Surface groups in uniform lattices of some semi-simple groups

Abstract

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.

91 pages, 21 figures, v5: many typos, inaccuracies have been corrected. Section 9 and 11 underwent major changes. Th. 9.2.4 (in v4) was wrong. This theorem was used in Th. 11.4.1.itself used in Th. 16.3.1, In v5, Th.11.4.1. (and 11.4.4) is now weaker. Using a new argument section 16.2 we get Th. 16.3.1 with the same generality

References in corpus (2)

• Cross Ratios and Identities for Higher Thurston Theory
• La conjecture des sous-groupes de surfaces (d'après Jeremy Kahn et Vladimir Markovic)