Peripheral subgroups of Kleinian groups

TL;DR

This paper studies the deformation space of Kleinian groups, providing a computable region where peripheral structures are stable under small deformations, and covering the entire deformation space with controlled topological complexity.

Contribution

It introduces a polynomial inequality-based region in the representation space where peripheral structures are stable, and covers the deformation space with these regions for different pleating laminations.

Findings

Identifies a computable polynomial inequality region in the representation space.

Shows peripheral structures are coarsely similar within this region.

Provides a countable cover of the entire deformation space with controlled topology.

Abstract

The conformal boundary of a hyperbolic $3$-manifold $M$ is a union of Riemann surfaces. If any of these Riemann surfaces has a nontrivial Teichm\"uller space, then the hyperbolic metric of $M$ can be deformed quasi-isometrically. These deformations correspond to small pertubations in the matrices of the holonomy group $ \pi_1(M) \subset \mathsf{PSL}(2,\mathbb{C}) $, which together give an island of discrete representations around the identity map in $ X=\operatorname{Hom}(\pi_1(M), \mathsf{PSL}(2,\mathbb{C})) $. Determining the extent of this island is a hard problem. If $M$ is geometrically finite and its convex core boundary is pleated only along simple closed curves, then we cut up its conformal boundary in a way governed by the pleating combinatorics to produce a fundamental domain for $ \pi_1(M) $ that is combinatorially stable under small deformations, even those which change the pleating structure. We give a computable region in $X$, cut out by polynomial inequalities over $\mathbb{R}$, within which this fundamental domain is valid: all the groups in the region have peripheral structures that look `coarsely similar', in that they come from real-algebraically deforming a fixed conformal polygon and its side-pairings. The union of all these regions for different pleating laminations gives a countable cover, with sets of controlled topology, of the entire quasi-isometric deformation space of $ \pi_1(M) $ -- which is known to be topologically wild.