Asymptotic dimension of discrete groups

By Alexander Dranishnikov, J. Smith. 2006

99 citations recorded by the source corpus.

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in (B-D2) to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3881f1b024af1f38821fb3f2481a62e0df596354

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Some notes on Property A (2006, background): “[ BD01 ] covers some ideas relating to finite asymptotic dimension of discrete groups.”
Asymptotic Dimension (2007, background): “There is a remark in [48] that the definition of asymptotic dimension of arbitrary groups given above coincides with the asymptotic dimension of the coarse space (G, E) where E ∈ E if and only if {x−1y | (x, y) ∈ E} is finite.”
A notion of geometric complexity and its application to topological rigidity (2010, unknown): “le X2. On the other hand Z=pZoZ, which can be similarly realized as a subgroup of SL(2;Z=pZ[X;X 1]), has nite asymptotic dimension by results of Bell and Dranishnikov [BD3] and Dranishnikov and Smith [DS]. Indeed, it is an extension with both quotient Z and kernel Z=pZ having nite asymptotic dimensi”
A Hurewicz theorem for the Assouad–Nagata dimension (2006, methodology): “In the process of proving the analogs of the Hurewicz theorem we produce a much simpler proof than that in [1] and a stronger result: the function is only assumed to be large-scale uniform instead of Lipschitz, and the domain is not required to be geodesic.”