Surface subgroups on 1-vertex and 3-vertex polyhedra forming triangular hyperbolic buildings

TL;DR

This paper investigates surface subgroups within certain hyperbolic triangular buildings by analyzing 2-cycles in polyhedra, discovering surface subgroups in three specific torsion-free groups, thus advancing understanding of these complex geometric structures.

Contribution

It introduces a novel approach by examining 2-cycles in polyhedra and their covers to identify surface subgroups, expanding the known examples in hyperbolic building groups.

Findings

Surface subgroups found in three of the 23 torsion-free groups.

Method based on analyzing 2-cycles in polyhedra and their covers.

Provides new examples of surface subgroups in hyperbolic triangular buildings.

Abstract

In this article we study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. Kangaslampi and Vdovina have constructed and classified all groups acting simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. These groups gave the first examples of cocompact lattices acting simply transitively on vertices of hyperbolic triangular Kac-Moody buildings that are not right-angled. The hyperbolic buildings arise as universal covers of polyhedra glued together from with 15 triangular faces with words written on the boundary. Later they proved, that most of the obtained 23 torsion free groups do not admit periodic planes of genus 2. Here we take another approach to study surface subgroups of these groups. Namely, we consider first 2-cycles in the 1-vertex polyhedron defined by the triangles, then in the 3-vertex cover of this polyhedron. As a result we find surface subgroups in three of the 23 torsion free groups.