#### Volume 13, issue 1 (2013)

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

### Hyungryul Baik and Lucien Clavier

Algebraic & Geometric Topology 13 (2013) 549–576
##### Abstract

We give a complete description of the closure of the space of one-generator closed subgroups of ${PSL}_{2}\left(ℝ\right)$ for the Chabauty topology, by computing explicitly the matrices associated with elements of $Aut\left(\mathbb{D}\right)\cong {PSL}_{2}\left(ℝ\right)$, and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps ${\phi }_{n}:X\to Y$ 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 ${PSL}_{2}\left(ℝ\right)$ only by looking at the Hausdorff limit of some closed subsets of $ℂ$.

##### Keywords
PSL2(R), Chabauty topology, Hausdorff topology, geometric limit
Primary: 30F40
Secondary: 20H10
##### Publication
Received: 24 February 2012
Revised: 18 July 2012
Accepted: 4 October 2013
Published: 10 March 2013
##### Authors
 Hyungryul Baik Department of Mathematics Cornell University 310 Malott Hall Ithaca New York 14853-4201 USA http://www.math.cornell.edu/~hrbaik/ Lucien Clavier Department of Mathematics Cornell University 310 Malott Hall Ithaca New York 14853-4201 USA http://www.math.cornell.edu/~lpc49/