Two simultaneous actions of big mapping class groups

Abstract

We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed.The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromovboundary of the graph using the embedding of its vertices in the circle.The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.

An infinite clique of high-filling rays in the plane minus a Cantor set

“…Nested disks. We fix a complete hyperbolic metric of the first kind on R 2 − Cantor (see for example the first part of [3] to construct such a metric). We see R 2 − Cantor as a sphere minus {∞} ∪ Cantor and speak about ∞ and the Cantor set as if they were marked points of the sphere.…”

“…Second, the Gromov-boundary of the loop graph can be identified with the set of cliques of high-filling rays (see [2]). Moreover, the two attractive and repulsive points of the boundary which are fixed by a loxodromic element of the mapping class group are necessarily finite cliques, and have the same cardinal (see [3]). On finite type surfaces, these high-filling rays would roughly correspond to the singular leaves of a minimal foliation.…”

“…On finite type surfaces, these high-filling rays would roughly correspond to the singular leaves of a minimal foliation. Using this correspondence and taking finite type subsurfaces of the plane minus a Cantor set give us cliques with any finite number of high-filling rays (see [3]). But because this construction uses only finite type subsurfaces, these cliques are not specific to infinite type surfaces.…”

Stable commutator length on big mapping class groups

“…Calegari's question was answered by J. Bavard in [4], where she proves that the second bounded cohomology of the mapping class group of the plane minus a Cantor set is in fact infinite dimensional. The study of the second bounded cohomology of big mapping class groups has recently been taken considerably further (see [1], [5], [14], and [17]). In particular, if S has a non-displaceable subsurface, in the terminology of Mann-Rafi [16], then the second bounded cohomology of its mapping class group, H 2 b (Map(S); R), is known to be infinite dimensional (see [14]).…”

Isometry groups of infinite-genus hyperbolic surfaces

“…Proof Let n be the number of planar ends of S. Let G be a non-abelian simple finite group of order greater than n! and suppose G is the isometry group of some hyperbolic structure on S. We first observe that G must act on the planar ends of S trivially: indeed, the action of G on the planar ends of S induces a homomorphism from G to the symmetric group on n letters, but the assumption on the cardinality of G implies that the kernel is non-empty and hence, by the simplicity of G, must be all of G. Using again the work of Bavard-Walker [8], we must have that G is cyclically orderable and thus cyclic, which contradicts our assumption that G is non-abelian.…”