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On Tit’s conjecture and other questions concerning Artin and generalized Artin groups. (English) Zbl 0633.20021
Let \(\Gamma\) be a graph every edge in which is determined by two vertices x, y from the vertex set X not excepting \(x=y\). Suppose that there is a function \(\phi\) which assigns to each edge \(\{\) x,y\(\}\) of \(\Gamma\) a word of the form \((xyx...)(yxy...)^{-1}\), where the two bracketed words have the same length \(>1\). The group G(\(\Gamma\),\(\phi)\) with the presentation \(<X\|\phi\{\) x,y\(\}\), (\(\{\) x,y\(\}\) an edge in \(\Gamma)>\) is called an Artin group. More generally let the function \(\phi\) assign to each edge \(\{\) x,y\(\}\) of \(\Gamma\) a word of the form \(x^{\epsilon_ 1}y^{\delta_ 1}x^{\epsilon_ 2}y^{\delta_ 2}...x^{\epsilon_ n}y^{\delta_ n}\) where \(n\geq 2\), \(| \epsilon_ i| =| \delta_ i| =1\) for \(i=1,2,...,n\). The group \(G=G(\Gamma,\phi)\) defined as above is called a generalized Artin group. The author gives necessary and sufficient conditions for the group \(G=G(\Gamma,\phi)\) to satisfy the small cancellation conditions C(4) and T(4). If it is so then the finitely presented group G(\(\Gamma\),\(\phi)\) has solvable word and conjugacy problems. Also every relation between the elements \(x^ 2\) (x\(\in X)\) is a consequence of the relations \(y^ 2z^ 2=z^ 2y^ 2\) where \(\{\) y,z\(\}\) ranges over all edges of \(\Gamma\) for which \(\phi\{\) y,z\(\}\) is conjugated to one of the words \([y^{\epsilon},z^{\delta}]\), \(| \epsilon | =| \delta | =1\), or \((st)^ 2(st^{-1})^ 2\).
Reviewer: V.A.Roman’kov

MSC:
20F05 Generators, relations, and presentations of groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
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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References:
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