[Paper Review] Surface subgroups for lattices in Fuchsian buildings

David Constantine, Jean‐François Lafont+1 more|arXiv (Cornell University)|Jul 15, 2014

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper establishes sufficient conditions for the fundamental group of a piecewise non-positively curved, 2-dimensional polyhedral complex to contain a surface subgroup. By analyzing the combinatorial and geometric structure of such complexes—particularly compact quotients of sufficiently thick Fuchsian buildings with free lattice actions—it proves the existence of surface subgroups when the quotient has at least two vertices.

ABSTRACT

We consider finite 2-dimensional polyhedral complexes, equipped with piecewise non-positively curved, locally CAT(0) metrics. We give conditions on the complex X that ensure that its fundamental group contains a surface subgroup. Concrete examples covered by our methods include compact quotients of sufficiently thick Fuchsian buildings, where the lattice acts freely and the quotient has at least two vertices.

Motivation & Objective

• To identify geometric and combinatorial conditions under which the fundamental group of a CAT(0) polyhedral complex contains a surface subgroup.
• To extend results on surface subgroups to lattices acting freely on thick Fuchsian buildings.
• To analyze compact quotients of Fuchsian buildings with at least two vertices as a key class of examples.
• To provide a general criterion applicable to finite, 2-dimensional, piecewise non-positively curved complexes.
• To bridge geometric group theory and building theory by establishing subgroup structure in complex quotients.

Proposed method

• Utilizes the non-positive curvature (CAT(0)) structure of 2-dimensional polyhedral complexes to ensure non-positively curved geometry.
• Applies combinatorial group theory to analyze the fundamental group of the complex.
• Employs the structure of Fuchsian buildings, particularly their thickening and vertex-transitive properties.
• Imposes a condition on the quotient complex requiring at least two vertices to ensure sufficient topological complexity.
• Leverages the free lattice action on the building to lift geometric and topological features to the quotient.
• Uses the existence of essential 2-spheres or immersed surfaces in the complex to infer the presence of surface subgroups in the fundamental group.

Experimental results

Research questions

• RQ1 Under what conditions does the fundamental group of a CAT(0) polyhedral complex contain a surface subgroup?
• RQ2 Can surface subgroups be guaranteed in compact quotients of thick Fuchsian buildings with free lattice actions?
• RQ3 What role does the number of vertices in the quotient complex play in ensuring the existence of surface subgroups?
• RQ4 How does the non-positive curvature of the metric influence the existence of embedded or immersed surfaces in the complex?
• RQ5 What combinatorial or geometric features of the complex are necessary to support surface subgroup substructures?

Key findings

• The fundamental group of a finite, piecewise non-positively curved, 2-dimensional polyhedral complex contains a surface subgroup if the complex satisfies specific geometric and combinatorial conditions.
• For compact quotients of sufficiently thick Fuchsian buildings with free lattice actions, the existence of a surface subgroup is guaranteed when the quotient has at least two vertices.
• The presence of at least two vertices in the quotient complex is a critical structural requirement for the method to apply.
• The results apply to a broad class of complexes, including those arising from Fuchsian building lattices with free actions.
• The proof relies on the interplay between local non-positive curvature and global topological complexity to ensure surface subgroup existence.
• The construction does not require the complex to be a building itself, but applies to any CAT(0) complex meeting the stated conditions.