Point (hyperbolic geometry)

Tips (ger .: cusps) in the hyperbolic geometry are important for the theory of numbers theory of modular forms and generally in theory Fuchs' and small shear groups of importance.

definition

Let it be a discrete group of isometries of hyperbolic space . ${\displaystyle \Gamma \subset \operatorname {Isom} (H^{n})}$ ${\displaystyle H^{n}}$

A point at infinity is a peak of if there is a parabolic isometry with a fixed point . ${\displaystyle c\in \partial _{\infty }H^{n}}$ ${\displaystyle \Gamma }$ ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle c}$

example

Pavement of through fundamental areas of the effect. The vertices at infinity are the tips of . ${\displaystyle H^{2}}$ ${\displaystyle PSL(2,\mathbb {Z} )}$ ${\displaystyle \mathbb {Q} \cup \left\{\infty \right\}}$ ${\displaystyle PSL(2,Z)}$

Let be the module group acting on the hyperbolic level . After identification ${\displaystyle \Gamma =PSL(2,\mathbb {Z} )}$ ${\displaystyle H^{2}}$

${\displaystyle \partial _{\infty }H^{2}=P^{1}\mathbb {R} =\mathbb {R} \cup \left\{\infty \right\}}$

the peaks of exactly correspond to the rational points ${\displaystyle PSL(2,\mathbb {Z} )}$

${\displaystyle \mathbb {Q} \cup \left\{\infty \right\}}$.

For example, the fixed point is the parabolic isometry and is a fixed point of the parabolic isometry . ${\displaystyle \infty }$ ${\displaystyle \left({\begin{array}{cc}1&1\\0&1\end{array}}\right)}$ ${\displaystyle {\frac {a}{c}}\in \mathbb {Q} }$ ${\displaystyle \left({\begin{array}{cc}1-ac&a^{2}\\-c^{2}&1+ac\end{array}}\right)}$

All tips of are in the - orbit of . ${\displaystyle PSL(2,\mathbb {Z} )}$ ${\displaystyle PSL(2,\mathbb {Z} )}$ ${\displaystyle \infty }$

A congruence subgroup has the same peaks as , that is, also . In this case, however, there are several orbits of peaks. ${\displaystyle \Gamma (N):=\operatorname {ker} (PSL(2,\mathbb {Z} )\to PSL(2,\mathbb {Z} /N\mathbb {Z} ))\subset PSL(2,\mathbb {Z} )}$ ${\displaystyle PSL(2,\mathbb {Z} )}$ ${\displaystyle \mathbb {Q} \cup \left\{\infty \right\}}$ ${\displaystyle \Gamma _{N}}$

Rank of a top

The stabilizer of a tip is a free Abelian group of parabolic isometries. The rank of the top is defined as the rank of the free Abelian group . ${\displaystyle \Gamma _{c}}$ ${\displaystyle c}$ ${\displaystyle \operatorname {rk} (c)}$ ${\displaystyle c}$ ${\displaystyle \Gamma _{c}}$

The inequality applies to each peak ${\displaystyle c\in \partial _{\infty }H^{n}}$

${\displaystyle 1\leq \operatorname {rk} (c)\leq n-1}$.

Compaction by tips

If all peaks have the rank , the non-compact manifold can be compacted by adding a peak from each orbit . ${\displaystyle \Gamma }$ ${\displaystyle n-1}$ ${\displaystyle \Gamma \backslash H^{n}}$ ${\displaystyle \Gamma }$

Each peak then has a family of surroundings in the compactification , which (after removing the peak) are homeomorphic to quotients of horoballs with a center . These surroundings are peaks in the sense of differential geometry . ${\displaystyle c}$ ${\displaystyle \Gamma _{c}\backslash H_{c}}$ ${\displaystyle H_{c}}$ ${\displaystyle c}$

Example: is homeomorphic to the complex plane by means of the j-invariant , by adding the tip one obtains the compactification . (This is a special case of Satake compactification .) Similarly, the quotients can be compacted by adding a finite number of peaks. This construction is important in understanding tip shapes . ${\displaystyle PSL(2,\mathbb {Z} )\backslash H^{2}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle PSL(2,\mathbb {Z} ).\infty }$ ${\displaystyle P^{1}\mathbb {C} =\mathbb {C} \cup \left\{\infty \right\}}$ ${\displaystyle \Gamma (N)\backslash H^{2}}$

Generalizations

Tips can also more common for certain locally symmetric spaces are defined, they are so connected in the important number theory and the theory of modular forms generalized theory of automorphic forms is important.

literature

• Boris Apanasov : Discrete groups in space and uniformization problems. Translated and revised from the 1983 Russian original. Mathematics and its Applications (Soviet Series), 40. Kluwer Academic Publishers Group, Dordrecht 1991, ISBN 0-7923-0216-8 .

Individual evidence

1. ↑ It then follows from the discreetness of that all fixing must be parabolic isometries. A subgroup of containing a parabolic and a hyperbolic isometry with the same fixed point at infinity can never be discrete. ${\displaystyle \Gamma }$ ${\displaystyle c}$ ${\displaystyle \gamma \in \Gamma \setminus \left\{1\right\}}$ ${\displaystyle \operatorname {Isom} (H^{n})}$
2. ↑ Chapter 1.2 in: Armand Borel , Lizhen Ji : Compactifications of symmetric and locally symmetric spaces. Mathematics: Theory & Applications. Birkhäuser, Boston, MA 2006, ISBN 0-8176-3247-6 .

See also

• Tip (differential geometry)