Surfaces Mapping to Wedges of Spaces. A Topological Variant of the Grushko-Neumann Theorem

Subgroup decomposition in Out(F_n), Part I: Geometric Models

“…For the proofs of (1) and (2), by restricting to F ′ we may assume that F ′ = F n and so Y = Y 0 = X and G = G r = G 0 r . Item (1) will follow once we prove the reverse inclusion F supp (π 1 S) ⊏ F supp [∂S], which we do using Stallings' method from [Sta00]. That inclusion is obvious if…”

“…As mentioned above, the subgroup system [π 1 L] associated to complementary subgroup L of a geometric model X is never be a free factor system, as we show in Lemma 2.5 (5). In the special case of a geometric outer automorphism, where X = S is a compact surface and L = ∂S = ∅, this follows from a result due to Stallings [Sta00] which shows that for any free factorization π 1 S = A 1 * • • • * A K there exists a component of ∂S which is not conjugate into any factor A k , and our argument is an adaptation of Stallings' proof.…”

Scl, sails, and surgery

“…An interesting precursor of this latter approach is Stallings' last paper [18], which uses topological methods to factorize products of commutators in a free product of groups into terms that are localized in the factors.…”

“…An interesting precursor of this latter approach is John Stallings' last paper [18], which uses topological methods to factorize products of commutators in a free product of groups into terms which are localized in the factors. It is a pleasure to acknowledge my own great intellectual debt to John, and it seems especially serendipitous to discover, in relatively unheralded work he did in the later part of his life, some beautiful new ideas which continue to inform and inspire.…”

Elementary theory of free non-abelian groups

“…Lemma 9. [35] Let G be a fundamental group of a compact connected surface T , and either the boundary of T is empty or it consists of several components E 1 , . .…”