# 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 H\"older. 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.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-02991946
Contributor : François Labourie <>
Submitted on : Friday, November 6, 2020 - 11:03:56 AM
Last modification on : Saturday, November 7, 2020 - 3:26:24 AM

### Identifiers

• HAL Id : hal-02991946, version 1
• ARXIV : 1805.10189

### Citation

François Labourie, Jeremy Kahn, Shahar Mozes. Surface groups in uniform lattices of some semi-simple groups. 2020. ⟨hal-02991946⟩

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