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On Artin groups admitting retractions to parabolic subgroups | ||
| International Journal of Group Theory | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 08 خرداد 1404 | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2025.143611.1935 | ||
| نویسندگان | ||
| Islam Foniqi* 1؛ María Cumplido2؛ Bruno Aarón Cisneros de la Cruz3 | ||
| 1University of East Anglia | ||
| 2Universidad de Sevilla | ||
| 3National Autonomous University of Mexico | ||
| چکیده | ||
| This paper extends the theory of retractions from even Artin groups to broader families, including FC-type Artin groups and others with mixed labelling on their Coxeter graphs. Retractions are group homomorphisms that preserve standard parabolic subgroups, and we show that such mappings uniquely extend to all parabolic subgroups, independent of their presentation. This property enables a systematic generalization of prior work by Antolín and Foniqi, simplifying the conditions for determining intersections of parabolic subgroups. A core result of this paper is the identification of conditions under which FC-type Artin groups admit retractions. We classify these groups based on the structure of their Coxeter graphs, proving that triangles in these graphs must satisfy specific label constraints, such as avoiding certain odd and mixed label configurations. Furthermore, we demonstrate that retractions simplify the study of subgroup intersections by reducing them to well-defined subsets of generators, tied to the interaction of two parabolic subgroups. The paper also addresses the unique extensions of retractions to arbitrary parabolic subgroups, leveraging combinatorial and geometric properties of FC-type Artin groups. By employing retractions, we reduce the problem of determining parabolic subgroup intersections to cases involving smaller sub-graphs or subgroups, providing a recursive framework that facilitates computation. These results advance the understanding of Artin groups with mixed labels and provide new tools for analyzing their parabolic subgroup structures. They also highlight the power of retractions as a unifying technique for addressing longstanding questions in Artin group theory. | ||
| کلیدواژه‌ها | ||
| Artin group؛ FC type؛ parabolic؛ retraction | ||
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آمار تعداد مشاهده مقاله: 25 |
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