## Duke Mathematical Journal

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

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

#### 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

$${\sum}_{x\in U\backslash F/W}\overline{rk}(U\cap xW{x}^{-1})\le \overline{rk}\left(U\right)\overline{\mathrm{rk}}\left(W\right),\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\overline{rk}\left(U\right)=max\hspace{0.17em}\{rk(U)-1,0\}.$$ 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