Coronas for properly combable spaces. (English) Zbl 1535.51012
This paper presents a systematic method for constructing coronas, specifically Higson-dominated boundaries at infinity, of combable spaces. The authors introduce three additional properties for combings: properness, coherence, and expandingness. Properness is a crucial condition for the successful construction of the corona. Assuming coherence and expandingness, attaching our corona to a Rips complex construction results in a contractible \(\sigma\)-compact space, where the corona acts as a \(Z\)-set. This leads to the bijectivity of transgression maps, injectivity of the coarse assembly map, and surjectivity of the coarse co-assembly map. In the case of groups, the authors provide an estimate of the cohomological dimension of the corona based on the asymptotic dimension. Moreover, if the group has a finite model for its classifying space \(\mathrm{BG}\), the constructions yield a \(Z\)-structure for the group.
Reviewer: Sonia Pérez Díaz (Madrid)
MSC:
| 51F30 | Lipschitz and coarse geometry of metric spaces |
| 57M07 | Topological methods in group theory |
| 20F65 | Geometric group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
Keywords:
combings; \(\mathcal{Z}\)-boundaries; coronas; coarse assembly map; asymptotic dimension; coarse homology theories; transgression maps; non-positively curved spaces and groupsSoftware:
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