Hyperbolic groups have finite asymptotic dimension

By J. Roe. 2005

71 citations recorded by the source corpus.

We detail a proof of a result of Gromov, that hyperbolic groups (and metric spaces) have finite asymptotic dimension. This fact has become important in recent work on the Novikov conjecture.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa9ed7001f09f1de79482c6ff02ee19063393424

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Expanders, exact crossed products, and the Baum–Connes conjecture (2013, background): “Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.”
A metric approach to limit operators (2014, background): “Suffice to say for now that many natural examples of metric spaces have property A: for example, many negatively curved spaces [16], finite dimensional non-positively curved cube complexes [2], and all countable subgroups of groups of invertible matrices (over any field) [8] have property A.”
The asymptotic dimension of a curve graph is finite (2005, background): “In fact, Higson and Roe show that X has property A1, which is equivalent to property A for such spaces.”
Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$C∗-algebras (2015, background): “On the other hand, part (iv) implies that many interesting classes of groups—for example, word hyperbolic groups [32], CAT(0) cubical groups [43], lattices in Lie groups, and many solvable groups [4]—admit at least some actions with finite dynamic asymptotic dimension.”
Some notes on Property A (2006, result): “This follows from the above and the result of [Roe05b] (due originally to Gromov) that such groups have finite asymptotic dimension.”
Equivariant covers for hyperbolic groups (2006, unknown): “31], [Roe05].”
On rigidity of Roe algebras (2011, background): “Property A holds for a very large class of spaces: notable examples of discrete groups with property A are amenable groups, hyperbolic groups [26], and linear groups [13].”