**Coarse Geometric Coherence**

In our recent papers, 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.

## List of Current Projects & Topics of Interest

- enlarge the collection of known coarsely coherent groups and metric spaces
- develop permanence properties of coarse coherence
- construct property analogues for coarse coherence with certain desirable features and consequences
- understand the relationship (if any) between other coarse invariants and coarse coherence
- compute explicit examples of groups that are not coarsely coherent

## Publications

2020 B. Goldfarb, J. Grossman, ``Colimit theorems for coarse coherence with applications." Submitted. arxiv: 2001.09731

Abstract:

We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It is known that coarse coherence of the fundamental group has important implications for classification of high-dimensional manifolds. The Family Colimit Theorem is one of the permanence theorems that give structure to the class of coarsely coherent groups. In fact, all known permanence theorems follow from the Family Colimit Theorem. We also use this theorem to construct new groups from this class.

Abstract:

We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It is known that coarse coherence of the fundamental group has important implications for classification of high-dimensional manifolds. The Family Colimit Theorem is one of the permanence theorems that give structure to the class of coarsely coherent groups. In fact, all known permanence theorems follow from the Family Colimit Theorem. We also use this theorem to construct new groups from this class.

2018 B. Goldfarb, J. Grossman, ``Coarse coherence of metric spaces and groups and its permanence properties."

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.

*Bull. Aust. Math. Soc.*, 422-433. doi:10.1017/S0004972718000977Abstract:

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.