### Abstract

Assume that G is a virtually torsion-free solvable group of finite rank and A is a ℤG-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on A that ensures that H^{n}(G,A) and H_{n}(G,A) are finite for all n ≥ 0. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.

Original language | English (US) |
---|---|

Pages (from-to) | 6441-6459 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 9 |

DOIs | |

State | Published - Jan 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*367*(9), 6441-6459. https://doi.org/10.1090/S0002-9947-2015-06262-X

}

*Transactions of the American Mathematical Society*, vol. 367, no. 9, pp. 6441-6459. https://doi.org/10.1090/S0002-9947-2015-06262-X

**The cohomology of virtually torsion-free solvable groups of finite rank.** / Kropholler, Peter; Lorensen, Karl.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The cohomology of virtually torsion-free solvable groups of finite rank

AU - Kropholler, Peter

AU - Lorensen, Karl

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Assume that G is a virtually torsion-free solvable group of finite rank and A is a ℤG-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on A that ensures that Hn(G,A) and Hn(G,A) are finite for all n ≥ 0. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.

AB - Assume that G is a virtually torsion-free solvable group of finite rank and A is a ℤG-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on A that ensures that Hn(G,A) and Hn(G,A) are finite for all n ≥ 0. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.

UR - http://www.scopus.com/inward/record.url?scp=84923272847&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923272847&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2015-06262-X

DO - 10.1090/S0002-9947-2015-06262-X

M3 - Article

AN - SCOPUS:84923272847

VL - 367

SP - 6441

EP - 6459

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -