Nondisplaceable Subsurfaces

12 Aug 2022

This is the second post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. The purpose of this post is to introduce the main obstruction to coarse boundedness of subsets of a big mapping class group: nondisplaceable finite-type subsurfaces. I also define the mapping class group in this post.

The mapping class group

Mapping class groups of finite-type surfaces are central objects in modern mathematics, bridging fields as disparate as algebraic geometry and low-dimensional topology. Informally one should think of the mapping class group as the group of “symmetries” of a space. The following definition makes sense for a topological space \(X\), but is not the only reasonable definition for a surface. The mapping class group of a surface \(\Sigma\) is the group of isotopy classes of orientation-preserving homeomorphisms of \(\Sigma\),

\[\operatorname{Map}(\Sigma) = \pi_0(\operatorname{Homeo}^+(\Sigma)) = \operatorname{Homeo}^+(\Sigma)/\operatorname{Homeo}_0(\Sigma),\]

where \(\operatorname{Homeo}^+(\Sigma)\) is the group of homeomorphisms \(f \colon \Sigma \to \Sigma\) and \(\operatorname{Homeo}_0(\Sigma)\) is the normal subgroup comprising those homeomorphisms \(f\) which are isotopic to the identity, in the sense that there exists a continuous one-parameter family of homeomorphisms \(f_t\colon \Sigma \to \Sigma\) with \(f_0 = f\) and \(f_1\) equal to the identity homeomorphism. If we give \(\operatorname{Homeo}^+(\Sigma)\) the compact–open topology, an isotopy is a path in \(\operatorname{Homeo}(\Sigma)\), so if we care only about isotopy classes of homeomorphisms, we are taking path-components in \(\operatorname{Homeo}^+(\Sigma)\), hence the notation above.

For a closed (hence finite-type) surface \(\Sigma\), every homotopy equivalence \(f\colon \Sigma \to \Sigma\) is homotopic to a homeomorphism, every homeomorphism is isotopic to a diffeomorphism, two homeomorphisms are isotopic if and only if they are homotopic, and every outer automorphism of \(\pi_1(\Sigma)\) can be realized as a homotopy equivalence of \(\Sigma\), so there are a number of equivalent definitions of \(\operatorname{Map}(\Sigma)\). Not all of these facts remain true for infinite-type surfaces.

I punted on defining the compact–open topology to Wikipedia; for us the essential feature of the topology on \(\operatorname{Map}(\Sigma)\) is the following: a neighborhood basis of the identity is given by

\[U_S = \{[f] \in \operatorname{Map}(\Sigma) : f|_S = 1 \},\]

where \(S\) is a finite-type subsurface of \(\Sigma\).

Nondisplaceable subsurfaces

Recall from post “zero” that \(A \subset \operatorname{Map}(\Sigma)\) is coarsely bounded if and only if \(A\) has bounded orbits in any continuous action of \(\operatorname{Map}(\Sigma)\) by isometries on a metric space.

A connected, finite-type subsurface \(S\) of \(\Sigma\) is nondisplaceable if for every homeomorphism \(f \colon \Sigma \to \Sigma\), we have \(f(S) \cap S \ne \varnothing\). If \(S\) is not connected, we say \(S\) is nondisplaceable if for each component \(C\) of \(S\) and every homeomorphism \(f \colon \Sigma \to \Sigma\), we have \(f(C) \cap S \ne \varnothing\). Mann and Rafi prove that the existence of nondisplaceable finite-type subsurfaces obstructs coarse boundedness.

Say that a mapping class is supported on a subsurface \(S\) if there exists a representative homeomorphism \(f\colon \Sigma \to \Sigma\) such that

\[\{x \in \Sigma : f(x) \ne x\} \subset S.\]

Here is a precise statement of Mann–Rafi’s result.

