The problem
For nontrivial finitely generated subgroups \(H, K\) of a free group, \(\mathrm{rank}(H \cap K) - 1 \le (\mathrm{rank}(H) - 1)(\mathrm{rank}(K) - 1)\). (Resolved 2011, independently twice.)
The Hanna Neumann conjecture
historicgeometric-group-theory free-groups
History & significance
Hanna Neumann (1957) bounded the rank of an intersection of subgroups; Burns strengthened the conjectured bound by a factor of two, and the strengthened form resisted all combinatorial attack for fifty years — Tardos (1992) got within a hair but not all the way. In 2011 two independent proofs arrived months apart: Mineyev's Hilbert-module argument and Friedman's sheaf-theoretic one, both establishing the sharp bound. A case study in a problem falling twice at once after decades of nothing.
Connected problems
More in Geometric group theory
- same area Cannon's conjecture
- same area Amenability of Thompson's group F
- same area Residual Finiteness of Hyperbolic Groups
References
- primaryRead the primary source →
- sourceFriedman: sheaves proof (arXiv)
The resolution (human proof)
Mineyev (2011, published 2012) and independently Friedman (2011, published 2015) proved the strengthened Hanna Neumann bound: two entirely different machineries — Hilbert modules vs sheaves on graphs — converging on the same sharp inequality.
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