JP Journal of Algebra, Number Theory and Applications
Volume 38, Issue 4, Pages 327 - 337
(August 2016) http://dx.doi.org/10.17654/NT038040327 |
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A NOTE ON FINITELY GENERATED GROUPS OF COHOMOLOGICAL DIMENSION AT MOST ONE
Rasheed M. S. Mahmood and Mohammad Al-Hawari
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Abstract: Call a finitely generated group G to possess the property P with respect to a nonzero commutative ring R with unit element, if the cohomological dimension of G is not greater than 1. In this paper, we show that if G is a group acting on a tree X without inversions such that for each vertex v of X, the vertex group of v has the property P, the edge group of each edge e of X is finite and the quotient graph for the action of G on X is finite, then G has the property P. As an application, we show that if is a tree product of the groups with amalgamation subgroups such that each has the property P, is finite, and I is finite, then A has the property P. Furthermore, if is the HNN group of basis G and associated pairs of isomorphic subgroups of G such that G has the property P, is finite, and I is finite, then has the property P. |
Keywords and phrases: groups acting on trees, cohomology of groups, tree product of groups, HNN groups. |
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