Gromov product

In mathematics , the Gromov product , named after Mikhail Leonidowitsch Gromow , is a concept from the theory of metric spaces . It clearly measures how long two geodesics starting at a point remain "close together".

definition

Let it be a metric space and . The Gromov product of and in is defined as ${\displaystyle (X,d)}$ ${\displaystyle x,y,z\in X}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle x}$

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

Example of a metric tree, all edges have length 1.

Euclidean plane: ${\displaystyle (A,B)_{C}=p}$

Examples

Trees

In a metric tree , exactly the length of the intersection of the (unique) shortest connections from to and from to . In the picture on the right (all edges should have length 1) is ${\displaystyle (y,z)_{x}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle z}$

${\displaystyle (b,w)_{m}=0,(b,i)_{m}=1,(e,i)_{m}=2}$.

Euclidean plane

For a Euclidean triangle ABC is just the length of the section on the line (or ) from to the point of contact of the line with the inscribed circle of the triangle. In the picture below right is . ${\displaystyle (A,B)_{C}}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle {\overline {BC}}}$ ${\displaystyle C}$ ${\displaystyle (A,B)_{C}=p}$

properties

• Symmetry: . ${\displaystyle (y,z)_{x}=(z,y)_{x}}$

• Degeneration in endpoints: . ${\displaystyle (y,z)_{y}=(y,z)_{z}=0}$

• For everyone and , ${\displaystyle p,q,x,y}$ ${\displaystyle z}$

${\displaystyle d(x,y)=(x,z)_{y}+(y,z)_{x},}$

${\displaystyle 0\leq (y,z)_{x}\leq \min {\big \{}d(y,x),d(z,x){\big \}},}$

${\displaystyle {\big |}(y,z)_{p}-(y,z)_{q}{\big |}\leq d(p,q),}$

${\displaystyle {\big |}(x,y)_{p}-(x,z)_{p}{\big |}\leq d(y,z).}$

• The Gromov product measures how long geodesics remain close to each other: if and three points of a -hyperbolischen metric space, then remove the segments of the length of the two geodesics of after and after no more than a distance from one another. ${\displaystyle x,y}$ ${\displaystyle z}$ ${\displaystyle \delta }$ ${\displaystyle (y,z)_{x}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle z}$ ${\displaystyle 2\delta }$

• A metric space is exactly then -hyperbolisch if for all and in true ${\displaystyle \delta }$ ${\displaystyle p,x,y}$ ${\displaystyle z}$ ${\displaystyle X}$

${\displaystyle (x,z)_{p}\geq \min {\big \{}(x,y)_{p},(y,z)_{p}{\big \}}-\delta .}$

Gromov edge

The Gromov boundary of a δ- hyperbolic metric space is defined as the set of equivalence classes of sequences with (so-called admissible sequences , clearly it is about infinitely diverging sequences) with respect to the equivalence relation ${\displaystyle \partial _{\infty }X}$ ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in N}\subset X}$ ${\displaystyle \lim _{i,j\to \infty }(x_{i},x_{j})_{0}=\infty }$

${\displaystyle (x_{n})_{n\in N}\sim (y_{n})_{n\in N}\Longleftrightarrow \lim(x_{n},y_{n})_{o}=\infty }$

for any (fixed) base point . The topology of the Gromov boundary is determined by the environment base ${\displaystyle o\in X}$

${\displaystyle U(\xi ,r):=\left\{\eta \in \partial _{\infty }X:\exists (x_{i}),(y_{j})\ s.d.\ \xi =\left[x_{i}\right],\eta =\left[y_{j}\right],\liminf _{i,j\to \infty }(x_{i},y_{j})_{o}\geq r\right\},\xi \in \partial _{\infty }X,r\geq 0}$

The Gromov product can be turned into a continuous function

${\displaystyle (.,.)_{o}\colon \partial _{\infty }X\times \partial _{\infty }X\to \left[0,\infty \right]}$

continue.

literature

• Sur les groupes hyperboliques d'après Mikhael Gromov. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. Edited by É. Ghys and P. de la Harpe. Progress in Mathematics, 83. Birkhauser Boston, Inc., Boston, MA, 1990. ISBN 0-8176-3508-4