Peripheral subgroups of Kleinian groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915692030197760 |
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| author | Elzenaar, Alex |
| author_facet | Elzenaar, Alex |
| contents | 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 $ π_1(M) \subset \mathsf{PSL}(2,\mathbb{C}) $, which together give an island of discrete representations around the identity map in $ X=\operatorname{Hom}(π_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 $ π_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 $ π_1(M) $ -- which is known to be topologically wild. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Peripheral subgroups of Kleinian groups Elzenaar, Alex Geometric Topology Complex Variables Group Theory 20H10 (Primary) 20F65, 30F40, 51B10, 57K32 (Secondary) 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 $ π_1(M) \subset \mathsf{PSL}(2,\mathbb{C}) $, which together give an island of discrete representations around the identity map in $ X=\operatorname{Hom}(π_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 $ π_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 $ π_1(M) $ -- which is known to be topologically wild. |
| title | Peripheral subgroups of Kleinian groups |
| topic | Geometric Topology Complex Variables Group Theory 20H10 (Primary) 20F65, 30F40, 51B10, 57K32 (Secondary) |
| url | https://arxiv.org/abs/2508.00297 |