Submultiplicativity and the Hanna Neumann conjecture

In this article, we dene submultiplicativity of ℓ²-numbers in the category of Γ-complexes over a given Γ-complex X̂, which generalizes the statement of the Strengthened Hanna Neumann Conjecture (SHNC). In the case when Γ- is a left-orderable group and X is a free Γ-complex, we prove submulti-plicativity for the subcategory consisting of Γ-ordered leafages over X̂ with an additional analytic assumption called the deep-fall property. We show that the deep-fall property is satised for graphs. This implies SHNC.