On the Atiyah problem on hyperbolic configurations of four points

Abstract

Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic $3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\dots ,p_n$ of a variable $t\in \mathbb{C}{P}^1$, of degree $n-1$, and conjectured that they are linearly independent over $\mathbb{C}$, no matter which configuration $\mathbf{x}$ one starts with. We prove this conjecture for $n=4$ in two cases: in case the $4$ points are non-coplanar, and in case one of the points lies in the hyperbolic convex hull of the other three.

5 pages, 2 figures. Final version with many minor corrections and improvements, to appear in Geometriae Dedicata

References in corpus (1)

• The Geometry of Point Particles