## Algebraic & Geometric Topology

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

#### Abstract

We give a complete description of the closure of the space of one-generator closed subgroups of $PSL2(ℝ)$ for the Chabauty topology, by computing explicitly the matrices associated with elements of $Aut(D)≅PSL2(ℝ)$, and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps $φn:X→Y$ 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 $PSL2(ℝ)$ only by looking at the Hausdorff limit of some closed subsets of $ℂ$.

#### Article information

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

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

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

#### 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

#### References

• H Akiyoshi, M Sakuma, M Wada, Y Yamashita, Punctured torus groups and 2-bridge knot groups. I, Lecture Notes in Mathematics 1909, Springer, Berlin (2007)
• C Chabauty, Limite d'ensembles et géométrie des nombres, Bull. Soc. Math. France 78 (1950) 143–151
• P de la Harpe, Spaces of closed subgroups of locally compact groups
• J H Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY (2006)
• J H Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY (2006)
• B Kloeckner, The space of closed subgroups of $\Bbb R\sp n$ is stratified and simply connected, J. Topol. 2 (2009) 570–588
• A Marden, Outer circles, Cambridge Univ. Press (2007)
• Y N Minsky, On Dynamics of $\mathrm{Out(F}_n)$ on $\mathrm{PSL}(2,\mathbb{C})$ characters (2010)
• I Pourezza, J Hubbard, The space of closed subgroups of ${\bf R}\sp{2}$, Topology 18 (1979) 143–146
• R M Schori, J E West, The hyperspace of the closed unit interval is a Hilbert cube, Trans. Amer. Math. Soc. 213 (1975) 217–235
• W P Thurston, The geometry and topology of three-manifolds, Lecture Notes, Princeton Univ. (1980)