We will present a proof of the Strengthened Hanna Neumann Conjecture (SHNC), and some more general results. We will mention trees, forests, flowers, gardens, and leafages. Submultiplicativity is a generalization of the statement of SHNC from graphs to complexes, and from free groups to more general groups. Submultiplicativity holds for complexes under an additional assumption: the deep-fall property. This property is related to the Atiyah Conjecture, a problem from analysis.
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