Submultiplicativity and the Hanna Neumann conjecture. (Q764063)

The famous Hanna Neumann Conjecture (HNC) from 1956-1957 says that if \(F\) is a nontrivial free group and \(A\) and \(B\) are two nontrivial finitely generated subgroups of \(F\), then \[ r(A\cap B)-1\leq (r(A)-1)(r(B)-1), \] where \(r(F)\) is the rank of a free group \(F\). The Strengthened Hanna Neumann Conjecture (SHNC), formulated by her son Walter Neumann (1989-1990), states that in the above situation \[ \sum_{AgB\in A\backslash F/B}\overline r(g^{-1}Ag\cap B)\leq\overline r(A)\overline r(B), \] where \(\overline r(F):=\max\{0,r(F)-1\}\) is the reduced rank of a free group \(F\) and \(A\backslash F/B=\{AgB\mid g\in F\}\) is the set of all double cosets \(AgB\) for \(g\in F\). The author of the paper under review by discussing `submultiplicativity' which is the term for a general SHNC-like property for \(F\)-complexes and based on his earlier work [J. Topol. Anal. 3, No. 3, 307-376 (2011; Zbl 1236.20030)] gives a nice proof for SHNC as follows: Let \(F\) be a free group and \(X\) be a finite graph with \(\pi_1(X)\cong F\). Consider immersions of finite graphs \(\alpha\colon Y\to X\) and \(\beta\colon Z\to X\) representing the subgroups \(A,B\leq F\), respectively. Let \(p_X\colon\widehat X\to X\) be the universal covering. Since free groups are left-orderable, the systems generated by \(\alpha,\beta,p_X\) are deep fall. Hence \[ b_i^{(2)}(\widehat S;F)\leq b_i^{(2)}(\widehat Y;F)\cdot b_i^{(2)}(\widehat Z;F), \] where \(\widehat S,\widehat Y,\widehat Z\) are in correspondence with \(S,Y,Z\), respectively, in the pull-back of the fiber-product diagram for \(Y,Z\) and \(b_i^{(2)}(\widehat Y;F):=\dim_FH_i^{(2)}(\widehat Y)\) is called the \(i\)-th `\(\ell^2\)-Betti number' in which \(\dim_F\) is the Murray-von Neumann dimension of Hilbert \(F\)-module. Finally by the earlier work of the author [loc. cit.] the last inequality is equivalent to SHNC.