# On the intersection of tame subgroups in groups acting on trees

## TL;DR

This paper investigates the intersection properties of tame subgroups within groups acting on trees, providing bounds on the complexity of their intersections, especially in cases where subgroups act freely on edges.

## Contribution

It introduces bounds for the complexity, including Kurosh rank, of intersections of tame subgroups in groups acting on trees, under specific conditions.

## Key findings

- Bounds for Kurosh rank of subgroup intersections.
- Results for subgroups acting freely on edges.
- Insights into subgroup complexity in tree actions.

## Abstract

Let $G$ be a group acting on a tree $T$ with finite edge stabilizers of bounded order. We provide, in some very interesting cases, upper bounds for the complexity of the intersection $H\cap K$ of two tame subgroups $H$ and $K$ of $G$ in terms of the complexities of $H$ and $K$. In particular, we obtain bounds for the Kurosh rank $Kr(H\cap K)$ of the intersection in terms of Kurosh ranks $Kr(H)$ and $Kr(K)$, in the case where $H$ and $K$ act freely on the edges of $T$.