A conjecture about Artin–Tits groups

Abstract. We conjecture that the word problem of Artin–Tits groups can be solved without introducing trivial factors ss−1 or s−1s. Here we make this statement precise and explain how it can be seen as a weak form of hyperbolicity. We prove the conjecture in the case of Artin–Tits groups of type FC, and we discuss various possible approaches for further extensions, in particular a syntactic argument that works at least in the right-angled case. Artin–Tits groups, also known as Artin groups, are those groups that are defined by relations of the form (1) sts... = tst... where both terms consist of two alternating letters and have the same length. These groups were first investigated by J. Tits in the late 1960’s [4]. Two seminal references are [5] and [15]. At the moment many questions involving these groups remain open, including the decidability of the word problem in the general case [8]. The aim of this paper is to discuss a conjecture about the word problem of Artin–Tits groups, namely that, when a word w represents 1, hence is equivalent to the empty wor