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Partially commutative Artin-Coxeter groups and their arboreal structure. (English) Zbl 1050.20028
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.
##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F36 Braid groups; Artin groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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