Surface groups in uniform lattices of some semi-simple groups

We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are Hölder. Using this notion, we show a quantitative version of our surface subgroup theorem, and in particular that one can obtain $K$-Sullivan limit maps, as close as one wants to smooth round circles. All these results use the coarse geometry of “path of triangles” in a certain flag manifold, and we prove an analogue to the Morse Lemma for quasi-geodesics in that context.

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J. K. and F. L. acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures And Representation varieties” (the GEAR Network). J. K. acknowledges support by the National Science Foundation under Grant no.DMS 1352721. F. L. was partially supported by the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. FP7-246918 as well as by the ANR grant DynGeo ANR-11-BS01-013. S. M. acknowledges support by ISF grant 1003/11, ISF-Moked grant 2095/15 and ISF-Moked grant 2919/19.

Received 9 August 2019

Received revised 21 December 2020

Accepted 8 June 2023

Published 10 May 2024