TY - JOUR

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

AU - Kropholler, Peter

AU - Lorensen, Karl

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2015

Y1 - 2015

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.

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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 -