Continuity of the bending map

TL;DR

The work extends the bending map from convex cocompact to geometrically finite hyperbolic metrics by introducing an ${\cal R}$-quotient on measured laminations and proving the continuity of the quotient bending map ${b_{\cal R}}$ under a careful convergence framework. It builds a foundation using convex pleated surfaces to analyze bending laminations and employs both algebraic and geometric convergence of representations to establish convergence of bending data. An erratum (Oct 2025) clarifies that the original continuity claim hinges on the topology of ${\cal ML}(\partial M)/{\cal R}$, introducing a tubular topology that restores continuity of ${b_{\cal R}}$ and distinguishes it from the quotient topology. The corrected results sharpen our understanding of how bending data behaves under geometric limits and parabolic changes, with implications for strong convergence criteria of geometrically finite Kleinian representations.

Abstract

The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the space of measured geodesic laminations and shows that the quotient map obtained from the bending map is continuous. October 2025, an erratum has been added that corrects an error in the main Theorem. The topology used in the quotient space should not be the quotient topology but the "tubular topology" defined in this erratum.

Figures (2)

• Figure 1: The section $\Pi\cap C(\rho_n)$
• Figure 2: Cutting $d$

Theorems & Definitions (36)

• Claim 2.1
• proof
• Claim 2.2
• proof
• Lemma 2.3
• proof
• Claim 2.4
• proof
• Lemma 3.1
• proof