Topology of finite graphs

Algebraic Extensions in Free Groups

“…One shows that this process terminates, that it does not matter in which order identifications take place, and that the resulting A-labeled graph is reduced and equal to Γ A (H). In particular, it does not depend on the choice of a set of generators of H. Also, this shows that Γ A (H) is finite if and only if H is finitely generated (see one of [16,20,4,22,11,7,15] for more details). Example 2.1 Let A = {a, b, c}.…”

“…In his 1983 paper [16], Stallings showed how many of the algorithmic constructions introduced in the first half of the 20th century to handle finitely generated subgroups of free groups, can be clarified and simplified by adopting a graph-theoretic language. This method has been used since then in a vast array of articles, including work by the co-authors of this paper.…”

“…ending) at v have distinct images. Further following [16], we say that the morphism ϕ : Γ → ∆ is a cover if it is locally bijective, that is, if the following holds: for each vertex v of Γ, each edge of ∆ starting (resp. ending) at ϕ(v) is the image under ϕ of an edge of Γ starting (resp.…”

“…We will use some well-known facts (see [16]). If H is a finitely generated subgroup of F (A), then the rank of H is given by the formula…”

Profinite semigroups and applications

“…To test whether an endomorphism ψ of the free group Ω κ A G is an automorphism, it suffices to check whether the subgroup generated by ψ(A) is all of Ω κ A G. There is a well-known algorithm to check this property, namely Stallings' folding algorithm applied to the "flower automaton", whose petals are labeled with the words ψ(A) [82,46].…”