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Surface subgroup conjecture

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Let M{\cal M} be a closed hyperbolic 33-manifold. A surface subgroup is the image of the fundamental group of a closed surface of genus at least 22 under a map inducing an injection on fundamental groups. The Surface Subgroup Conjecture. There exists a π1\pi_1-injective map

j:S→Mj:S\to {\cal M}

from a closed surface SS of genus at least 22 into M{\cal M}. This conjecture asks whether every closed hyperbolic 33-manifold contains a surface subgroup; its status is not specified in the source.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Surface subgroup conjecture

    In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.

    source: Wikipedia

References

Primary source

Lewis Bowen, “Immersions of Pants into a Fixed Hyperbolic Surface”, arXiv:math/0505480 (2005).

hyperbolic-3-manifoldssurface-subgroupsunsolved-problem
Posted

Progress summary

Refreshed
Claimed solved

The conjecture was proved: every closed hyperbolic three-manifold contains a closed surface whose fundamental group injects into its own.

Friedhelm Waldhausen posed the conjecture, later listed as Problem 3.75 in Kirby's problem list. It asks whether every closed hyperbolic 33-manifold contains a closed surface of genus at least 22 with an injective fundamental-group map.

Known results

  • Cooper, Long, and Reid (1997): the conjecture in the relevant boundary case.
  • Lackenby (2008): arithmetic manifolds.
  • Bowen (2005): conditional surface constructions and a reduction to whether the constructed surface can be closed.
  • Kahn and Markovic (2009): the closed case, via many quasiconvex surface subgroups.

Kahn–Markovic proof, 2009–2012

Kahn and Markovic announced a proof in 2009; their paper appeared in the Annals of Mathematics in 2012. Later literature continues to cite existence and ubiquity of surface subgroups in the closed case as their theorem, with no reported gap or retraction.

Current status (as of August 2026): The Surface Subgroup Conjecture is resolved by the Kahn–Markovic theorem; no remaining case is recorded.

Sources

Solutions 0

No solutions have been posted yet.