Surface groups in uniform lattices of some semi-simple groups

Jeremy Kahn; François Labourie; Shahar Mozes

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arXiv · 1805.10189

Surface groups in uniform lattices of some semi-simple groups

Jeremy Kahn·François Labourie·Shahar Mozes

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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.

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Jeremy Kahn, François Labourie, Shahar Mozes. 2018-05-25. Surface groups in uniform lattices of some semi-simple groups. https://arxiv.org/abs/1805.10189

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