The space of vectored hyperbolic surfaces is path-connected

Warakkagun, Sangsan

doi:10.1090/ecgd/395

The space of vectored hyperbolic surfaces is path-connected
HTML articles powered by AMS MathViewer

by Sangsan Warakkagun;

Conform. Geom. Dyn. 28 (2024), 115-130

DOI: https://doi.org/10.1090/ecgd/395

Published electronically: October 4, 2024

PDF | Request permission

Abstract:

In the space $\mathcal {H}^2$ of hyperbolic surfaces decorated with a base unit vector, the topology induced by the Gromov–Hausdorff convergence coincides with the Chabauty topology on the space of discrete torsion-free subgroups of $\mathrm {PSL}_2(\mathbb {R})$. Using paths constructed from changing the Fenchel–Nielsen coordinates and shrinking simple closed curves to cusps, we demonstrate path-connectivity of $\mathcal {H}^2$ and some of its subspaces.

References

Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, and Weixu Su, On various Teichmüller spaces of a surface of infinite topological type, Proc. Amer. Math. Soc. 140 (2012), no. 2, 561–574. MR 2846324, DOI 10.1090/S0002-9939-2011-10918-3
Venancio Álvarez and José M. Rodríguez, Structure theorems for Riemann and topological surfaces, J. London Math. Soc. (2) 69 (2004), no. 1, 153–168. MR 2025333, DOI 10.1112/S0024610703004836
Hyungryul Baik and Lucien Clavier, The space of geometric limits of one-generator closed subgroups of $\textrm {PSL}_2(\Bbb R)$, Algebr. Geom. Topol. 13 (2013), no. 1, 549–576. MR 3116379, DOI 10.2140/agt.2013.13.549
Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992. MR 1219310, DOI 10.1007/978-3-642-58158-8
Ian Biringer, Metrizing the Chabauty topology, Geom. Dedicata 195 (2018), 19–22. MR 3820493, DOI 10.1007/s10711-017-0274-5
Ian Biringer, Nir Lazarovich, and Arielle Leitner, On the Chabauty space of $\mathrm {PSL}_2(\mathbb {R})$, I: lattices and grafting, 2021, To appear in Mem. Amer. Math. Soc.
Martin R. Bridson, Pierre de la Harpe, and Victor Kleptsyn, The Chabauty space of closed subgroups of the three-dimensional Heisenberg group, Pacific J. Math. 240 (2009), no. 1, 1–48. MR 2485473, DOI 10.2140/pjm.2009.240.1
Peter Buser, Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2010. Reprint of the 1992 edition. MR 2742784, DOI 10.1007/978-0-8176-4992-0
Peter Buser, Eran Makover, Björn Mützel, and Robert Silhol, Quasiconformal embeddings of Y-pieces, Comput. Methods Funct. Theory 14 (2014), no. 2-3, 431–452. MR 3265371, DOI 10.1007/s40315-014-0062-2
R. D. Canary, D. B. A. Epstein, and P. L. Green, Notes on notes of Thurston [MR0903850], Fundamentals of hyperbolic geometry: selected expositions, London Math. Soc. Lecture Note Ser., vol. 328, Cambridge Univ. Press, Cambridge, 2006, pp. 1–115. With a new foreword by Canary. MR 2235710
Yves Cornulier, On the Chabauty space of locally compact abelian groups, Algebr. Geom. Topol. 11 (2011), no. 4, 2007–2035. MR 2826931, DOI 10.2140/agt.2011.11.2007
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
Tsachik Gelander, A lecture on invariant random subgroups, New directions in locally compact groups, London Math. Soc. Lecture Note Ser., vol. 447, Cambridge Univ. Press, Cambridge, 2018, pp. 186–204. MR 3793288
Benoît Kloeckner, The space of closed subgroups of $\Bbb R^n$ is stratified and simply connected, J. Topol. 2 (2009), no. 3, 570–588. MR 2546586, DOI 10.1112/jtopol/jtp022
Nir Lazarovich and Arielle Leitner, Local limits of connected subgroups of $\rm SL_3(\Bbb R)$, C. R. Math. Acad. Sci. Paris 359 (2021), 363–376 (English, with English and French summaries). MR 4264320, DOI 10.5802/crmath.16
Arielle Leitner, Conjugacy limits of the diagonal Cartan subgroup in $SL_3(\Bbb {R})$, Geom. Dedicata 180 (2016), 135–149. MR 3451461, DOI 10.1007/s10711-015-0095-3
William P. Thurston, The geometry and topology of three-manifolds. Vol. IV, American Mathematical Society, Providence, RI, [2022] ©2022. Edited and with a preface by Steven P. Kerckhoff and a chapter by J. W. Milnor. MR 4554426