Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, \({\mathbb{C}})\)
type: Item
instance of: scholarly article
title: Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, \({\mathbb{C}})\)
zbMATH Open document ID: 0627.57003
DOI: 10.1016/0022-4049(87)90034-X
author: John G. Ratcliffe
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 20J99
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 4020003
zbMATH Keywords: finitely generated groups
zbMATH Keywords: isomorphic to fundamental groups of 3- manifolds
zbMATH Keywords: discrete subgroups of \(SL_ 2\,{bbfC}\)
zbMATH Keywords: fundamental group of a finite, aspherical CW-complex
zbMATH Keywords: Euler characteristic
zbMATH Keywords: SQ-universal
zbMATH Keywords: rank
zbMATH Keywords: deficiency
zbMATH Keywords: hyperbolic 3-manifolds
zbMATH Keywords: hyperbolic volume
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4020003