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Register a new userCoarse direct products and property {C}
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We show that coarse property C is preserved by finite coarse direct products. We also show that the coarse analog of Dydaks countable asymptotic dimension is equivalent to the coarse version of straight finite decomposition complexity and is therefore preserved by direct products.
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We develop a coarse notion of bundle and use it to understand the coarse geometry of group extensions and, more generally, groups acting on proper metric spaces. The results are particularly sharp for groups acting on (locally finite) trees with Abelian stabilizers, which we are able to classify completely.
When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of C$^*$-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in C$^*$-categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of W$^*$-categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any W$^*$-category of normal representations of a W$^*$-algebra. Finding a universal property for the more general case of direct integrals remains an open problem.
In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several conjectures from [LS14a]. Finally, combining these results with computations from [LS14c] and the refined s-invariant from [LS14b] we obtain new results about the slice genera of certain knots.
The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten the proof of the Dahmani-Guirardel-Osin theorem.
We generalize Bestvinas notion of a $mathcal{Z}$-boundary for a group to that of a coarse $mathcal{Z}$-boundary. We show that established theorems about $mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $mathcal{Z}$-boundaries become theorems when applied to coarse $mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a model $mathcal{Z}$-geometry. In accordance with the existing theory, we also develop an equivariant version of the above -- that of a coarse $Emathcal{Z}$-boundary.
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