an:01864813
Zbl 1050.20028
Basarab, ??erban A.
Partially commutative Artin-Coxeter groups and their arboreal structure
EN
J. Pure Appl. Algebra 176, No. 1, 1-25 (2002).
00089242
2002
j
20F55 20F05 20F10 20F36 05C25
partially commutative Artin-Coxeter groups; presentations; word problem; normal forms
Summary: Given a group \(G\) and a set \(S\subseteq G\) of generators, set \(S^{-1}=\{s^{-1}\mid s\in G\}\) and \(\widetilde S=S\cup S^{-1}\). For \(g\in G\), let \(l(g)\) denote the minimum length of any expression \(g=s_1\cdots s_d\) with \(s_i\in\widetilde S\). For \(g,h\in G\), set \(g\subset h\) if \(l(g)+l(g^{-1}h)=l(h)\).
The paper is devoted to the study of the pairs \((G,S)\) for which \(1\notin S\), \(S\cap S^{-1}=S_1:=\{s\in S\mid s^2=1\}\), and the partial order \(\subset\) satisfies the following conditions: (i) \((G,\subset)\) is a semilattice; denote by \(g\cap h\) the greatest lower bound w.r.t. the order \(\subset\) for any pair \((g,h)\) of elements of \(G\), (ii)
\(g^{-1}(g\cap h)\subset g^{-1}h\) for all \(g,h\in G\), and (iii) \(gh=hg\) is the least upper bound \(g\cup h\) w.r.t. \(\subset\) for the pair \((g,h)\) whenever \(g\cap h=1\) and there exists \(u\in G\) such that \(g\subset u\) and \(h\subset u\).
It is shown that the pairs above are exactly those for which \(G\) admits the presentation \(G=\langle S;\;s^2=1\) for \(s\in S_1\), and \(sts^{-1}t^{-1}=1\) for those \(s,t\in S\), \(s\neq t\), for which the commuting relation \(st=ts\) holds in \(G\rangle\). Call the groups defined by such presentations `partially commutative Artin-Coxeter groups'.
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.