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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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