The Hanna Neumann conjecture for surface groups

Abstract

The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.

On double coset separability and the Wilson–Zalesskii property

“…Hence, by Corollary 1.2, such groups have the Wilson–Zalesskii property. For limit groups this answers a question of Antolín and Jaikin–Zapirain from [2, Subsection 2.2].…”

“…The opposite inclusion is obvious, so (ii) has been established. We will now prove that (ii) implies (i) (in the case of profinite topology this was done in [2, Corollary 10.4]). Suppose that (ii) holds and $$M \lhd _o G$$ is arbitrary.…”

“…Acknowledgements. I am grateful to Pavel Zalesskii for fruitful discussions and for drawing my attention to the paper [2], which motivated this note.…”

“…Condition (1) played an important role in the papers of Ribes and Zalesskii [10], Ribes, Segal and Zalesskii [9], Wilson and Zalesskii [13] and Antolín and Jaikin‐Zapirain [2], to mention a few. In all of these papers, this condition was established along with (and after) the double coset separability condition, stating that for all $$H,K \in \mathcal {S}$$ $$$$\begin{equation} HK \text{ is separable in } G. \end{equation}$$$$…”

“…Following [2], we say that a group $$G$$ has the Wilson–Zalesskii property if (1) holds for arbitrary finitely generated subgroups $$H, K \leqslant G$$ this property is named after Wilson and Zalesskii, who established it in the case of finitely generated virtually free groups in [13]. We will call a group $$G$$ double coset separable if (2) holds for all finitely generated subgroups $$H,K \leqslant G$$.…”