Coronas for properly combable spaces

TL;DR

This paper develops a systematic method for constructing coronas (boundaries at infinity) of combable spaces, leading to new insights into their topological and algebraic properties, especially for groups with certain combings.

Contribution

It introduces properness, coherence, and expandingness properties for combings and shows how these enable the construction of coronas with desirable topological features and applications to group theory.

Findings

Constructed coronas as Z-sets in contractible spaces.

Established bijectivity of transgression maps and (co)homological properties.

Provided Z-structures for groups with finite classifying spaces.

Abstract

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 $\sigma$-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.