Fenchel's conjecture on NEC groups

TL;DR

This work extends Fenchel's conjecture from co-compact Fuchsian groups to proper NEC groups by examining the existence of torsion-free normal subgroups of finite index that contain orientation-reversing elements, yielding a subgroup that uniformises a compact unbordered non-orientable surface. The authors develop systematic kernel-construction methods: for various signatures, they define surjections $\Gamma\to G$ to carefully chosen finite groups $G$ (e.g., $G_{[m_1,...,m_r]}$, $G_{(n_1,...,n_s)}$) so that the kernel is torsion-free and contains orientation-reversing elements; they often combine multiple maps via intersections to handle complex signature data. They obtain positive results in broad families, notably when the orbit space $\mathcal{H}/\Gamma$ is nonorientable or bordered orientable of positive genus, and in many genus-zero subcases, while certain low-genus, single-cycle cases remain open. The results connect NEC group theory with orbifold coverings, non-orientable regular maps, hypermaps, and abstract regular polytopes, illustrating a rich interplay between algebraic, geometric, and combinatorial structures and opening avenues for further exploration of orientation-reversing actions on unbordered non-orientable surfaces.

Abstract

A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable surface, or in algebraic terms, that $F$ has a normal subgroup of finite index that contains no element of finite order other than the identity. In this paper we initiate and make progress on an extension of Fenchel's conjecture by considering the following question: Does every planar non-Euclidean crystallographic group $Γ$ containing transformations that reverse orientation have a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered non-orientable surface? We answer this question in the affirmative in the case where the orbit space of $Γ$ is a nonorientable surface, and also in the case where this orbit space is a bordered orientable surface of positive genus. In the case where the genus of the quotient is $0$, we have an affirmative answer in many subcases, but the question is still open for others.

Theorems & Definitions (18)

• Theorem 1.1: nielsenbundgaard-nielsenfox
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: buj-martinez
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• Theorem 4.1
• Remark 4.2
• Theorem 4.3