Giving a quadratic solution for the word problem in 3-free Artin groups

We will explain a method of rewriting words that allows us to obtain geodesic representatives for elements in Artin groups that do not have a relation of length 3 and, as a direct consequence, we will solve the word problem in this (big) family of Artin groups. Our techniques are based on detecting a certain type of words, which we call critical, and defining from them transformation rules that we call tau-moves. In particular, we show that in every non-geodesic word there is a sequence of tau-moves that ends with a reduction in the word. Additionally, we provide a recursive method to apply these sequences, so that we obtain an algorithm with quadratic complexity in the length of the input word. This is a joint work with Rubén Blasco-García and Rose Morris-Wright.

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