Extending properties to relatively hyperbolic groups. (English) Zbl 1446.19005
Let \(G\) be a finitely generated group that is relatively hyperbolic with respect to subgroups \(H_1,\ldots,H_n\). Given a property of metric spaces that is invariant under coarse invariance, the authors study the question when this property extends from the subgroups \(H_i\) to the whole group \(G\).
They show that every property that satisfies certain permanence properties and is satisfied by all spaces with finite asymptotic dimension is extendable in the above sense. These permanence properties are coarse inheritance, the finite union theorem, the union theorem and the transitive fibering theorem. In particular, the result applies to finite asymptotic dimension, coarse embeddability, exactness, finite decomposition complexity and straight finite decomposition complexity.
For some of these notion, extendability to relatively hyperbolic groups was known before, for example see [M. Dardalat and E. Guentner, J. Reine Angew. Math. 612, 1–15 (2007; Zbl 1146.46047)]. For more information on permanence properties of metric spaces see [E. Guentner et al., Recent progress in general topology III. Amsterdam: Atlantis Press. 507–533 (2014; Zbl 1300.54003)].
They show that every property that satisfies certain permanence properties and is satisfied by all spaces with finite asymptotic dimension is extendable in the above sense. These permanence properties are coarse inheritance, the finite union theorem, the union theorem and the transitive fibering theorem. In particular, the result applies to finite asymptotic dimension, coarse embeddability, exactness, finite decomposition complexity and straight finite decomposition complexity.
For some of these notion, extendability to relatively hyperbolic groups was known before, for example see [M. Dardalat and E. Guentner, J. Reine Angew. Math. 612, 1–15 (2007; Zbl 1146.46047)]. For more information on permanence properties of metric spaces see [E. Guentner et al., Recent progress in general topology III. Amsterdam: Atlantis Press. 507–533 (2014; Zbl 1300.54003)].
Reviewer: Daniel Kasprowski (Bonn)
MSC:
| 19D50 | Computations of higher \(K\)-theory of rings |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 20F69 | Asymptotic properties of groups |
Keywords:
relative hyperbolicity; finite decomposition complexity; asymptotic dimension; Cayley graphReferences:
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