# Linear programming and the intersection of subgroups in free groups

Research paper by **Sergei V. Ivanov**

Indexed on: **27 Jul '16**Published on: **27 Jul '16**Published in: **Mathematics - Group Theory**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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 Walter Neumann
coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed in
deterministic exponential time of size of $H_1$. This coefficient $\sigma(H_1)$
is a 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 a
generalized intersection of $H_1, H_2$.