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Indecomposable continua as Higson coronae. (English) Zbl 1464.54019

It is known that the Higson corona \(\nu\mathbb{N}\) of the space of natural numbers \(\mathbb{N}\) is an indecomposable continuum. The main purpose of the paper is to characterize a space that is coarsely equivalent to \(\mathbb{N}\). More precisely, the main theorem states: for any non-compact proper metric space \(X\) that is coarsely geodesic and has coarse bounded geometry, the Higson corona \(\nu X\) is an indecomposable continuum if and only if \(X\) is coarsely equivalent to \(\mathbb{N}\). In a similar way, the author gives a characterization of a space that is coarsely equivalent to the space of integers \(\mathbb{Z}\). The author then applies the result to finitely generated groups. He shows that for any finitely generated group with the word length metric, \(G\) has exactly one end (respectively, two ends) if and only if the Higson corona \(\nu G\) is a decomposable continuum (respectively, a topological sum of two non-metrizable indecomposable continua, each of which is homeomorphic to \(\nu\mathbb{N}\)).

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20E34 General structure theorems for groups
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