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

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