## Duke Mathematical Journal

### Approximation by subgroups of finite index and the Hanna Neumann conjecture

Andrei Jaikin-Zapirain

#### 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

$$\sum_{x\in U\backslash F/W}\overline{\operatorname{rk}}(U\cap xWx^{-1})\le\overline{\operatorname{rk}}(U)\overline{\mathrm{rk}}(W),\quad \mbox{where }\overline{\operatorname{rk}}(U)=\max\{\operatorname{rk}(U)-1,0\}.$$ 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
Revised: 17 November 2016
First available in Project Euclid: 16 March 2017

https://projecteuclid.org/euclid.dmj/1489629612

Digital Object Identifier
doi:10.1215/00127094-0000015X

Mathematical Reviews number (MathSciNet)
MR3679885

Zentralblatt MATH identifier
1375.20035

#### 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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