On the rank of the intersection of free subgroups in virtually free groups. (Q406345)

Let \(F\) be a free group and \(H,K\) finitely generated subgroups of \(F\). In [\textit{J. Friedman}, Mem. Am. Math. Soc. 1100 (2015; Zbl 1327.20025)] and [\textit{I. Mineyev}, J. Topol. Anal. 4, No. 1, 1-12 (2012; Zbl 1257.20034)] it is proved Hanna Neumann's conjecture which states that NEWLINE\[NEWLINE\overline r(H\cap K)\leq\overline r(H)\overline r(K);NEWLINE\]NEWLINE where \(\overline r(H)=\max (r(H)-1,0)\) is the reduced rank of a free group \(H\) of (free) rank \(r(H)\).NEWLINENEWLINE In [\textit{S. V. Ivanov}, Int. J. Algebra Comput. 9, No. 5, 521-528 (1999; Zbl 1028.20021)] a relevant result is proved for free products. Namely if \(G=G_1*G_2\) is the free product of the groups \(G_1\) and \(G_2\), \(H\) and \(K\) are finitely generated subgroups of \(G\) which intersect trivially with all the conjugates to the factors \(G_1\) and \(G_2\) (therefore are free), then the following estimate holds: NEWLINE\[NEWLINE\overline r(H\cap K)\leq 6\overline r(H)\overline r(K).NEWLINE\]NEWLINE In [Sb. Math. 204, No. 2, 223-236 (2013); translation from Mat. Sb. 204, No. 2, 73-86 (2013; Zbl 1285.20027)] the author generalized the result above and proved an analogous result for the rank of the intersection of free subgroups in free products with amalgamation of a finite normal subgroup.NEWLINENEWLINE Here the author extents the previous results in a larger class of groups. More precisely he proves the Theorem. Suppose \(G\) is the fundamental group of a finite graph of groups \((\Gamma,Y)\) with finite edge groups \(G_e\). Let \(H,K\leq G\) be finitely generated subgroups which intersect trivially with the conjugates to all the vertex groups of \((\Gamma,Y)\) (and are, therefore, free). Then the following estimate holds: NEWLINE\[NEWLINE\overline r(H\cap K)\leq 6m\cdot\overline r(H)\overline r(K),\tag{*}NEWLINE\]NEWLINE where NEWLINE\[NEWLINEm=\max\{|g^{-1}G_eg\cap HK|,\;e\in E(Y),\;g\in G\}.NEWLINE\]NEWLINE In particular, NEWLINE\[NEWLINE\overline r(H\cap K)\leq 6m'\cdot\overline r(H)\overline r(K),NEWLINE\]NEWLINE where \(m'\) is the maximum of the orders of the edge groups of \((\Gamma,Y)\).NEWLINENEWLINE The proof of this theorem is based on the Bass-Serre theory. The author proves some interesting lemmas which guide to an equivalent inequality of (*) in terms of the degrees of vertices of certain graphs. Then he uses the idea of \textit{S. V. Ivanov} [loc. cit.] and terminates the proof.NEWLINENEWLINE Recalling the well known characterization of \textit{A. Karrass A. Pietrowski} and \textit{D. Solitar} [in J. Aust. Math. Soc. 16, 458-466 (1973; Zbl 0299.20024)] for the finitely generated virtually free groups, the author hints to possible applications of his result. -- Also it is pointed out, by an example, that the above estimate is sharp.