Theorem (cf. Mann–Rafi Theorem 1.9, Proposition 2.8, Lemma 5.2). Suppose that \(\Sigma\) contains a finite-type nondisplaceable subsurface \(S\). Then \(\operatorname{Map}(\Sigma)\) is not coarsely bounded. More generally, if \(S\) is sufficiently complex as measured by the Euler characteristic of each of its components, any subgroup of \(\operatorname{Map}(\Sigma)\) containing the mapping classes supported on \(S\) is not coarsely bounded in \(\operatorname{Map}(\Sigma)\).

We finish this post by sketching the idea of the proof, with the simplifying assumption that \(S\) is a connected finite-type subsurface of \(\Sigma\) with the property that every homeomorphism of \(\Sigma\) leaves \(S\) invariant up to isotopy and such that the curve graph of \(S\) is connected and has infinite diameter.

The curve graph \(\mathcal{C}(S)\) of a finite-type surface \(S\) is the simplicial graph with vertex set the set of isotopy classes of essential, simple closed curves on \(S\), where a curve is essential if it is not null-homotopic nor homotopic to a puncture or boundary component of \(S\), and simple if it does not intersect itself. Two vertices (essential simple closed curves) \(\gamma_1\) and \(\gamma_2\) are adjacent in \(\mathcal{C}(S)\) if and only if \(\gamma_1\) and \(\gamma_2\) have disjoint representatives. It is an important fact in geometric group theory that if \(S\) is not drawn from a finite list of exceptions, \(\mathcal{C}(S)\) with the graph metric assigning each edge length one is connected, infinite-diameter and Gromov hyperbolic, that \(\operatorname{Map}(S)\) acts on \(\mathcal{C}(S)\) by isometries and that a certain class of elements \(f\) of \(\operatorname{Map}(S)\) called pseudo-Anosov mapping classes act on \(\mathcal{C}(S)\) with positive stable translation length, in the sense that

\[\lim_{n\to\infty}\frac{d(\gamma, f^n(\gamma))}{n} > 0\]

for some and hence every vertex \(\gamma\) of \(\mathcal{C}(S)\). There exist such a pseudo-Anosov mapping classes with representatives that may be taken to restrict to the identity on the boundary of \(S\).

Anyway, in the special case where every homeomorphism of \(\Sigma\) leaves the isotopy class of \(S\) invariant, restriction to \(S\) defines a continuous action of \(\operatorname{Map}(\Sigma)\) on \(\mathcal{C}(S)\). If a homeomorphism \(f \colon S \to S\) restricts to the identity on \(\partial S\), extending by the identity on \(\Sigma - S\) defines a homeomorphism (and thus a mapping class) of \(\Sigma\) whose action on \(\mathcal{C}(S)\) agrees with the action of \([f]\). Taking \(f\) to be pseudo-Anosov on \(S\) proves that \(\operatorname{Map}(\Sigma)\) does not have bounded orbits in the action on \(\mathcal{C}(S)\), so \(\operatorname{Map}(\Sigma)\) and any subgroup containing the extension of \(f\) to \(\Sigma\), for example \(U_R\) for some finite-type subsurface \(R\) whose isotopy class is disjoint from that of \(S\), is not coarsely bounded in \(\operatorname{Map}(\Sigma)\).


Suppose as a first case that \(0 < \operatorname{genus}(\Sigma) < \infty\). Then any finite-type subsurface \(S\) of \(\Sigma\) with genus equal to that of \(\Sigma\) is nondisplaceable for the reason that it contains all the isotopy classes of essential simple closed curves of \(\Sigma\) that are non-separating, and homeomorphisms send non-separating curves to non-separating curves.

As another example, suppose \(\Sigma\) has at least three but finitely many maximal ends, in the sense of the preorder introduced in the previous post. Then any finite-type subsurface which separates each maximal end from each other is nondisplaceable. As an example, the subsurface indicated in the figure below separates the maximal ends of the subsurface considered in the previous post. It is certainly not the only such subsurface.

The indicated finite-type subsurface separates the three maximal ends of the surface