Peripheral subgroups of Kleinian groups

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 Teichmüller 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 $\ddot{I}{€}_1(M)\subset \mathsf{PSL}(2,\mathbb{C})$, which together give an island of discrete representations around the identity map in $X=\mathrm{Hom}(\ddot{I}{€}_1(M),\mathsf{PSL}(2,\mathbb{C}))$. 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 $\ddot{I}{€}_1(M)$ 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 $\ddot{I}{€}_1(M)$ -- which is known to be topologically wild.

39 pages, 10 figures. v2: updated introduction; corrected errors in section 3 on semi-algebraic descriptions of Teichmüller space; added more detail to some proofs. Numbering has changed, correspondence welcome