Homological Dimension of Solvable Groups
Peter Kropholler, Conchita Mart\'inez-P\'erez
TL;DR
This paper characterizes the homological dimension of elementary amenable groups over any commutative ring, showing it is either infinite or equal to the Hirsch length, extending Stammbach's 1970 results for solvable groups.
Contribution
It generalizes Stammbach's result by establishing the homological dimension for elementary amenable groups over arbitrary rings.
Findings
Homological dimension equals the Hirsch length or is infinite.
Provides criteria for finiteness of homological dimension.
Complete classification for elementary amenable groups.
Abstract
In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.
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Taxonomy
TopicsFinite Group Theory Research · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models