A universal Cannon-Thurston map and the surviving curve complex

TL;DR

This paper constructs a universal Cannon-Thurston map for a new 'surviving curve complex' associated with punctured surfaces, proving its hyperbolicity and boundary structure, and extending previous boundary map results.

Contribution

It introduces the surviving curve complex, proves its hyperbolicity, and constructs a universal Cannon-Thurston map for it, extending prior boundary map work.

Findings

Proved hyperbolicity of the surviving curve complex

Identified the boundary as a lamination space

Extended Cannon-Thurston map to the ordinary curve complex boundary

Abstract

Using the Birmanexact sequence for pure mapping class groups, we construct a universal Cannon--Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon--Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer.