The \(L^2\)-torsion polytope of amenable groups. (English) Zbl 1483.20078
Summary: We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the \(L^2\)-torsion polytope among \(G\)-CW-complexes for these groups. As another application we prove that the \(L^2\)-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.
MSC:
| 20F65 | Geometric group theory |
| 57Q10 | Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. |
| 16S85 | Associative rings of fractions and localizations |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
Keywords:
\(L^2\)-torsion polytope; amenable groups; polytope class; Atiyah conjecture; \(3\)-manifoldsReferences:
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