(Springer, 2025-08-20) Bujalance García, Emilio; Cirre Torres, Francisco Javier; Conder, Marston D. E.; Costa González, Antonio Félix; Agencia Estatal de Investigación
A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable surface, or in algebraic terms, that 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 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 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 , we have an affirmative answer in many subcases, but the question is still open for others.
