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Coarse structures on groups
Andrew Nicas,David Rosenthal +1 more
TL;DR: In this paper, the group-compact coarse structure on a Hausdorff topological group was introduced and the asymptotic dimension of the coarse structure was shown to be 1.
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Abstract: We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete groups. We show that the asymptotic dimension in our sense of the free topological group on a non-empty topological space that is homeomorphic to a closed subspace of a Cartesian product of metrizable spaces is 1.
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References
Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups
TL;DR: In this paper, the integral K-theoretic Novikov conjecture holds for torsion free arithmetic subgroups of linear algebraic groups, and for groups with torsions.
Asymptotic dimension of discrete groups
Alexander Dranishnikov,J. Smith +1 more
TL;DR: In this article, the authors extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups, and use this extension to prove a formula for the (1 − ε)-asymptotics of solvable groups in terms of their Hirsch length.
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Asymptotic Dimension
Gregory C. Bell,Alexander Dranishnikov +1 more
- 26 Mar 2007
TL;DR: The asymptotic dimension theory was founded by Gromov in the early 90s and has been studied extensively in the literature as mentioned in this paper, where the authors give a survey of its recent history.