Asymptotic dimension

In mathematics , the asymptotic dimension is an invariant of metric spaces , which is particularly important in geometric group theory .

definition

The asymptotic dimension of a metric space is the smallest natural number with the following property: ${\displaystyle \operatorname {asdim} (X)}$ ${\displaystyle X}$ ${\displaystyle n}$

for each there is a covering of by open sets of restricted diameter , so that for each the metric sphere intersects at most these sets . ${\displaystyle R>0}$ ${\displaystyle X}$ ${\displaystyle U_{\alpha }}$ ${\displaystyle x\in X}$ ${\displaystyle B(x,R)}$ ${\displaystyle n+1}$

Examples

• The asymptotic dimension of a compact space is 0.
• The asymptotic dimension of Euclidean space is . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$
• The asymptotic dimension of a Gromov hyperbolic space is , where the edge denotes at infinity . ${\displaystyle 1+\dim(\partial _{\infty }X)}$ ${\displaystyle \partial _{\infty }X}$

properties

• From follows . ${\displaystyle Y\subset X}$ ${\displaystyle \operatorname {asdim} (Y)\leq \operatorname {asdim} (X)}$
• The asymptotic dimension is invariant under quasi-isometrics and more generally under coarse isometries .
• The following applies to product rooms . ${\displaystyle \operatorname {asdim} (X\times Y)\leq \operatorname {asdim} (X)+\operatorname {asdim} (Y)}$
• Set of Bell Dranishnikov: Be a geodesic metric space , a Lipschitz continuous mapping and for all and all was then applies . ${\displaystyle X}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle R>0}$ ${\displaystyle y\in Y}$ ${\displaystyle \operatorname {asdim} (f^{-1}(B(y,R)))\leq n}$ ${\displaystyle \operatorname {asdim} (X)\leq \operatorname {asdim} (Y)+n}$

Web links

• Bell-Dranishnikov: Asymptotic Dimension