Constructing geometrically infinite groups on boundaries of deformation spaces

TL;DR

This paper proves the existence of algebraic limits of quasi-conformal deformations of geometrically finite Kleinian groups with prescribed boundary data, advancing understanding of Kleinian group limits and boundary behavior.

Contribution

It establishes a new theorem constructing algebraic limits of Kleinian groups with specified boundary laminations or conformal structures, linking boundary data to group limits.

Findings

Existence of algebraic limits with prescribed boundary conditions

Unrealisability of given laminations in the limit groups

Supports the Bers-Thurston conjecture for finitely generated Kleinian groups

Abstract

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup \Omega_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit $\Gamma$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\mathrm{Int} M$ to $\H^3/\Gamma$ compatible with the natural isomorphism from $G$ to $\Gamma$, the given laminations are unrealisable in $\H^3/\Gamma$, and the given conformal structures are pushed forward by $h$ to those of $\H^3/\Gamma$. Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden's conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.