Asymptotic cone

In mathematics , the asymptotic cone of a metric space is a construction that formalizes the idea of ​​a border area after rescaling the metric (arbitrarily small) and thus generalizes the concept of the Gromov-Hausdorff limit value .

The construction depends on the choice of “scaling constants” and an ultrafilter . In the following, a free ultrafilter is always assumed. The index set is usually . Furthermore, it is with positive is a fixed sequence numbers ( "scaling constants"). ${\displaystyle {\mathcal {F}}}$ ${\displaystyle I=\mathbb {N} }$ ${\displaystyle (\lambda _{i})_{i}}$ ${\displaystyle \lim _{\mathcal {F}}\lambda _{i}=\infty }$

Ultralimes of metric spaces

Be a sequence of metric spaces . By means of the equivalence relation to define the Ultra product and on this a pseudo metric by ${\displaystyle \left\{(A_{i},d_{i})\right\}_{i\in I}}$ ${\displaystyle (a_{i})\sim (b_{i})\Leftrightarrow \left\{i\in I\colon a_{i}=b_{i}\right\}\in {\mathcal {F}}}$ ${\displaystyle \Pi _{i}A_{i}/\sim }$

${\displaystyle D(a,b)=\lim _{\mathcal {F}}d_{i}(x_{i},y_{i})}$,

d. i.e., an element is off such that for every neighborhood of : ${\displaystyle D(a,b)}$ ${\displaystyle \left[0,\infty \right]}$ ${\displaystyle V}$ ${\displaystyle D(a,b)}$

${\displaystyle \left\{i\in I\colon d_{i}(x_{i},y_{i})\in V\right\}\in {\mathcal {F}}}$.

One then considers the subset of the ultra product, consisting of the (equivalence classes of) sequences with . On this, the pseudometric only assumes finite values. ${\displaystyle a=(a_{i})_{i\in I}}$ ${\displaystyle D(a,p)<\infty }$ ${\displaystyle D}$

The metric space that is obtained as the quotient of this subset under the equivalence relation is called the ultralimes of the sequence relative to the observation point . The pseudometric induces the metric on the Ultralimes. ${\displaystyle \lim _{\mathcal {F}}(A_{i},d_{i},p_{i})}$ ${\displaystyle \left\{(A_{i},d_{i})\right\}_{i\in I}}$ ${\displaystyle p=(p_{i})_{i\in I}}$ ${\displaystyle a\sim b\Leftrightarrow D(a,b)=0}$ ${\displaystyle D}$

Asymptotic cone

Be a metric space and . Then one defines the asymptotic cone of (with respect to the ultrafilter and the scaling constants) by ${\displaystyle (X,d)}$ ${\displaystyle x\in X}$ ${\displaystyle X}$

${\displaystyle \operatorname {Cone} (X,d,x):=\lim _{\mathcal {F}}(X,d/\lambda _{i},x)}$.

Occasionally, the ultrametric asymptotic cone is also considered. This is defined as . ${\displaystyle \lim _{\mathcal {F}}(X,d^{1/\lambda _{i}},x)}$

properties

• If there is a geodetic metric space , then likewise. ${\displaystyle X}$ ${\displaystyle \operatorname {Cone} (X)}$
• If there is a Hadamard room , so is it . ${\displaystyle X}$ ${\displaystyle \operatorname {Cone} (X)}$
• If there is a CAT (0) space , then likewise. ${\displaystyle X}$ ${\displaystyle \operatorname {Cone} (X)}$
• If there is a CAT (κ) -space for a , then is a metric tree . ${\displaystyle X}$ ${\displaystyle \kappa <0}$ ${\displaystyle \operatorname {Cone} (X)}$
• If the orbits of the isometric group have a restricted Hausdorff distance of , then it is a homogeneous metric space . ${\displaystyle X}$ ${\displaystyle \operatorname {Cone} (X)}$
• A - quasiisometry induces a - Bilipschitz mapping . ${\displaystyle (L,C)}$ ${\displaystyle \phi \colon X\to Y}$ ${\displaystyle L}$ ${\displaystyle \operatorname {Cone} (\phi )\colon \operatorname {Cone} (X)\to \operatorname {Cone} (Y)}$

Examples

• For (Euclidean space), is . ${\displaystyle X=\mathbb {R} ^{n}}$ ${\displaystyle \operatorname {Cone} (\mathbb {R} ^{n})=\mathbb {R} ^{n}}$
• For (the hyperbolic space), is a -tree. ${\displaystyle X=\mathbb {H} ^{n}}$ ${\displaystyle \operatorname {Cone} (\mathbb {H} ^{n})}$ ${\displaystyle \mathbb {R} }$
• For a symmetrical space of the non-compact type is a Euclidean building . ${\displaystyle X}$ ${\displaystyle \operatorname {Cone} (X)}$

Connection with Gromov-Hausdorff convergence

If a family is precompact in the Gromov-Hausdorff topology , then there is an accumulation point of this sequence. In particular, the Gromov-Hausdorff limit value, if it exists, agrees with. ${\displaystyle \left\{(X,d/\lambda _{i},x)\colon i\in I\right\}}$ ${\displaystyle \operatorname {Cone} (X)}$ ${\displaystyle \operatorname {Cone} (X)}$

literature

• vd Dries - Wilkie : On Gromov's Theorem concerning groups of polynomial growth and elementary logic . J. Alg. 89: 349-374 (1984).
• Kleiner - Leeb : Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings , Inst. Hautes Etudes Sci. Publ. Math. (1997), no. 86, 115-197 (1998).

Web links

• Robert Young: Notes on asymptotic cones

Individual evidence

1. ↑ Kleiner-Leeb, op. Cit.
2. ↑ Kleiner-Leeb, op. Cit.
3. ↑ Kleiner-Lebb, op. Cit.