View source for Acyclic group

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{{MSC|20J05}}

A group having the same constant coefficient homology as the trivial group (cf. also [[Homology|Homology]]). This means that its [[Classifying space|classifying space]] is an acyclic space. In the literature the earliest examples are Higman's four-generator four-relator group {{Cite|Hi}}

$$\langle x_0, x_1, x_2, x_3 : x_{i+1}x_ix_{i+1}^{-1} = x_i^2, i\in \mathbb{Z}/4\rangle$$

and others found in combinatorial group theory {{Cite|BaGr}}, {{Cite|BaDyHe}}, {{Cite|BeMi}}. Further examples arise in geometry ({{Cite|Ep}}, {{Cite|Ma}}, {{Cite|Se}}, {{Cite|SaVa}}, {{Cite|GrSe}}) or as automorphism groups of large objects ({{Cite|HaMc}}; for example, the group of all bijections of an infinite set). Algebraically closed groups are acyclic.

Many proofs of acyclicity of infinitely generated groups rely on the property that all binate groups are acyclic {{Cite|Be3}} (cf. also [[Binate group|Binate group]]). An important result in the [[Plus-construction|plus-construction]] approach to the higher [[Algebraic K-theory|algebraic $K$-theory]] of rings and operator algebras is that the infinite [[General linear group|general linear group]] of the [[Cone|cone]] of a ring is acyclic {{Cite|Wa}}, {{Cite|Be}}. Topologically, the plus-construction of a topological space is completely determined by a certain perfect, locally free, and hence acyclic, group {{Cite|BeCa}}.

Ubiquity results for acyclic groups include the following:

Every perfect group is the homomorphic image of an acyclic group {{Cite|He}}.

Every group is a normal subgroup of a normal subgroup of an acyclic group. This result has applications to [[Algebraic topology|algebraic topology]] {{Cite|KaTh}}.

Every [[Abelian group|Abelian group]] is the centre of an acyclic group {{Cite|BaDyHe}}, {{Cite|Be2}}.

In contrast to the above are results indicating that acyclic groups have "few" normal subgroups. Thus, the following acyclic groups admit no non-trivial finite-dimensional linear representations over any field:

algebraically closed groups;

Higman's group {{Cite|Hi}};

torsion-generated acyclic groups {{Cite|Be4}};

binate groups {{Cite|AlBe}};

the automorphisms groups of {{Cite|HaMc}}, see {{Cite|Be5}}, {{Cite|Be6}}.

Moreover, many of the above groups are simple modulo the centre.

