Connectivity of the space of ending laminations

TL;DR

This paper proves the connectivity of the space of ending laminations for certain surfaces, confirming a conjecture and impacting understanding of the complex of curves and Kleinian groups.

Contribution

It establishes the connectedness of the space of ending laminations for surfaces of genus at least four or punctured surfaces of genus at least two, confirming a conjecture of P. Storm.

Findings

Connectedness of ending laminations space for specified surfaces

Homeomorphism to the Gromov boundary of the complex of curves

Implications for rigidity of the complex of curves and Kleinian groups

Abstract

We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups.