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Injective simplicial maps of the arc complex. (English) Zbl 1206.57018
If $$R$$ is a compact, connected, orientable surface with boundary, an arc $$A$$ in $$R$$ is properly embedded if $$\partial A\subset\partial R$$ and $$A$$ is transversal to $$R$$; and it is essential if it cannot be deformed into $$\partial R$$ keeping its end points in $$\partial R$$ throughout the deformation. The arc complex $$\mathcal{A}(R)$$ of $$R$$ is the abstract simplicial complex whose simplices are collections of isotopy classes of properly embedded essential arcs in $$R$$ which can be represented by disjoint arcs.
This paper studies injective simplicial self-maps of the arc complex. The authors prove that every injective simplicial self-map of $$\mathcal{A}(R)$$ is induced by a homeomorphism of the surface; and that, with the exception of a few low genus surfaces (disk, annulus, pair of pants, and a torus with one hole), the group $$\operatorname{Aut}(\mathcal{A}(R))$$ of simplicial automorphisms of $$\mathcal{A}(R)$$ is naturally isomorphic to the group $$\Gamma^*(R)$$ of isotopy classes of self-homeomorphisms of $$R$$. In the exceptional cases, $$\operatorname{Aut}(\mathcal{A}(R))$$ is naturally isomorphic to the quotient of $$\Gamma^*(R)$$ by its center.
Reviewer: Marco Abate (Pisa)

##### MSC:
 57M99 General low-dimensional topology 30F60 Teichmüller theory for Riemann surfaces 57M60 Group actions on manifolds and cell complexes in low dimensions
##### Keywords:
Mapping class group; arc complex; Riemann surface
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