Algebraic & Geometric Topology

The space of geometric limits of one-generator closed subgroups of $\mathrm{PSL}_2(\mathbb{R})$

Hyungryul Baik and Lucien Clavier

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Abstract

We give a complete description of the closure of the space of one-generator closed subgroups of PSL2() for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D)PSL2(), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps φn:XY transform convergent sequences of closed subsets of the domain X into convergent sequences of closed subsets of the range Y. In particular, this allows us to compute certain geometric limits of PSL2() only by looking at the Hausdorff limit of some closed subsets of .

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 1 (2013), 549-576.

Dates
Received: 24 February 2012
Revised: 18 July 2012
Accepted: 4 October 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715506

Digital Object Identifier
doi:10.2140/agt.2013.13.549

Mathematical Reviews number (MathSciNet)
MR3116379

Zentralblatt MATH identifier
1266.30031

Subjects
Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Keywords
PSL2(R) Chabauty topology Hausdorff topology geometric limit

Citation

Baik, Hyungryul; Clavier, Lucien. The space of geometric limits of one-generator closed subgroups of $\mathrm{PSL}_2(\mathbb{R})$. Algebr. Geom. Topol. 13 (2013), no. 1, 549--576. doi:10.2140/agt.2013.13.549. https://projecteuclid.org/euclid.agt/1513715506


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