Abstract
An Artin group is defined with a presentation with a finite set of generators S𝑆 and, for each pair of generators s,t∈S𝑠,𝑡∈𝑆, a single relation (or none) having the form sts⋯=tst⋯𝑠𝑡𝑠⋯=𝑡𝑠𝑡⋯ where the number of letters on each side of the equality is the same and is denoted by ms,t𝑚𝑠,𝑡. Despite their simple definition, little is generally known about Artin groups (for example, the general solution to any Dehn problem is unknown). When the length ms,t𝑚𝑠,𝑡 of all existing relations is even, the Artin group is said to be even. Also, given an Artin group with its standard presentation, a standard parabolic subgroup is the subgroup generated by a subset of generators. In even Artin groups, there exists an obvious retraction to their standard parabolic subgroups that sends a generator to itself if it belongs to the subgroup and to 1 otherwise. Recently, Antolín and Foniqi showed that these retractions are immensely useful for obtaining new results in even Artin groups. This joint work with Bruno A. Cisneros de la Cruz and Islam Foniqi aims to answer the natural question: what happens when we introduce odd-length relations? How can we define retractions, and how do they generalize the existing results?