Geodetic compactification

In the mathematical field of differential geometry , geodetic compactification or geometrical compactification is a compactification of hyperbolic spaces or generally non-positively curved spaces by a sphere at infinity .

This compactification also works for general Hadamard spaces , but then the boundary at infinity does not generally have to be a sphere . Due to the construction of the edge points as (lying at infinity) end points of geodesics this edge is at infinity as a visible frame (if it is a sphere is) or as visible sphere (engl .: visibility sphere hereinafter).

This article deals with the infinite boundary of negatively curved, simply connected, Riemannian manifolds. The definition can also be transferred to Gromov hyperbolic spaces and in particular to hyperbolic groups, see Gromov hyperbolic space # Gromov boundary and Hyperbolic group # boundary at infinity .

definition

Let it be a simply connected Riemannian manifold of non-positive section curvature (or more generally a Hadamard space ). ${\displaystyle X}$

We define two geodesic rays as equivalent if ${\displaystyle \gamma _{1},\gamma _{2}\colon \left[0,\infty \right[\to X}$

${\displaystyle \limsup _{t\to \infty }d(\gamma _{1}(t),\gamma _{2}(t))<\infty }$

applies. We denote the set of equivalence classes with , another common name is . A geodesic is said to be asymptotic at a point if it belongs to the equivalence class . ${\displaystyle \partial _{\infty }X}$ ${\displaystyle X(\infty )}$ ${\displaystyle \gamma \colon \mathbb {R} \to X}$ ${\displaystyle p\in \partial _{\infty }X}$ ${\displaystyle \gamma \mid _{\left[0,\infty \right[}}$ ${\displaystyle p}$

For manifolds of non-positive sectional curvature there is a bijection between the unit sphere in (for any ) and , which is why the geodetic edge is also called "Sphere in Infinity" or "Visible Sphere". For any Hadamard spaces (which are not a manifold) there does not have to be a sphere. ${\displaystyle T_{x}X}$ ${\displaystyle x\in X}$ ${\displaystyle \partial _{\infty }X}$ ${\displaystyle \partial _{\infty }X}$

The geodetic compactification of is the union with the topology defined in the following section. ${\displaystyle X}$ ${\displaystyle X\cup \partial _{\infty }X}$

topology

The topology on the compactification is defined by the following conditions. ${\displaystyle X\cup \partial _{\infty }X}$

A sequence converges to a point represented by a geodesic ray if the sequence of geodesics converges from a (fixed) base point to a geodesic in the equivalence class . ${\displaystyle x_{n}\in X}$ ${\displaystyle \gamma }$ ${\displaystyle \left[\gamma \right]\in \partial _{\infty }X}$ ${\displaystyle \gamma _{n}}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{n}}$ ${\displaystyle \left[\gamma \right]}$

A surrounding base of is given by the broken-off cone family with . Here the "cone" is the set of those points for which the angle between and the geodesic through the base point and passing through is smaller than and the "broken cone" ${\displaystyle \left[\gamma \right]}$ ${\displaystyle C(\gamma ,\epsilon _{j},t_{j})}$ ${\displaystyle \epsilon _{j}\to 0,t_{j}\to \infty }$ ${\displaystyle C(\gamma ,\epsilon _{j})}$ ${\displaystyle x}$ ${\displaystyle \gamma }$ ${\displaystyle x_{0}}$ ${\displaystyle x}$ ${\displaystyle \epsilon _{j}}$

${\displaystyle C(\gamma ,\epsilon _{j},t_{j})=C(\gamma ,\epsilon _{j})\setminus B(\gamma (0),t_{j})}$.

The topology created by this environment base is called the cone topology .

Isometries (and more generally quasi-isometries ) of have a constant effect on compactification . ${\displaystyle X}$ ${\displaystyle X\cup \partial _{\infty }X}$

Tits metric

The angular distance is a metric that gives the standard metric of the sphere in the case of Euclidean spaces, but a discrete metric (two points are spaced apart ) in the case of negatively curved spaces . The topology produced by this metric (except for the flat one ) does not match the cone topology. ${\displaystyle \partial _{\infty }X}$ ${\displaystyle \pi }$ ${\displaystyle \mathbb {R} ^{n}}$

The angular distance (engl .: angle metric ) of two of geodetic rays with representing points is defined as ${\displaystyle \gamma _{1},\gamma _{2}}$ ${\displaystyle \gamma _{1}(0)=\gamma _{2}(0)=x_{0}}$ ${\displaystyle \left[\gamma _{1}\right],\left[\gamma _{2}\right]\in \partial _{\infty }X}$

${\displaystyle d_{<}(\left[\gamma _{1}\right],\left[\gamma _{2}\right])=\lim _{t\to \infty }\angle _{x}(\gamma _{1}(t),\gamma _{2}(t))}$.

(If there is no Riemannian manifold but only a Hadamard space, the angle is the angle in the respective comparison triangle.) ${\displaystyle X}$

In particular, it is true if and only if there is a geodesic that is for too or asymptotic. ${\displaystyle d_{<}(\left[\gamma _{1}\right],\left[\gamma _{2}\right])=\pi }$ ${\displaystyle t\to \pm \infty }$ ${\displaystyle \left[\gamma _{1}\right]}$ ${\displaystyle \left[\gamma _{2}\right]}$

The Tits metric is the inner metric associated with the angular distance . ${\displaystyle d_{T}}$

${\displaystyle (\partial _{\infty }X,d_{T})}$is a CAT (1) space .

Tits building

Let it be a symmetrical space of a non-compact type . We consider the effect of the isometric group on . The stabilizer of each point is a parabolic subgroup of , conversely, every parabolic subgroup of occurs as a stabilizer of a point in . ${\displaystyle X=G/K}$ ${\displaystyle G=Isom(X)}$ ${\displaystyle \partial _{\infty }X}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \partial _{\infty }X}$

For a true parabolic subgroup, let the set of all points fixed by in . One can show that a simplex and that the interior of this simplex is equal to the set of points with stabilizer . The decomposition as a simplicial complex ${\displaystyle P\subset G}$ ${\displaystyle \Sigma _{P}}$ ${\displaystyle P}$ ${\displaystyle \partial _{\infty }X}$ ${\displaystyle \Sigma _{P}}$ ${\displaystyle P}$

${\displaystyle \partial _{\infty }X=\bigcup _{P\ parabolische\ echte\ Untergruppe}\Sigma _{P}}$

gives the structure of a spherical tits building . The apartments of the Tits building correspond to the edges of maximum flax . ${\displaystyle \partial _{\infty }X}$ ${\displaystyle \partial _{\infty }F}$ ${\displaystyle F}$

literature

• Armand Borel , Lizhen Ji : Compactifications of symmetric and locally symmetric spaces. (= Mathematics: Theory & Applications). Birkhäuser, Boston, MA 2006, ISBN 0-8176-3247-6
• Martin R. Bridson , André Haefliger : Metric spaces of non-positive curvature. (= Basic Teachings of Mathematical Sciences. 319). Springer-Verlag, Berlin 1999. ISBN 3-540-64324-9
• Bruce Kleiner , Bernhard Leeb : Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. In: Inst. Hautes Études Sci. Publ. Math. No. 86, 1997, pp. 115-197 (1998).

Individual evidence

1. ↑ We use that in a CAT (0) -space two points can be connected by a unique geodesic.
2. ^ Bridson-Haefliger, op.cit, Theorem 9.20
3. ↑ Borel-Ji, op.cit., Proposition I.2.6