Complex-hyperbolic ideal triangle group

Complex ideal triangle groups are reflection groups to the ideal triangles of the complex hyperbolic geometry. The Goldman-Parker conjecture , proven by Richard Evan Schwartz , describes the discrete complex-hyperbolic groups of triangles.

definition

The ideal edge of the complex hyperbolic plane is the 3-dimensional unit sphere . An ideal triangle is a triangle with corners in the ideal edge. A complex ideal triangle group is the group created by the reflections on the sides of an ideal triangle. The products of two producers are each parabolic isometrics . ${\displaystyle \mathbb {C} \mathbb {H} ^{2}}$ ${\displaystyle S^{3}=\left\{(z,w)\in \mathbb {C} ^{2}:\|z\|^{2}+\|w\|^{2}=1\right\}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

Parameterization of the complex ideal triangle groups

The reflection groups belonging to isometric ideal triangles are conjugated in . Every ideal triangle can be mapped to a triangle of form using a suitable isometric drawing ${\displaystyle Isom(\mathbb {C} \mathbb {H} ^{2})=PU(2,1)}$ ${\displaystyle g\in PU(2,1)}$

${\displaystyle (\beta ,{\overline {\beta }}),(\beta ,\beta ),({\overline {\beta }},{\overline {\beta }})}$

bring where

${\displaystyle \beta ={\frac {1}{\sqrt {2}}}{\frac {s+i}{\sqrt {1+s^{2}}}}}$

for one is. (Because of this , the three points are on top.) The ideal triangles and, accordingly, also the ideal groups of triangles, are parameterized by the non-negative real number . ${\displaystyle s\in \left[0,\infty \right)}$ ${\displaystyle \|\beta \|={\frac {1}{\sqrt {2}}}}$ ${\displaystyle S^{3}}$ ${\displaystyle s}$

Goldman-Parker conjecture

The Goldman-Parker conjecture, proven by Richard Schwartz, states that an ideal triangle group is a discrete subgroup of the isometric group if and only if the inequality for the parameter described above ${\displaystyle PU(2,1)}$

${\displaystyle s\leq {\sqrt {\frac {125}{3}}}}$

applies.

literature

• W. Goldman , J. Parker : Complex hyperbolic ideal triangle groups , J. Reine Angew. Math. 425, 71-86 (1992)
• Richard Schwartz: Ideal triangle groups, dented tori and numerical analysis , Ann. Math. 153, 533-598 (2001)
• Richard Schwartz: A better proof of the Goldman-Parker conjecture , Geom. Top. 9, 1539-1601 (2005)