# The Hanna Neumann conjecture for Demushkin Groups

## TL;DR

This paper proves the Hanna Neumann conjecture for certain subgroups within nonsolvable Demushkin groups, confirming a key inequality related to subgroup intersections and their generating sets.

## Contribution

It establishes the Hanna Neumann conjecture for topologically finitely generated subgroups of nonsolvable Demushkin groups, a significant class of pro-p groups.

## Key findings

- Confirmed the conjecture for Demushkin groups
- Derived an inequality relating subgroup intersections and generators
- Extended the understanding of subgroup structure in pro-p groups

## Abstract

We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that \begin{equation*} \sum_{g \in U \backslash G/W} \bar d(U \cap gWg^{-1}) \leq \bar d(U) \bar d(W) \end{equation*} where $\bar d(K) = \max\{d(K) - 1, 0\}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.