The Alexander Method for Infinite-Type Surfaces

Abstract

We prove that for any infinite-type orientable surface S there exists a collection of essential curves Γ in S such that any homeomorphism that preserves the isotopy classes of the elements of Γ is isotopic to the identity. The collection Γ is countable and has infinite complement in C(S), the curve complex of S. As a consequence we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.

Big Torelli groups: generation and commensuration

“…By the Alexander method for infinite-type surfaces, due to Hernández-Moralez-Valdez [13], we deduce that f is the identity in MCG(Σ).…”

“…Therefore ψ(g)f (γ) = f g(γ). By use of the Alexander method [13] we conclude that ψ(g) = f gf −1 . This shows that every abstract commensurator of I(Σ) is defined by conjugation by a mapping class of Σ, and in particular, so is every automorphism of I(Σ).…”

“…Choose a complete hyperbolic metric on Σ without half planes and realize all curves as geodesics. The assumption on the metric implies that we can decompose Σ into pairs of pants with geodesic boundary, funnels and/or cusps (as shown in [13]). If the geodesic representative of σ has an accumulation point, then we can find a pair of pants P of Σ containing such a point.…”

An Alexander method for infinite-type surfaces

“…In this paper, we extend the notion of an Alexander system to include curve systems on infinite-type surfaces (surfaces whose fundamental groups are infinitely generated), and then prove our main result: a generalization of the Alexander method of Farb-Margalit to include the case of infinite-type surfaces (including non-orientable surfaces) [7]. A version of the Alexander method was proven for orientable surfaces by Hernández-Morales-Valdez [11] and for nonorientable surfaces by Hernández-Hidber [10]. In both of these papers, the authors prove that the Alexander method applies to a family of Alexander systems which they construct.…”

Groups of proper homotopy equivalences of graphs and Nielsen Realization

“…We will show first that Ψ is injective if X is a core graph (meaning X = X g ), and in general we will describe the kernel of Ψ. The next theorem is analogous to the fact that a homeomorphism of a surface with nonabelian fundamental group that induces identity in π 1 is isotopic to the identity, see [8,15]. Theorem 3.1.…”