15 February 2021 Nearly Fuchsian surface subgroups of finite covolume Kleinian groups
Jeremy Kahn, Alex Wright
Author Affiliations +
Duke Math. J. 170(3): 503-573 (15 February 2021). DOI: 10.1215/00127094-2020-0049

Abstract

Let Γ<PSL2(C) be discrete, of cofinite volume, and noncocompact. We prove that, for all K>1, there is a subgroup H<Γ that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.

Acknowledgments

We thank Darryl Cooper and David Futer for explaining their work, and Vladimir Markovic for valuable suggestions including the idea to use some sort of wheel component. Much of the work on this project took place during the Mathematical Sciences Research Institute (MSRI) program in spring 2015 and the Institute for Advanced Study (IAS) program in fall 2015, and we thank MSRI and IAS. This research was conducted during the period when Wright served as a Clay Research Fellow.

The authors acknowledge support from U.S. National Science Foundation (NSF) grants DMS 1107452, 1107263, and 1107367, “RNMS: GEometric structures And Representation varieties” (the GEAR Network). This material is based upon work supported by NSF grant DMS 1352721. This work was also supported by a grant from the Simons Foundation/SFARI (500275, Kahn).

Citation

Download Citation

Jeremy Kahn. Alex Wright. "Nearly Fuchsian surface subgroups of finite covolume Kleinian groups." Duke Math. J. 170 (3) 503 - 573, 15 February 2021. https://doi.org/10.1215/00127094-2020-0049

Information

Received: 13 January 2019; Revised: 9 March 2020; Published: 15 February 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1215/00127094-2020-0049

Subjects:
Primary: 57M50
Secondary: 30F40

Keywords: hyperbolic three manifold , Kleinian group , quasi-Fuchsian , surface subgroup

Rights: Copyright © 2021 Duke University Press

Vol.170 • No. 3 • 15 February 2021
Back to Top