The intersection of subgroups in free groups and linear programming

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 σ(H 1 ) of H 1 is rational and can be computed in deterministic exponential time in the size of H 1 . This coefficient σ(H 1

A list of applications of Stallings automata

“…It is worth remarking that, in the free group case, it is not even known whether the supremum in the definition of degree of inertia is a maximum, i.e., whether it is always achieved in a particular subgroup K. Interestingly, S. V. Ivanov showed in [53] that this is the case if we replace the numerator rk(H ∩ K) by the corresponding disconnected version x∈H\F/K rk(H ∩ x −1 Kx) (in [53], this related notion is called the Walter Neumann coefficient of Walter Neumann coefficient H, in clear connection with the Strengthened Hanna Neumann inequality). This result is proved using quite unusual techniques: the author codifies the Stallings automata of all the subgroups K that must be taken into account Proposition 4.81.…”

Degrees of compression and inertia for free-abelian times free groups

“…therefore, it can be interpreted in the following way: "the smallest possible multiplicative constant [8] already considered and studied the strengthened version of what we call here the degree of inertia. He defined the Walter Neumann coefficient of…”

“…Theorem 2.2 (Ivanov, [8]). For any finitely generated free group F n , the function σ is computable and the supremum is a maximum; more precisely, there is an algorithm which, on input h 1 , .…”

“…The question whether di Fn is computable (related to the question whether the corresponding supremum is a maximum or not) in free groups seems to be much more delicate. In Section 2 we refer to a quite similar question, which was successfully solved recently by S. Ivanov in [8]. However, at the time of writing, we do not know how to use this result to eventually compute di Fn .…”

The Equalizer Conjecture For The Free Group of Rank Two

“…There are inert subgroups which are not fixed subgroups, for example a, b 2 is inert in F(a, b) but not a fixed subgroup as it is not root-closed, while more exotic examples were found by Rosenmann [28,Example 3.1]. Recent work on inert subgroups has focused on trying to algorithmically determine inertness by quantifying it [20,29] as well as generalizing the concept to other groups [34][35][36].…”