Asymptotic dimension

In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced my Mikhail Gromov in his 1993 monograph "Asymptotic invariants of infinite groups" in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture. Asymptotic dimension has important applications in geometric analysis and index theory.
Formal definition.
Let formula_1 be a metric space and formula_2 be an integer. We say that formula_3 if for every formula_4 there exists a uniformly bounded cover formula_5 of formula_1 such that every closed formula_7-ball in formula_1 intersects at most formula_9 subsets from formula_5. Here 'uniformly bounded' means that formula_11.
We then define the "asymptotic dimension" formula_12 as the smallest integer formula_2 such that formula_3, if at least one such formula_15 exists, and define formula_16 otherwise.
Also, one says that a family formula_17 of metric spaces satisfies formula_18 "uniformly" if for every formula_4 and every formula_20 there exists a cover formula_21 of formula_22 by sets of diameter at most formula_23 (independent of formula_24) such that every closed formula_7-ball in formula_22 intersects at most formula_9 subsets from formula_28.
Asymptotic dimension in geometric group theory.
Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu
, which proved that if formula_63 is a finitely generated group of finite homotopy type (that is with a classifying space of the homotopy type of a finite CW-complex) such that formula_64, then formula_63 satisfies the Novikov conjecture. As was subsequently shown, finitely generated groups with finite asymptotic dimension are "topologically amenable", i.e. satisfy Guoliang Yu's "Property A" introduced in and equivalent to the exactness of the reduced C*-algebra of the group.