Weak order on groups generated by involutions

In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the Cayley graph of $(W,S)$. In the case of a Coxeter system $(W,S)$, Bj\"orner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their related structures. In this article, we discuss the following question: For which involution systems is the weak order a complete meet-semilattice? The class of involution systems that satisfies this condition is larger than the class of Coxeter systems (it contains, for instance, Cactus groups). In the case of an involution system with sign character, we provide a finite presentation by generators and relations and a classification in rank 3. We also obtain new characterizations of Coxeter systems in terms of the weak order, and prove a number of results on certain subclasses of these involution systems. Finally, we discuss further works and open problems in relation to biautomatic structures, geometric representations, mediangle graphs, and more.