Duke Mathematical Journal

Approximation by subgroups of finite index and the Hanna Neumann conjecture

Andrei Jaikin-Zapirain

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1489629612

Digital Object Identifier
doi:10.1215/00127094-0000015X

Mathematical Reviews number (MathSciNet)
MR3679885

Zentralblatt MATH identifier
1375.20035

Subjects
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]

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

Citation

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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