The universal Cannon--Thurston maps and the boundary of the curve complex

TL;DR

This paper constructs a universal Cannon--Thurston map linking the boundary of a closed surface group to the boundary of the curve complex of a once-punctured surface, revealing new topological properties.

Contribution

It introduces a universal Cannon--Thurston map for genus two and higher surfaces and proves the boundary of the curve complex is locally path-connected.

Findings

Constructed a universal Cannon--Thurston map for surface groups.

Proved the boundary of the curve complex is locally path-connected.

Extended previous work on boundary maps in hyperbolic geometry.

Abstract

In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected.