The intersection of subgroups in free groups and linear programming

TL;DR

This paper introduces a linear programming approach to analyze the intersection properties of finitely generated subgroups in free groups, providing rational coefficients and efficient algorithms for their computation.

Contribution

It establishes the rationality of the WN-coefficient for subgroups and presents an exponential-time algorithm to compute it, along with constructing a subgroup achieving this bound.

Findings

The WN-coefficient $\sigma(H_1)$ is rational and computable in exponential time.

Existence of a subgroup $H_2^*$ that attains the minimal intersection bound.

The Stallings graph of $H_2^*$ can be constructed with at most doubly exponential size in exponential time.

Abstract

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if $H_1$ is a finitely generated subgroup of a free group $F$, then the WN-coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed in deterministic exponential time in the size of $H_1$. This coefficient $\sigma(H_1)$ is the minimal nonnegative real number such that, for every finitely generated subgroup $H_2$ of $F$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar{ {\rm r}} (H) := \max ( {\rm r} (H)-1,0)$ is the reduced rank of $H$, ${\rm r} (H)$ is the rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is the reduced rank of the generalized intersection of $H_1$ and $H_2$. We also show the existence of a subgroup $H_2^* = H_2^*(H_1)$ of $F$ such that $\bar {\rm r}(H_1, H_2^*) = \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2^*)$, the Stallings graph $\Gamma(H_2^*)$ of $H_2^*$ has at most doubly exponential size in the size of $H_1$ and $\Gamma(H_2^*)$ can be constructed in exponential time in the size of $H_1$.