Configuration spaces of points on the circle and hyperbolic Dehn fillings, II

TL;DR

This paper proves the global injectivity of a deformation map relating weighted point configurations on a circle to hyperbolic structures, extending previous local results to a global setting for n > 4.

Contribution

It establishes the global injectivity of the weight-to-structure correspondence for configuration spaces on the circle, generalizing earlier local results.

Findings

Proved global injectivity of the deformation map

Extended local results to a global context

Connected configuration space deformations to hyperbolic structures

Abstract

In our previous paper, we discussed the hyperbolization of the configuration space of n(> 4) marked points with weights in the projective line up to projective transformations. A variation of the weights induces a deformation. It was shown that this correspondence of the set of the weights to the Teichm\"uller space when n = 5 and to the Dehn filling space when n= 6 is locally one-to-one near the equal weight. In this paper, we establish its global injectivity.