Injective simplicial maps of the arc complex. (English) Zbl 1206.57018

Irmak, Elmas; McCarthy, John D.

Turk. J. Math. 34, No. 3, 339-354 (2010).

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