Immersing almost geodesic surfaces in a closed hyperbolic three manifold

By Jeremy A. Kahn, V. Marković. 2009

262 citations recorded by the source corpus.

Let M 3 be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into M 3 so that for each, one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7c3c2c246bdf245f4a5818384d0555d1a5be9b8

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A boundary criterion for cubulation (2009, background): “In fact, they proved the following [12, Thm. 1.1]: P ROPOSITION 4.1.”
The Thurston norm and twisted Alexander polynomials (2012, methodology): “(iii) Agol [2], building on the Surface Subgroup Theorem of Kahn–Markovic [23] and on work of Wise [49] showed that in fact the fundamental group of any closed hyperbolic 3-manifold is virtually RFRS.”
AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, methodology): “In recent years there has been a revived interest from number theory in discrete Zariski dense subgroups, which are contained in arithmetic lattices without being lattices themselves. Such groups have been coined thin groups Sarnak [2014]. Hitchin representations and maximal representations, and mor”
Determining Fuchsian groups by their finite quotients (2014, background): “The first question, due to Gromov, was motivated by the case of hyperbolic 3-manifolds, and in this special case the question was settled recently by Kahn and Markovic [27].”
The virtual Haken conjecture (2012, methodology): “For a closed hyperbolic 3-manifold, we have the following result of BergeronWise based on work of Kahn-Markovic [24] (and making use of seminal results of Sageev on cubulating groups containing codimension-one subgroups [40]).”
An invitation to higher Teichm\"uller theory (2018, methodology): “For more general constructions of surface subgroups in lattices of Lie groups, following the construction of Kahn and Markovic surface subgroups in three-manifold groups [71] we refer to work of Hamenstädt and Kahn [65], and forthcoming work of Labourie, Kahn, and Mozes [80].”
Distinguishing geometries using finite quotients (2014, methodology): “The proof of Theorem A uses the dramatic recent developments of Agol [1], Kahn and Markovic [27] and Wise [50], whose work implies that the fundamental groups of closed hyperbolic 3–manifolds are also fundamental groups of compact, virtually special cube complexes (see [3] for a summary of these dev”