====References====
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|valign="top"|{{Ref|AlBe}}||valign="top"| R.C. Alperin, A.J. Berrick, "Linear representations of binate groups" ''J. Pure Appl. Algebra'', '''94''' (1994) pp. 17–23 {{MR|1277521}} {{ZBL|0813.20060}}
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|valign="top"|{{Ref|BaDyHe}}||valign="top"| G. Baumslag, E. Dyer, A. Heller, "The topology of discrete groups" ''J. Pure Appl. Algebra'', '''16''' (1980) pp. 1–47 {{MR|0549702}} {{ZBL|0419.20026}}
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|valign="top"|{{Ref|BaGr}}||valign="top"| G. Baumslag, K.W. Gruenberg, "Some reflections on cohomological dimension and freeness" ''J. Algebra'', '''6''' (1967) pp. 394–409 {{MR|0232827}} {{ZBL|}}
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|valign="top"|{{Ref|Be}}||valign="top"| A.J. Berrick, "An approach to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110270/a1102704.png" />-theory", Pitman (1982) {{MR|649409}} {{ZBL|}}
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|valign="top"|{{Ref|Be2}}||valign="top"| A.J. Berrick, "Two functors from abelian groups to perfect groups" ''J. Pure Appl. Algebra'', '''44''' (1987) pp. 35–43 {{MR|0885094}} {{ZBL|}}
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|valign="top"|{{Ref|Be3}}||valign="top"| A.J. Berrick, "Universal groups, binate groups and acyclicity", ''Proc. 1987 Singapore Group Theory Conf.'', W. de Gruyter (1989) {{MR|0981847}} {{ZBL|0663.20053}}
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|valign="top"|{{Ref|Be4}}||valign="top"| A.J. Berrick, "Remarks on the structure of acyclic groups" ''Bull. London Math. Soc.'', '''22''' (1990) pp. 227–232 {{MR|1041135}} {{ZBL|0749.20001}}
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|valign="top"|{{Ref|Be5}}||valign="top"| A.J. Berrick, "Groups with no nontrivial linear representations" ''Bull. Austral. Math. Soc.'', '''50''' (1994) pp. 1–11 {{MR|1285653}} {{ZBL|0815.20026}}
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|valign="top"|{{Ref|Be6}}||valign="top"| A.J. Berrick, "Corrigenda: Groups with no nontrivial linear representations" ''Bull. Austral. Math. Soc.'', '''52''' (1995) pp. 345–346 {{MR|1348495}} {{ZBL|}}
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|valign="top"|{{Ref|BeCa}}||valign="top"| A.J. Berrick, C. Casacuberta, "A universal space for plus-constructions" ''Topology'' (to appear) {{MR|1670384}} {{ZBL|0933.55016}}
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|valign="top"|{{Ref|BeMi}}||valign="top"| A.J. Berrick, C.F. Miller, III, "Strongly torsion generated groups" ''Proc. Cambridge Philos. Soc.'', '''111''' (1992) pp. 219–229 {{MR|1142741}} {{ZBL|0762.20017}}
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|valign="top"|{{Ref|Ep}}||valign="top"| D.B.A. Epstein, "A group with zero homology" ''Proc. Cambridge Philos. Soc.'', '''68''' (1968) pp. 599–601 {{MR|0229692}} {{ZBL|0162.27502}} {{ZBL|0157.30703}}
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|valign="top"|{{Ref|GrSe}}||valign="top"| P. Greenberg, V. Sergiescu, "An acyclic extension of the braid group" ''Comment. Math. Helv.'', '''66''' (1991) pp. 109–138 {{MR|1090167}} {{ZBL|0736.20020}}
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|valign="top"|{{Ref|HaMc}}||valign="top"| P. de la Harpe, D. McDuff, "Acyclic groups of automorphisms" ''Comment. Math. Helv.'', '''58''' (1983) pp. 48–71 {{MR|}} {{ZBL|0522.20034}}
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|valign="top"|{{Ref|He}}||valign="top"| A. Heller, "On the homotopy theory of topogenic groups and groupoids" ''Ill. Math. J.'', '''24''' (1980) pp. 576–605 {{MR|0586797}} {{ZBL|0458.18006}}
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|valign="top"|{{Ref|Hi}}||valign="top"| G. Higman, "A finitely generated infinite simple group" ''J. London Math. Soc.'', '''26''' (1951) pp. 61–64 {{MR|0038348}} {{ZBL|0042.02201}}
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|valign="top"|{{Ref|KaTh}}||valign="top"| D.M. Kan, W.P. Thurston, "Every connected space has the homology of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110270/a1102705.png" />" ''Topology'', '''15''' (1976) pp. 253–258 {{MR|0413089}} {{ZBL|0355.55004}}
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|valign="top"|{{Ref|Ma}}||valign="top"| J.N. Mather, "The vanishing of the homology of certain groups of homeomorphisms" ''Topology'', '''10''' (1971) pp. 297–298 {{MR|0288777}} {{ZBL|0207.21903}}
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|valign="top"|{{Ref|SaVa}}||valign="top"| P. Sankaran, K. Varadarajan, "Acyclicity of certain homeomorphism groups" ''Canad. J. Math.'', '''42''' (1990) pp. 80–94 {{MR|1043512}} {{ZBL|0711.57022}}
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|valign="top"|{{Ref|Se}}||valign="top"| G.B. Segal, "Classifying spaces related to foliations" ''Topology'', '''17''' (1978) pp. 367–382 {{MR|0516216}} {{ZBL|0398.57018}}
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|valign="top"|{{Ref|Wa}}||valign="top"| J.B. Wagoner, "Developping classifying spaces in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110270/a1102706.png" />-theory" ''Topology'', '''11''' (1972) pp. 349–370
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