The work of Gromov

Gromov's work is the distance at which two geodesics starting at one point begin to diverge significantly.

Named after Gromov .

Gromov’s product is used, in particular, to define a metric on the absolute boundary of a metric space.

Definition

Let a fixed point be fixed $x\in X$ ${\ displaystyle x \ in X}$ metric space $(X,d)$ ${\ displaystyle (X, d)}$ . Then, the product of Gromov (relative to the point $x$ ${\ displaystyle x}$ ) points $y$ ${\ displaystyle y}$ and $z$ ${\ displaystyle z}$ this space is called the quantity

(y,z)_{x}:={\frac {1}{2}}(d(x,y)+d(x,z)-d(y,z)).

{\ displaystyle (y, z) _ {x}: = {\ frac {1} {2}} (d (x, y) + d (x, z) -d (y, z)).}

Properties

Gromov's work is non-negative and symmetrical: $\forall x,y,z\in X\quad (y,z)_{x}=(z,y)_{x},\quad (y,z)_{x}\geq 0.$ ${\ displaystyle \ forall x, y, z \ in X \ quad (y, z) _ {x} = (z, y) _ {x}, \ quad (y, z) _ {x} \ geq 0. }$
For a tree case, $(y,z)_{x}$ ${\ displaystyle (y, z) _ {x}}$ there is the length of the coincident part of the geodetic paths $[xy]$ ${\ displaystyle [xy]}$ and $[xz]$ ${\ displaystyle [xz]}$ .
For $\delta$ ${\ displaystyle \ delta}$ hyperbolic spaces the inequality holds.
$(x,z)_{p}\geq \min {\big \{}(x,y)_{p},(y,z)_{p}{\big \}}-\delta .$ {\ displaystyle (x, z) _ {p} \ geq \ min {\ big \ {} (x, y) _ {p}, (y, z) _ {p} {\ big \}} - \ delta .}

Literature

E. Gis, P. De la Arp. Hyperbolic groups according to Mikhail Gromov. - M .: World, 1992.

The work of Gromov

Definition

Properties

Literature

More articles: