Abstract :
Given a group G and a set S G of generators, set S−1={s−1s G} and . For g G, let l(g) denote the minimum length of any expression g=s1…sd with . For g,h G, set g h if l(g)+l(g−1h)=l(h).
The paper is devoted to the study of the pairs (G,S) for which 1 S, S∩S−1=S1 {s S s2=1}, and the partial order satisfies the following conditions:
(G, ) is a semilattice; denote by g∩h the greatest lower bound w.r.t. the order for any pair (g,h) of elements of G,g−1(g∩h) g−1h for all g,h G, andgh=hg is the least upper bound g h w.r.t. for the pair (g,h) whenever g∩h=1 and there exists u G such that g u and h u.
It is shown that the pairs above are exactly those for which G admits the presentation G= S; s2=1 for s S1, and sts−1t−1=1 for those s,t S, s≠t, for which the commuting relation st=ts holds in G . Call partially commutative Artin–Coxeter groups the groups defined by such presentations.
The pairs (G,S) above satisfy a “deletion condition” (D) analogous to the well-known deletion condition for Coxeter groups. It is shown that the pairs (G,S) satisfying (D) have solvable word problem, as is the case with usual Coxeter groups.
Normal forms for elements in partially commutative Artin–Coxeter groups are also described.
