Hyperplane complements of large type
Let \(\Omega\) be a convex open subset of \(V={\mathbb{R}}^ n\), and \b{M} a collection of linear hyperplanes which is locally finite on \(\Omega\). Then \(\Omega\) is partitioned using \b{M} in facets. \b{M} is of large type on \(\Omega\) if each plinth P (i.e. codimension 2 facet) has multiplicity {\#}\(\{\) \(M\in \underline M\); \(P\subset M\}\geq 3\). We prove that if \b{M} is of large type on \(\Omega\), the complex hyperplane complement \(Y=\{z\in V\otimes {\mathbb{C}}\); Im(z)\(\in \Omega \}-\cup \{M\otimes {\mathbb{C}}\); \(M\in M\}\) is an Eilenberg-MacLane space of type K(\(\pi\),1). This is analogous to a theorem of \textit{P. Deligne} [ibid. 17, 273-302 (1972; Zbl 0238.20034)]. The proof is homotopy theoretical. The most striking consequence of the condition of large type is that no facets of codimension greater than 2 exist. The method allows us to prove the following ''Freiheitssatz'': Let G be an Artin group, i.e. G has finitely many generators \(s_ i\) and relations \(s_ is_ js_ i\cdot \cdot \cdot =s_ js_ is_ j\cdot \cdot \cdot\) \((m_{ij}\) factors on both sides, \(m_{ij}=\infty\) having no consequence). Suppose G is of large type, i.e. \(3\leq m_{ij}\leq \infty\) for all i,j. Then the set \(\{s^ 2_ i;i\}\) freely generates a free subgroup of G. The condition of large type is taken from \textit{K. I. Appel} and \textit{P. E. Schupp} [ibid. 72, 201-220 (1983; Zbl 0536.20019)]. In that paper small cancellation theory is used and (a special case of) the above ''Freiheitssatz'' is obtained.
scientific article; zbMATH DE number 3899768