Parabolic isometry

In mathematics , parabolic isometries are important in hyperbolic geometry and more generally in the theory of CAT (0) spaces .

definition

Let it be a complete CAT (0) space , for example a hyperbolic space . For an isometry, let it be defined by ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle d_{f}\colon X\to \mathbb {R} _{\geq 0}}$

${\displaystyle d_{f}(x)=d(f(x),x)}$.

Isometry is parabolic when there is no with ${\displaystyle x_{0}\in X}$

${\displaystyle d_{f}(x_{0})=\inf \left\{d_{f}(x)\colon x\in X\right\}}$

if the infimum is not accepted.

Fixed point at infinity

A parabolic isometry has a fixed point at infinity . It leaves all horospheres around this point invariant .

example

Let be the half-plane model of the hyperbolic plane and one through ${\displaystyle X={\mathbf {H} }^{2}=\left\{z\in \mathbb {C} \colon \operatorname {Im} (z)>0\right\}}$ ${\displaystyle f\colon X\to X}$

${\displaystyle f(z)=z+a}$

with given figure. From the definition of the hyperbolic metric it follows that an isometry is and applies. In particular is ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle d_{f}(z)={\frac {a}{\operatorname {Im} (z)}}}$

${\displaystyle \inf \left\{d_{f}(z)\colon z\in \mathbf {H} ^{2}\right\}=0}$.

Because in the hyperbolic plane no fixed point is, there is no but with , the infimum is therefore not accepted. The isometry is parabolic. ${\displaystyle f}$ ${\displaystyle z\in \mathbf {H} ^{2}}$ ${\displaystyle d_{f}(z)=0}$ ${\displaystyle f}$

More generally, isometries of the hyperbolic plane can be described by matrices and isometries of the 3-dimensional hyperbolic space by matrices . In both cases the isometry described by a matrix is parabolic if and only if for the trace of the matrix ${\displaystyle A\in SL(2,\mathbb {R} )}$ ${\displaystyle A\in SL(2,\mathbb {C} )}$ ${\displaystyle A}$

${\displaystyle \operatorname {Sp} (A)\in \left\{-2,2\right\}}$

applies. The above example corresponds to the matrix . ${\displaystyle \left({\begin{matrix}1&a\\0&1\end{matrix}}\right)}$

See also

• Elliptical isometry
• Hyperbolic isometry

literature

• Martin Bridson , André Haefliger : Metric spaces of non-positive curvature. Basic teaching of the mathematical sciences 319. Springer-Verlag, Berlin 1999, ISBN 3-540-64324-9 .
• Koji Fujiwara, Koichi Nagano, Takashi Shioya: Fixed point sets of parabolic isometries of CAT (0) -spaces. Comment. Math. Helv. 81 (2006), no. 2, 305-335.

Web links

• Koji Fujiwara: CAT (0) spaces for Riemannian geometers. (PDF file; 116 kB)

Individual evidence

1. ↑ Bridson-Haefliger, op. Cit., Proposition 8.25