X\mid R\rangle{\text{ satisfies }}P\}|}\over{|Pr(X,n_1,\dots,r_l)|}}=1, \] where \(n=\min\{n_1,\dots,r_l\}\). The authors prove the following fact. Let \(G=\langle X\mid R\rangle\) be a finite presentation with \(k\) generators, \(l\) relations and with any fixed \(x_0\in X\). Then the fact that the set \(X-\{x_0\}\) generates a free group in \(G\) is generic in the sense of Gromov (Theorem 1.1). This result is closely related to a result of \textit{G. N. Arzhantseva} and \textit{A. Yu. Ol'shanskij} [Mat. Zametki 59, No. 4, 489-496 (1996; Zbl 0877.20021)] who proved a more general statement but for a different notion of genericity.
scientific article; zbMATH DE number 1558717