The Gromov boundary of the ray graph

Abstract

Abstract. The ray graph is a Gromov-hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray graph is homeomorphic to a quotient of a subset of the circle.

Big mapping class groups and rigidity of the simple circle

“…The description of the simple circle is closely related to Bavard and Walker's conical circle. Our analysis of the dynamics in this case sharpens some of the results of [3,4] and gives new proofs of others.…”

“…). The conical circle and this action were introduced in [3], although another closely related action was introduced earlier in [5].…”

“…This group and its subgroups arise naturally in dynamics (see e.g. [5,8]) and has been studied in detail by Bavard [2] and Bavard and Walker [3,4]. It is the best understood and most studied example of a so-called 'big' mapping class group; see e.g.…”

“…Calegari [5] constructed a faithful action of on the circle by homeomorphisms. Bavard and Walker [3] constructed a different action which is more closely related to the action of on the Gromov boundary of the ray graph; see [2]. These actions are both constructed D. Calegari and L. Chen via hyperbolic geometry but they are in fact different.…”

“…Statement of results. In § §2-3 we introduce, following Bavard and Walker [3], the action of on the conical circle.…”