The Topology and Analysis of the Hanna Neumann Conjecture

The statement of the Hanna Neumann Conjecture (HNC) is purely algebraic: for a free group Γ and any nontrivial finitely generated subgroups A and B of Γ, rk (A ∩B)− 1 ≤ (rkA− 1)(rkB − 1). The goal of this paper is to systematically develop machinery that would allow for generalizations of HNC and to exhibit their relations with topology and analysis. On the topological side we define immersions of complexes, leafages, systems of complexes, flowers, gardens, and atomic decompositions of graphs and surfaces. The analytic part involves working with the classical Murray-von Neumann (!) dimension of Hilbert modules. This also gives an approach to the Strengthened Hanna Neumann Conjecture (SHNC) and to its generalizations. We present three faces of it named, respectively, the square approach, the diagonal approach, and the arrangement approach. Each of the three comes from the notion of a system, and each leads to questions beyond graphs and free groups. Partial results, sufficient conditions, and generalizations of the statement of SHNC are presented.