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Publication

Hyperbolic representations of PU(1,n)

Gonzalo Emiliano Ruiz Stolowicz
2023
Theses

Abstract

The work is about the study of group representations in the group of isometries of a separable complex hyperbolic space. The main part is the classification of the representations of the group of isometries of a finite dimensional complex hyperbolic space that preserve the type (parabolic, elliptic and hyperbolic) and that are irreducible in the hyperbolic sense. Also the relation between these representations and the functions of hyperbolic type defined by Nicolas Monod.

Official source

Gonzalo Emiliano Ruiz Stolowicz
2023
Theses

Abstract

Official source

Related concepts (24)

Hyperbolic group

In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory.

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.

Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.

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