Duke Mathematical Journal

Approximation by subgroups of finite index and the Hanna Neumann conjecture

Andrei Jaikin-Zapirain

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Let F be a free group (pro-p group), and let U and W be two finitely generated subgroups (closed subgroups) of F. The Strengthened Hanna Neumann conjecture says that

xU\F/Wrk¯(UxWx1)rk¯(U)rk¯(W),whererk¯(U)=max {rk(U)1,0}. This conjecture was proved independently in the case of abstract groups by J. Friedman and I. Mineyev in 2011.

In this paper we give the proof of the conjecture in the pro-p context, and we present a new proof in the abstract case. We also show that the Lück approximation conjecture holds for free groups.

Article information

Duke Math. J., Volume 166, Number 10 (2017), 1955-1987.

Received: 29 November 2015
Revised: 17 November 2016
First available in Project Euclid: 16 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E18: Limits, profinite groups
Secondary: 20E05: Free nonabelian groups 20J05: Homological methods in group theory 16A06 20C07: Group rings of infinite groups and their modules [See also 16S34] 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]

free groups and pro-$p$ groups the Hanna Neumann conjecture Lück’s approximation


Jaikin-Zapirain, Andrei. Approximation by subgroups of finite index and the Hanna Neumann conjecture. Duke Math. J. 166 (2017), no. 10, 1955--1987. doi:10.1215/00127094-0000015X. https://projecteuclid.org/euclid.dmj/1489629612

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