Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Surface Subgroups for Cocompact Lattices of Isometries of $H^{2n}$

Jeremy Kahn, Zhenghao Rao

TL;DR

The paper proves that every cocompact lattice $\Gamma<\mathrm{SO}^+(2n,1)$ with $n\ge 2$ contains surface subgroups, completing the rank-one uniform-lattice picture by addressing the $\mathbb{H}^{2n}$ case left open by Hamenstädt. It extends the Kahn–Markovic pant-program to higher even dimensions by developing the geometry of good pants, a quasi-uniform feet-measure framework, and a doubling trick, then applies Hall’s Marriage Theorem to match pants along cuffs. A core technical advance is the control of cuff monodromies and the distribution of feet through an estimated invariant measure $\mu_a$ which is quasi-uniform and closely tracks the actual feet distribution $\nu_a$. This leads to a global, π1-injective immersed surface obtained by assembling well-matched pants, yielding surface subgroups for all cocompact lattices in $\mathrm{SO}^+(2n,1)$ and contributing to Gromov’s question on surface subgroups in one-ended hyperbolic groups.

Abstract

We prove the existence of surface subgroups within any cocompact lattice $Γ$ in $\mathrm{SO}(2n,1)$ for $n\geq2$. This result addresses the cases missing from the work of Hamenstädt in 2015, who constructed surface subgroups in cocompact lattices for all other rank-one semisimple Lie groups of non-compact type.

Surface Subgroups for Cocompact Lattices of Isometries of $H^{2n}$

TL;DR

The paper proves that every cocompact lattice with contains surface subgroups, completing the rank-one uniform-lattice picture by addressing the case left open by Hamenstädt. It extends the Kahn–Markovic pant-program to higher even dimensions by developing the geometry of good pants, a quasi-uniform feet-measure framework, and a doubling trick, then applies Hall’s Marriage Theorem to match pants along cuffs. A core technical advance is the control of cuff monodromies and the distribution of feet through an estimated invariant measure which is quasi-uniform and closely tracks the actual feet distribution . This leads to a global, π1-injective immersed surface obtained by assembling well-matched pants, yielding surface subgroups for all cocompact lattices in and contributing to Gromov’s question on surface subgroups in one-ended hyperbolic groups.

Abstract

We prove the existence of surface subgroups within any cocompact lattice in for . This result addresses the cases missing from the work of Hamenstädt in 2015, who constructed surface subgroups in cocompact lattices for all other rank-one semisimple Lie groups of non-compact type.

Paper Structure

This paper contains 24 sections, 34 theorems, 109 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Some geometry of good pants
  3. The structure of
  4. Preliminary definitions
  5. Monodromy, good curves and normal bundles
  6. Orthogeodesics, connections and feet
  7. Pants and feet
  8. Tripods and framed tripods
  9. Good and bad pants
  10. The doubling trick
  11. Steiner graph of pants
  12. Good cuffs implies good or bad pants
  13. Quasi-uniformity for the measure on feet of good pants
  14. Lengths of cuffs determined by a third connection
  15. Monodromy estimate and feet distance
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose $\Gamma<\mathop{\mathrm{SO}}\nolimits^+(2n,1)$ is a cocompact lattice with $n>1$. Then $\Gamma$ contains surface subgroups.

Figures (6)

  • Figure 1: One possible obstruction: the "gap" between the two components
  • Figure 2: A worse case: the "bands"
  • Figure 3: A pair of well-connected tripods
  • Figure 4: A pair of badly-connected tripods
  • Figure 5: Lifts of $P'$ in the universal cover
  • ...and 1 more figures

Theorems & Definitions (69)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 59 more