Homeomorphic subsurfaces and the omnipresent arcs

Abstract

In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct an interesting action of the mapping class group on a subgraph of the arc graph. This subgraph naturally emerges from a new characterisation of infinite-type surfaces in terms of homeomorphic subsurfaces.Résumé. -Cet article s'intéresse à plusieurs aspects des arcs sur les surfaces. La première partie s'occupe des aspects topologiques des arcs et de leurs compléments. Nous utilisons les résultats de la première partie pour définir ensuite une action du groupe modulaire sur un sousgraphe du graphe des arcs. Ce sous-graphe ressort naturellement d'une nouvelle caractérisation des surfaces de type infini en termes de sous-surfaces homéomorphes.

Flip graphs for infinite type surfaces

“…Note that arc and curve type graphs are becoming more understood in the context of infinite type surfaces [5,9,11,1] have been studied in different contexts and for different uses but, to the best of our knowledge, graphs with vertices being triangulations of infinite type surfaces…”

Ideally, all infinite‐type surfaces can be triangulated

“…Arcs play an important role in dynamical and topological considerations on infinite‐type surfaces, see, for instance, [4, 6]. The definition above means that there exist ends $$\lbrace U_i\rbrace _{i\in {\mathbb {N}}}$$ and $$\lbrace V_j\rbrace _{j\in {\mathbb {N}}}$$ (not necessarily distinct) such that for all $$i_0$$, there exists $$t_0 \in {\mathbb {R}}$$ such that $$\alpha (t) \in U_{i_0}$$ for all $$t>t_0$$, and there exists $$\lbrace V_j\rbrace _{j\in {\mathbb {N}}}$$ such that for all $$j_0$$, there exists $$s_0$$ with $$\alpha (t) \in V_{j_0}$$ for all $$t<s_0$$.…”

“…Arcs play an important role in dynamical and topological considerations on infinite-type surfaces, see, for instance, [4,6]. The definition above means that there exist ends {𝑈 𝑖 } 𝑖∈ℕ and {𝑉 𝑗 } 𝑗∈ℕ (not necessarily distinct) such that for all 𝑖 0 , there exists 𝑡 0 ∈ ℝ such that 𝛼(𝑡) ∈ 𝑈 𝑖 0 for all 𝑡 > 𝑡 0 , and there exists {𝑉 𝑗 } 𝑗∈ℕ such that for all 𝑗 0 , there exists 𝑠 0 with 𝛼(𝑡) ∈ 𝑉 𝑗 0 for all 𝑡 < 𝑠 0 .…”

Big Mapping Class Groups With Hyperbolic Actions: Classification and Applications

“…These conjectures were answered positively by Bavard [6], which was the start of a lot of activity on big mapping class groups. Further developments, providing actions of Map(Σ) on hyperbolic graphs under various topological conditions on the surface, include [3,20,21], for instance. The study of the geometry of such graphs is still in constant expansion (see, e.g., [4,7,26]).…”