Linear programming and the intersection of free subgroups in free products of groups

Abstract

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1\ast G_2$ of two finite groups $G_1$, $G_2$, then the WN-coefficient $\sigma (H_1)$ of $H_1$ is rational and can be computed in exponential time in the size of $H_1$. This coefficient $\sigma (H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1\ast G_2$, it is true that $\bar{\mathrm{r}}(H_1,H_2)\le \sigma (H_1)\bar{\mathrm{r}}(H_1)\bar{\mathrm{r}}(H_2)$, where $\bar{\mathrm{r}}(H)$ = max (r $(H)-1,0)$ is the reduced rank of $H$, r$(H)$ is the rank of $H$, and $\bar{\mathrm{r}}(H_1,H_2)$ is the reduced rank of the generalized intersection of $H_1$ and $H_2$. In the case of the free product $G_1\ast G_2$ of two finite groups $G_1$, $G_2$, it is also proved that there exists a factor-free subgroup $H_2^{\ast }=H_2^{\ast }(H_1)$ such that $\bar{\mathrm{r}}(H_1,H_2^{\ast })=\sigma (H_1)\bar{\mathrm{r}}(H_1)\bar{\mathrm{r}}(H_2^{\ast })$, $H_2^{\ast }$ has at most doubly exponential size in the size of $H_1$, and $H_2^{\ast }$ can be constructed in exponential time in the size of $H_1$.