Fenchel's conjecture on NEC groups

TL;DR

This paper extends Fenchel's conjecture to planar non-Euclidean crystallographic groups with orientation-reversing elements, proving the existence of certain finite index subgroups in specific cases and exploring open questions.

Contribution

It advances the understanding of Fenchel's conjecture by addressing non-orientable and bordered cases for NEC groups with orientation-reversing transformations.

Findings

Confirmed the conjecture for nonorientable orbit spaces.

Confirmed the conjecture for bordered orientable surfaces of positive genus.

Partial results for genus zero cases, with some open questions remaining.

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 $\Gamma$ 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 $\Gamma$ 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.