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