The word problem for Artin groups of FC type (Q1295541)

Artin groups are generalizations of the braid groups. Each Artin group has a canonical homomorphism onto a Coxeter group (analogous to the homomorphism from the \(n\)-string braid group onto the symmetric group \(S_n\)). An Artin group is said to be of finite type if the corresponding Coxeter group is finite, and of infinite type otherwise. Quite a lot is known about Artin groups of finite type, in particular, they are biautomatic [\textit{R. Charney}, Math. Ann. 292, No. 4, 671-684 (1992; Zbl 0736.57001)], which implies that they have word problem which is solvable in quadratic time. Similar results are known for certain classes of infinite-type Artin groups (e.g., those of extra-large type and right-angled Artin groups; see the paper for definitions and references). The present author considers Artin groups of FC-type, i.e., those which may be decomposed as iterated free products of finite-type Artin groups, amalgamating subgroups generated by subsets of the canonical generators, a class introduced by \textit{R. Charney} and \textit{M. W. Davis} [J. Am. Math. Soc. 8, No. 3, 597-627 (1995; Zbl 0833.51006)]. He shows that such groups are asynchronously automatic, which implies they have solvable word problem. Further, by constructing a particular set of coset representatives, he shows that such groups in fact have word problem solvable in quadratic time.