Quotients of the curve complex (Q6594723)

Let \((X,d)\) be metric space and \(Y=\{Y_{i}\}_{i \in I}\) a collection of subsets of \(X\). The electrification of \(X\) with respect to \(Y\) is the metric space \(X_{Y}\) obtained by adding a new vertex \(y_{i}\) for each set \(Y_{i}\), and coning off \(Y_{i}\) by attaching edges of length \(1/2\) from each \(y\in Y_{i}\) to \(y_{i}\). The metric on \(X_{Y}\) is the induced path metric, and so the image of each set \(Y_{i}\) in \(X_{Y}\) has diameter at most one. If \(X\) is a Gromov hyperbolic space, and \(Y\) is a collection of uniformly quasiconvex subsets of \(X\), then the electrification \(X_{Y}\) is Gromov hyperbolic. Furthermore, if a group \(G\) acts on \(X\) by isometries, and \(Y\) is \(G\)-equivariant, then \(G\) also acts on the electrification \(X_{Y}\) by isometries.\N\NThe authors consider the action of the mapping class group of a finite type orientable surface \(S\) on the curve complex of the surface \(\mathcal{C}(S)\). The results proved in this paper apply to the following families of uniformly quasiconvex subsets. (1) The symmetric curve sets: If \(q: S \rightarrow S'\) is an orbifold quotient of \(S\), let \(C_{q}(S)\) be the collection of all \(q\)-equivariant curves in \(S\). The collection \(\{\mathcal{C}_{q}(S)\}\) as \(q\) runs over all orbifold quotients of \(S\), is a mapping class group invariant, uniformly quasiconvex family of subsets of \(\mathcal{C}(S)\). Its electrification is denoted by \(\mathcal{C}_{q}(S)\). (2) Non-maximal train track sets: A train track \(\tau\) on \(S\) is maximal if every complementary region is a triangle or a monogon containing a single puncture. Let \(\tau\) be a train track and let \(\mathcal{C}(\tau)\) be the set of all simple closed curves carried by \(\tau\) The set \(\{\mathcal{C}(\tau) \}\), as \(\tau\) runs over all non-maximal train tracks in \(S\), is a mapping class group invariant, uniformly quasiconvex family of subsets of \(\mathcal{C}(S)\). Its electrification is denoted by \(\mathcal{C}_{\tau}(S)\). (3) Compression body disc sets: Let \(V\) be a compression body with boundary \(S\), and let \(D()\) be the collection of isotopy classes of essential simple closed curves in \(S\) which bound discs in \(V\). The set \(\{D(V)\}\), as \(V\) runs over all compression bodies, forms a mapping class group invariant, uniformly quasiconvex family of subsets of \(\mathcal{C}(S)\). Its electrification is denoted by \(\mathcal{C}_{D}(S)\).\N\NThe main result is Theorem 1.1: All \(\mathcal{C}_{q}(S)\), \(\mathcal{C}_{\tau}(S)\) and \(\mathcal{C}_{D}(S)\) are infinite diameter Gromov hyperbolic spaces, and furthermore, the action of the mapping class group on these spaces is strongly WPD.