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. 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. Let X {displaystyle X} be a metric space and n ≥ 0 {displaystyle ngeq 0} be an integer. We say that a s d i m ( X ) ≤ n {displaystyle asdim(X)leq n} if for every R ≥ 1 {displaystyle Rgeq 1} there exists a uniformly bounded cover U {displaystyle {mathcal {U}}} of X {displaystyle X} such that every closed R {displaystyle R} -ball in X {displaystyle X} intersects at most n + 1 {displaystyle n+1} subsets from U {displaystyle {mathcal {U}}} . Here 'uniformly bounded' means that s u p U ∈ U d i a m ( U ) < ∞ {displaystyle sup_{Uin {mathcal {U}}}diam(U)<infty } . We then define the asymptotic dimension a s d i m ( X ) {displaystyle asdim(X)} as the smallest integer n ≥ 0 {displaystyle ngeq 0} such that a s d i m ( X ) ≤ n {displaystyle asdim(X)leq n} , if at least one such n {displaystyle n} exists, and define a s d i m ( X ) := ∞ {displaystyle asdim(X):=infty } otherwise. Also, one says that a family ( X i ) i ∈ I {displaystyle (X_{i})_{iin I}} of metric spaces satisfies a s d i m ( X ) ≤ n {displaystyle asdim(X)leq n} uniformly if for every R ≥ 1 {displaystyle Rgeq 1} and every i ∈ I {displaystyle iin I} there exists a cover U i {displaystyle {mathcal {U}}_{i}} of X i {displaystyle X_{i}} by sets of diameter at most D ( R ) < ∞ {displaystyle D(R)<infty } (independent of i {displaystyle i} ) such that every closed R {displaystyle R} -ball in X i {displaystyle X_{i}} intersects at most n + 1 {displaystyle n+1} subsets from U i {displaystyle {mathcal {U}}_{i}} . Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu, which proved that if G {displaystyle G} 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 a s d i m ( G ) < ∞ {displaystyle asdim(G)<infty } , then G {displaystyle G} 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.