Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, \({\mathbb{C}})\)
Let \({\mathcal C}\) denote the class of those finitely generated groups which are isomorphic to fundamental groups of 3-manifolds, and let v\({\mathcal C}\) be the class of those groups which are virtually in \({\mathcal C}\), i.e. which have a subgroup of finite index in \({\mathcal C}\). Natural examples of groups in v\({\mathcal C}\) are the discrete subgroups of \(SL_ 2 {\mathbb{C}}\) or of \(PSL_ 2 {\mathbb{C}}.\) The author shows first of all that every G in v\({\mathcal C}\) has a subgroup \(G_ 0\) of finite index that is the fundamental group of a finite, aspherical CW-complex; consequently \(G_ 0\), and hence G, have an Euler characteristic. The author discusses next the values of \(\chi\) (G): Theorems 2 and 5. If G is in v\({\mathcal C}\) then \(\chi\) (G)\(\leq 1\). Moreover: (a) \(\chi (G)>0\) if, and only if, G is finite. (b) \(\chi (G)=0\) if, and only if, G is virtually the fundamental group of an aspherical, compact, orientable 3-manifold whose boundary is empty or made up of tori and Klein bottles. (c) If \(\chi (G)<0\) then G is SQ-universal, i.e. each countable group can be embedded into a quotient G/N of G. In Section 2 the number 1-\(\chi\) (G) is compared with the rank of G, i.e. with the minimal number of generators of G, and with the deficiency of G, i.e. with the maximum of the differences m-n, where \(G\cong <x_ 1,...,x_ m\); \(r_ 1,...,r_ n>.\) Theorems 3 and 4. def \(G\leq 1-\chi (G)\leq rank G.\) Moreover, the author determines explicitly the cases where the bounds are attained: \(1-\chi (G)=rank G\) iff G is free, and \(1-\chi (G)=def G\) iff G is the fundamental group of a finite, aspherical and 2-dimensional CW- complex. The last section is devoted to hyperbolic 3-manifolds; one of the key results is Theorem 7. Assume G is a torsion-free, discrete subgroup of \(PSL_ 2 {\mathbb{C}}\). Then \(\chi\) (G)\(\geq 0\) if, and only if, G is abelian or \(H^ 3/G\) has finite hyperbolic volume.
isomorphic to fundamental groups of 3- manifolds
fundamental group of a finite, aspherical CW-complex
scientific article; zbMATH DE number 4020003
scientific article