[Paper Review] Surface groups in uniform lattices of some semi-simple groups

Jeremy Kahn, François Labourie+1 more|arXiv (Cornell University)|May 25, 2018

Geometric and Algebraic Topology16 references3 citations

TL;DR

This paper establishes the existence of surface subgroups in uniform lattices of center-free, complex semisimple Lie groups by introducing a new geometric framework based on $K$-Sullivan maps—generalizations of $K$-quasi-circles in higher-rank symmetric spaces. It proves that such maps are Hölder continuous and that equivariant $\zeta$-Sullivan maps imply Anosov representations, yielding a quantitative version of the surface subgroup theorem with limit maps arbitrarily close to smooth circles.

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 Hölder. 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.

Motivation & Objective

• To establish the existence of surface subgroups in uniform lattices of complex semisimple Lie groups beyond the classical $\mathrm{PSL}(2,\mathbb{C})$ case.
• To generalize the Kahn–Markovic theorem to higher-rank Lie groups using a new geometric framework based on flag manifolds and $\mathfrak{sl}_2$-triples.
• To provide a quantitative version of the surface subgroup theorem by introducing $\zeta$-Sullivan maps and proving their regularity and dynamical implications.
• To show that $\zeta$-Sullivan maps are $\alpha$-Hölder continuous and that equivariance under Fuchsian groups implies Anosov representations.
• To extend the theory to non-complex groups such as $\mathrm{PU}(p,q)$ with $q>p>0$, broadening the scope of the main result.

Proposed method

• Define $K$-Sullivan maps as maps from $\mathbb{P}^1(\mathbb{R})$ to a flag manifold $\mathbf{F}$, where each triple of points on the circle bounds a metric $d_\tau$ such that the map stays uniformly close to a circle in this metric.
• Use the coarse geometry of 'paths of triangles' in the flag manifold $\mathbf{F}$ to analyze the behavior of $\zeta$-Sullivan maps and establish a Morse-type lemma for quasi-geodesics in this context.
• Prove that any $\zeta$-Sullivan map is $\alpha$-Hölder continuous for some $\alpha > 0$, establishing regularity without assuming continuity a priori.
• Show that if a representation $\rho$ of a cocompact Fuchsian group $\Gamma$ admits a $\rho$-equivariant $\zeta$-Sullivan map, then $\rho$ is $\mathbf{F}$-Anosov and the map is its limit curve.
• Apply exponential mixing of lattice actions (via spectral gap results) to control the geometry of the flag manifold and ensure the existence of such maps in uniform lattices.
• Use induction and metric comparison techniques (e.g., $\ell_1$ product metrics and Lévy–Prokhorov distance) to extend results from tori to general compact symmetric spaces and prove the quantitative bound on the distance between measures.

Experimental results

Research questions

• RQ1 Can surface subgroups be constructed in uniform lattices of higher-rank semisimple Lie groups beyond $\mathrm{PSL}(2,\mathbb{C})$?
• RQ2 What is the appropriate generalization of $K$-quasi-circles and $K$-quasi-symmetric maps in higher-rank symmetric spaces?
• RQ3 How can one ensure the regularity (e.g., Hölder continuity) of equivariant maps into flag manifolds that generalize limit curves?
• RQ4 Under what conditions does the existence of a $\zeta$-Sullivan map imply that a representation is Anosov?
• RQ5 Can the construction be made quantitative so that the limit map is arbitrarily close to a smooth circle?

Key findings

• Any $\zeta$-Sullivan map is $\alpha$-Hölder continuous for some $\alpha > 0$, establishing regularity without assuming continuity.
• If a representation $\rho$ of a cocompact Fuchsian group admits a $\rho$-equivariant $\zeta$-Sullivan map, then $\rho$ is $\mathbf{F}$-Anosov and the map is its limit curve.
• For any uniform lattice $\Gamma$ in a center-free, complex semisimple Lie group $\mathsf{G}$, there exists a surface subgroup in $\Gamma$, generalizing the Kahn–Markovic theorem.
• The limit map of the surface subgroup can be made arbitrarily close to a smooth round circle in the flag manifold via $K$-Sullivan maps with $K$ close to 1.
• The construction is quantitative: for any $\kappa > 0$, there exists a $\zeta$ such that the $\zeta$-Sullivan map satisfies $d(\phi \cdot \mu, \overline{\phi} \cdot \mu) \leq \kappa \cdot \mathrm{diam}(\mathsf{T}^1)$, ensuring control over measure distortion.
• Exponential mixing of the lattice action (via spectral gap) is used to ensure the existence of such maps in the presence of uniformity and irreducibility conditions.