Quasi-Möbius image

In metric geometry, quasi-Möbius images generalize the term Möbius image known from complex geometry . They occur in particular as marginal images of quasi-isometrics of Gromov hyperbolic spaces .

definition

A mapping between metric spaces and is called quasi-Möbius if it contains the double ratio except for a controlled deviation, i.e. if there is a constant bijection with it ${\displaystyle f\colon X_{1}\to X_{2}}$ ${\displaystyle (X_{1},d_{1})}$ ${\displaystyle (X_{2},d_{2})}$ ${\displaystyle \eta \colon \left(0,\infty \right)\to \left(0,\infty \right)}$

${\displaystyle \left[f(x_{1}),f(x_{2}),f(x_{3}),f(x_{4})\right]\leq \eta (\left[x_{1},x_{2},x_{3},x_{4}\right])\ \ \forall x_{1},x_{2},x_{3},x_{4}\in X_{1}}$

gives. The double ratio of four points in a metric space is defined by . ${\displaystyle (X,d)}$ ${\displaystyle \left[x_{1},x_{2},x_{3},x_{4}\right]={\frac {d(x_{1},x_{3})d(x_{2},x_{4})}{d(x_{1},x_{4})d(x_{2},x_{3})}}}$

If even

${\displaystyle \left[f(x_{1}),f(x_{2}),f(x_{3}),f(x_{4})\right]=\left[x_{1},x_{2},x_{3},x_{4}\right]\ \ \forall x_{1},x_{2},x_{3},x_{4}\in X_{1}}$

holds, then one speaks (in generalization of the term from complex geometry) of a Möbius mapping.

properties

Swap and gives the opposite inequality (with a different function ). It follows from this that the inverse of a quasi-Möbius image is again a quasi-Möbius image. Furthermore, the combination of two quasi-Möbiusa images is a quasi-Möbius image. ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle \eta }$

Quasi-Möbius pictures are quasiconformal . The reverse applies in Loewner rooms .

Efremovich-Tichomirova's theorem

Every quasi-isometry between hyperbolic spaces can be continued to a homeomorphism of the edges at infinity , which is a quasi-Möbius image. ${\displaystyle f\colon X\to Y}$ ${\displaystyle \delta }$ ${\displaystyle \partial _{\infty }f\colon \partial _{\infty }X\to \partial _{\infty }Y}$

This theorem has an inversion for non-degenerate hyperbolic spaces, i.e. H. if there is a constant such that each point is at most all three sides of a triangle apart . With this assumption, every quasi-Möbius homeomorphism can be extended to a quasi-isometry . ${\displaystyle \delta }$ ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle \partial _{\infty }f\colon \partial _{\infty }X\to \partial _{\infty }Y}$ ${\displaystyle f\colon X\to Y}$

literature

• J. Väisälä: Quasi-Möbius maps , J. Anal. Math. 44, 218-234 (1984/85)
• V. Efremovich, E. Tichomirova: Epimorphisms of hyperbolic spaces , Izv. Ac. Nauk. 28, 1139-1144 (1964)
• F. Paulin: Un groupe hyperbolique est déterminé par son bord , J. Lond. Math. Soc. 54: 50-74 (1996)
• M. Bonk, O. Schramm: Embeddings of Gromov-hyperbolic spaces , Geom. Func. Anal. 10, 266-306 (2000)