The topology and analysis of the Hanna Neumann conjecture

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Abstract

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 ≤ (rk A - 1)(rk B - 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 Murrayvon 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.

Original languageEnglish (US)
Pages (from-to)307-376
Number of pages70
JournalJournal of Topology and Analysis
Volume3
Issue number3
DOIs
StatePublished - Sep 2011

Keywords

  • Atiyah Conjecture
  • Hanna Neumann Conjecture
  • Hilbert module
  • L Betti numbers
  • Murrayvon Neumann dimension
  • arrangement
  • atomic decomposition
  • ergodicity for vector space
  • flower
  • free group
  • garden
  • immersion
  • slide map
  • square map
  • surface group
  • system of complexes

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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