The Gromov boundary of the ray graph

Two simultaneous actions of big mapping class groups

“…In this paper, we generalize results of [6] to our more general class of surfaces. In particular, we define the loop graph L(S; p) of a surface S based at a puncture p and show it is hyperbolic and that MCG(S; p) acts on it by isometries.…”

“…Some rays are special: they are high-filling, which means they are not in the connected component of any loop. We prove results analogous to [6]; namely, that these long rays appear in cliques, and the cliques are in bijection with the Gromov boundary of the loop graph (Theorem 6.3.1). Moreover, with the topology that these rays inherit from (a quotient of) the circle, they are homeomorphic to the Gromov boundary of the loop graph (Theorem 6.6.1).…”

“…Results. Our main goal in this paper is to generalize and extend the results of [4] and [6] to the case of more general infinite type surfaces. That is, to understand the structure of mapping class groups of infinite type surfaces which fix an isolated marked puncture.…”

“…Equators for infinite type surfaces. One of the most important tools in [6] was an obvious equator in S 2 − Cantor: a locally finite family of mutually disjoint and proper simple arcs whose complement is two topological disks. The equator is essentially equivalent to a nice fundamental domain.…”

“…The second part of the paper deals with Gromovhyperbolic graphs of surfaces of finite or infinite type and with an isolated marked puncture and their Gromov boundaries. In [6], we gave a concrete description of the Gromov boundary of the loop graph of the plane minus a Cantor set. Namely, we showed it is homeomorphic, with suitable topologies, to the set of cliques in a graph containing the loop and ray graphs.…”