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Coronas for properly combable spaces
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-11-26, DOI: 10.1142/s1793525321500643
Alexander Engel, Christopher Wulff

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible $σ$-compact space in which the corona sits as a $Z$-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space $BG$, then our constructions yield a $Z$-structure for the group.