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 by Mikhail Gromov in his 1993 monograph Asymptotic invariants of infinite groups[1] 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.
[2] Asymptotic dimension has important applications in geometric analysis and index theory.
be a metric space and
be an integer.
there exists a uniformly bounded cover
Here 'uniformly bounded' means that
We then define the asymptotic dimension
as the smallest integer
exists, and define
Also, one says that a family
of metric spaces satisfies
uniformly if for every
there exists a cover
by sets of diameter at most
(independent of
Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu[2] , which proved that if
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
satisfies the Novikov conjecture.
As was subsequently shown,[6] finitely generated groups with finite asymptotic dimension are topologically amenable, i.e. satisfy Guoliang Yu's Property A introduced in[7] and equivalent to the exactness of the reduced C*-algebra of the group.
max
{\displaystyle \operatorname {asdim} (G)\leq 1+\max _{v\in VY}\operatorname {asdim} (A_{v}).}