Bujalance García, EmilioCirre Torres, Francisco JavierConder, Marston D. E.Costa González, Antonio Félix2025-08-202025-08-202025-08-20Emilio Bujalance, F. Javier Cirre, Marston D. E. Conder, Antonio F. Costa (2025). Fenchel’s conjecture on NEC groups. Revista Matemática Complutense. https://doi.org/10.1007/s13163-025-00540-w1139-1138 | eISSN 1988-2807https://doi.org/10.1007/s13163-025-00540-whttps://hdl.handle.net/20.500.14468/29917The registered version of this article, first published in “Revista Matemática Complutense, 2025", is available online at the publisher's website: Springer Nature, https://doi.org/10.1007/s13163-025-00540-w La versión registrada de este artículo, publicado por primera vez en “Revista Matemática Complutense, 2025", está disponible en línea en el sitio web del editor: Springer Nature, https://doi.org/10.1007/s13163-025-00540-wA 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.eninfo:eu-repo/semantics/embargoedAccess12 Matemáticasjournal article