Abstract
In this article, we define submultiplicativity of $\ell^2$-numbers in the category of $\Gamma$-complexes over a given $\Gamma$-complex $\hat{X}$, which generalizes the statement of the Strengthened Hanna Neumann Conjecture (SHNC). In the case when $\Gamma$ is a left-orderable group and $\hat{X}$ is a free $\Gamma$-complex, we prove submultiplicativity for the subcategory consisting of $\Gamma$-ordered leafages over $\hat{X}$ with an additional analytic assumption called the deep-fall property. We show that the deep-fall property is satisfied for graphs. This implies SHNC.
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