University at Albany
In our recent paper, Boris Goldfarb and I investigate two properties of metric spaces: ``coarse coherence'' and ``coarse regular coherence'', and apply these properties to finitely generated groups equipped with word metrics. Coarse coherence and coarse regular coherence are geometric counterparts of the classical notion of coherence in homological algebra and of the regular coherence property of groups defined and studied by Waldhausen, respectively. These coarse coherence notions are intelligible in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. Coarse regular coherence is in fact a weakening of Waldhausen's regular coherence, but can be used as effectively in K-theory computations. Coarse regular coherence implies weak regular coherence as defined by Carlsson and Goldfarb. Further, all groups known to be weakly regular coherent are also coarsely regular coherent. The class of coarsely regular coherent groups is therefore a large class of groups containing all groups with straight finite decomposition complexity as defined by Dranishnikov and Zarichnyi. My research is dedicated to proving structural results by developing coarse permanence properties for coarse coherence.
Publications
2018 B. Goldfarb, J. Grossman, ``Coarse coherence of metric spaces and groups and its permanence properties." Bull. Aust. Math. Soc., 1-12. doi:10.1017/S0004972718000977
Abstract:
We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by F. Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. We show that coarse regular coherence implies weak regular coherence, a weakening of regular coherence by G. Carlsson and the first author. The latter was introduced with the same goal as Waldhausen's, in order to perform computations of algebraic K-theory of group rings. However, all groups known to be weakly regular coherent are also coarsely regular coherent. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.
Abstract:
We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by F. Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. We show that coarse regular coherence implies weak regular coherence, a weakening of regular coherence by G. Carlsson and the first author. The latter was introduced with the same goal as Waldhausen's, in order to perform computations of algebraic K-theory of group rings. However, all groups known to be weakly regular coherent are also coarsely regular coherent. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.
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