On the fundamental groups of one-dimensional spaces

An alternate proof that the fundamental group of a Peano continuum is finitely presented if the group is countable

“…We give an alternate proof, using coarse geometry, that if the fundamental group of a compact, connected, locally connected metric space is countable, then the fundamental group is finitely presented. This result was first proved by Katsuya Eda and the argument can be found in [5]. …”

The fundamental groups of subsets of closed surfaces inject into their first shape groups

“…For example, the natural homomorphism ϕ : π 1 (X, x 0 ) →π 1 (X, x 0 ), from the fundamental group of any 1-dimensional (Hausdorff) compactum (X, x 0 ) to its first shape homotopy group, has been shown to be injective [8]. Consequently, the fundamental group of any 1-dimensional (metric) continuum is isomorphic to a subgroup of the limit of an inverse sequence of finitely generated free groups; a fact already proved in [6] and also in [1,Theorem 5.11]. Injectivity of the above homomorphism has also been established for certain fractal-like trees of manifolds, which need not be semilocally simply connected at any point [10].…”

On the fundamental groups of trees of manifolds

“…Other classes of spaces Z for which ϕ : π 1 (Z , z 0 ) →π 1 (Z , z 0 ) has been shown to be injective, include (i) one-dimensional compacta [Curtis and Fort 1959;Eda and Kawamura 1998;Cannon and Conner 1998] and (ii) subsets of closed surfaces [Fischer and Zastrow 2005].…”

“…(This procedure was inspired by [Cannon and Conner 1998]. ) We may assume that the set {β n −1 (S) | S ∈ n }, for n ≥ 2, consists of finitely many disjoint arcs, whose collection we will denote by Ꮾ n .…”

“…Lemma 6.25 [Cannon and Conner 1998]. Let Y be a compact one-dimensional connected metric space and f : S 1 → Y a null-homotopic map.…”